-(q`2/|.q.|-sn) by XREAL_1:24; then (-(q`2/|.q.|-sn))/(1+sn)<1 by A4,XREAL_1:191; then ((-(q`2/|.q.|-sn))/(1+sn))^2<1^2 by A6,SQUARE_1:50; then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>0 by XREAL_1:50; then sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>0 by SQUARE_1:25; then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)> 0 by XCMPLX_1:76; then sqrt(1-(q`2/|.q.|-sn)^2/(1+sn)^2)> 0; then sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)> 0 by XCMPLX_1:76; then A8: -sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)< -0 by XREAL_1:24; let p be Point of TOP-REAL 2; set qz=p; assume p=(sn-FanMorphW).q; then p=|[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-sn)/ (1+sn))]| by A2,A3,Th17; then A9: qz`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) & qz`2= |.q.|* ((q`2/ |.q.|- sn)/(1+sn)) by EUCLID:52; ((q`2/|.q.|-sn)/(1+sn))<0 by A1,A5,XREAL_1:141,148; hence thesis by A2,A9,A8,Lm1,JGRAPH_2:3,XREAL_1:132; end; theorem Th44: for sn being Real,q1,q2 being Point of TOP-REAL 2 st sn<1 & q1`1 <0 & q1`2/|.q1.|>=sn & q2`1<0 & q2`2/|.q2.|>=sn & q1`2/|.q1.|=sn and A4: q2`1<0 and A5: q2`2/|.q2.|>=sn and A6: q1`2/|.q1.| 0 by A1,A6,XREAL_1:9,149; let p1,p2 be Point of TOP-REAL 2; assume that A8: p1=(sn-FanMorphW).q1 and A9: p2=(sn-FanMorphW).q2; A10: |.p2.|=|.q2.| by A9,Th33; p2=|[ |.q2.|*(-sqrt(1-((q2`2/|.q2.|-sn)/(1-sn))^2)), |.q2.|* ((q2`2/|. q2.|-sn)/(1-sn))]| by A4,A5,A9,Th16; then A11: p2`2= |.q2.|* ((q2`2/|.q2.|-sn)/(1-sn)) by EUCLID:52; |.q2.|>0 by A4,Lm1,JGRAPH_2:3; then A12: p2`2/|.p2.|= ((q2`2/|.q2.|-sn)/(1-sn)) by A11,A10,XCMPLX_1:89; p1=|[ |.q1.|*(-sqrt(1-((q1`2/|.q1.|-sn)/(1-sn))^2)), |.q1.|* ((q1`2/|.q1 .|-sn)/(1-sn))]| by A2,A3,A8,Th16; then A13: p1`2= |.q1.|* ((q1`2/|.q1.|-sn)/(1-sn)) by EUCLID:52; A14: |.p1.|=|.q1.| by A8,Th33; |.q1.|>0 by A2,Lm1,JGRAPH_2:3; then p1`2/|.p1.|= ((q1`2/|.q1.|-sn)/(1-sn)) by A13,A14,XCMPLX_1:89; hence thesis by A12,A7,XREAL_1:74; end; theorem Th45: for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1 0 by A1,A6,XREAL_1:9,148; let p1,p2 be Point of TOP-REAL 2; assume that A8: p1=(sn-FanMorphW).q1 and A9: p2=(sn-FanMorphW).q2; A10: |.p2.|=|.q2.| by A9,Th33; p2=|[ |.q2.|*(-sqrt(1-((q2`2/|.q2.|-sn)/(1+sn))^2)), |.q2.|* ((q2`2/|. q2.|-sn)/(1+sn))]| by A4,A5,A9,Th17; then A11: p2`2= |.q2.|* ((q2`2/|.q2.|-sn)/(1+sn)) by EUCLID:52; |.q2.|>0 by A4,Lm1,JGRAPH_2:3; then A12: p2`2/|.p2.|= ((q2`2/|.q2.|-sn)/(1+sn)) by A11,A10,XCMPLX_1:89; p1=|[ |.q1.|*(-sqrt(1-((q1`2/|.q1.|-sn)/(1+sn))^2)), |.q1.|* ((q1`2/|.q1 .|-sn)/(1+sn))]| by A2,A3,A8,Th17; then A13: p1`2= |.q1.|* ((q1`2/|.q1.|-sn)/(1+sn)) by EUCLID:52; A14: |.p1.|=|.q1.| by A8,Th33; |.q1.|>0 by A2,Lm1,JGRAPH_2:3; then p1`2/|.p1.|= ((q1`2/|.q1.|-sn)/(1+sn)) by A13,A14,XCMPLX_1:89; hence thesis by A12,A7,XREAL_1:74; end; theorem for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1 =sn & q2`2/|.q2.|>=sn; hence thesis by A2,A3,A4,A5,A6,A7,Th44; end; case q1`2/|.q1.|>=sn & q2`2/|.q2.| =sn; then p2`2>=0 by A2,A4,A7,Th42; then A9: p2`2/|.p2.|>=0; p1`2<0 by A1,A3,A6,A8,Th43; hence thesis by A9,Lm1,JGRAPH_2:3,XREAL_1:141; end; case q1`2/|.q1.| 0 by A1,Lm1,JGRAPH_2:3; assume p=(sn-FanMorphW).q; then A4: p=|[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/ (1-sn))]| by A1,A2,Th16; then p`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52; hence thesis by A2,A4,A3,Lm13,EUCLID:52,XREAL_1:132; end; theorem for sn being Real holds 0.TOP-REAL 2=(sn-FanMorphW).(0.TOP-REAL 2) by Th16,JGRAPH_2:3; begin :: Fan Morphism for North definition let s be Real, q be Point of TOP-REAL 2; func FanN(s,q) -> Point of TOP-REAL 2 equals :Def4: |.q.|*|[(q`1/|.q.|-s)/(1 -s), sqrt(1-((q`1/|.q.|-s)/(1-s))^2)]| if q`1/|.q.|>=s & q`2>0, |.q.|*|[(q`1/|. q.|-s)/(1+s), sqrt(1-((q`1/|.q.|-s)/(1+s))^2)]| if q`1/|.q.| ~~0 otherwise q; correctness; end; definition let c be Real; func c-FanMorphN -> Function of TOP-REAL 2, TOP-REAL 2 means :Def5: for q being Point of TOP-REAL 2 holds it.q=FanN(c,q); existence proof deffunc F(Point of TOP-REAL 2)=FanN(c,$1); thus ex IT being Function of TOP-REAL 2, TOP-REAL 2 st for q being Point of TOP-REAL 2 holds IT.q=F(q) from FUNCT_2:sch 4; end; uniqueness proof deffunc F(Point of TOP-REAL 2)=FanN(c,$1); thus for a,b being Function of TOP-REAL 2, TOP-REAL 2 st (for q being Point of TOP-REAL 2 holds a.q=F(q)) & (for q being Point of TOP-REAL 2 holds b. q=F(q)) holds a = b from BINOP_2:sch 1; end; end; theorem Th49: for cn being Real holds (q`1/|.q.|>=cn & q`2>0 implies cn -FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(sqrt(1-((q`1/|.q.|-cn)/ (1-cn))^2))]|)& (q`2<=0 implies cn-FanMorphN.q=q) proof let cn be Real; hereby assume q`1/|.q.|>=cn & q`2>0; then FanN(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1-cn), sqrt(1-((q`1/|.q.|-cn)/(1- cn))^2)]| by Def4 .= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(sqrt(1-((q`1/|.q.|-cn)/(1 -cn))^2))]| by EUCLID:58; hence cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(sqrt(1-((q `1/|.q.|-cn)/(1-cn))^2))]| by Def5; end; assume A1: q`2<=0; cn-FanMorphN.q=FanN(cn,q) by Def5; hence thesis by A1,Def4; end; theorem Th50: for cn being Real holds (q`1/|.q.|<=cn & q`2>0 implies cn -FanMorphN.q= |[ |.q.|*((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/ (1+cn))^2))]|) proof let cn be Real; assume that A1: q`1/|.q.|<=cn and A2: q`2>0; per cases by A1,XXREAL_0:1; suppose q`1/|.q.|~~=cn & q`2>=0 & q<>0.TOP-REAL 2 implies cn-FanMorphN.q= |[ |.q.|*((q`1/|.q.|-cn)/(1-cn)), |.q .|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))]|) & (q`1/|.q.|<=cn & q`2>=0 & q<>0. TOP-REAL 2 implies cn-FanMorphN.q= |[ |.q.|*((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))]|) proof let cn be Real; assume that A1: -1 =cn & q`2>=0 & q<>0.TOP-REAL 2; per cases; suppose A4: q`2>0; then FanN(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1-cn), sqrt(1-((q`1/|.q.|-cn)/(1 -cn))^2)]| by A3,Def4 .= |[|.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/ (1-cn))^2))]| by EUCLID:58; hence thesis by A4,Def5,Th50; end; suppose A5: q`2<=0; then A6: cn-FanMorphN.q=q by Th49; A7: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; A8: 1-cn>0 by A2,XREAL_1:149; A9: q`2=0 by A3,A5; |.q.|<>0 by A3,TOPRNS_1:24; then |.q.|^2>0 by SQUARE_1:12; then (q`1)^2/|.q.|^2=1^2 by A7,A9,XCMPLX_1:60; then ((q`1)/|.q.|)^2=1^2 by XCMPLX_1:76; then A10: sqrt(((q`1)/|.q.|)^2)=1 by SQUARE_1:22; A11: now assume q`1<0; then -((q`1)/|.q.|)=1 by A10,SQUARE_1:23; hence contradiction by A1,A3; end; sqrt((|.q.|)^2)=|.q.| by SQUARE_1:22; then A12: |.q.|=q`1 by A7,A9,A11,SQUARE_1:22; then 1=q`1/|.q.| by A3,TOPRNS_1:24,XCMPLX_1:60; then (q`1/|.q.|-cn)/(1-cn)=1 by A8,XCMPLX_1:60; hence thesis by A2,A6,A9,A12,EUCLID:53,SQUARE_1:17,TOPRNS_1:24 ,XCMPLX_1:60; end; end; suppose A13: q`1/|.q.|<=cn & q`2>=0 & q<>0.TOP-REAL 2; per cases; suppose q`2>0; hence thesis by Th49,Th50; end; suppose A14: q`2<=0; A15: 1+cn>0 by A1,XREAL_1:148; A16: |.q.|<>0 by A13,TOPRNS_1:24; A17: q`2=0 by A13,A14; |.q.|>0 & 1>q`1/|.q.| by A2,A13,Lm1,XXREAL_0:2; then 1 *(|.q.|)>q`1/|.q.|*(|.q.|) by XREAL_1:68; then A18: (|.q.|)^2 =(q`1)^2+(q`2)^2 & (|.q.|)>q`1 by A13,JGRAPH_3:1,TOPRNS_1:24 ,XCMPLX_1:87; then A19: q`1= -(|.q.|) by A17,SQUARE_1:40; then -1=q`1/|.q.| by A13,TOPRNS_1:24,XCMPLX_1:197; then A20: (q`1/|.q.|-cn)/(1+cn) =(-(1+cn))/(1+cn) .=-1 by A15,XCMPLX_1:197; |.q.|=-q`1 by A17,A18,SQUARE_1:40; then |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/( 1+cn))^2))]| =q by A17,A20,EUCLID:53,SQUARE_1:17; hence thesis by A1,A14,A16,A19,Th49,XCMPLX_1:197; end; end; suppose q`2<0 or q=0.TOP-REAL 2; hence thesis; end; end; theorem Th52: for cn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st cn<1 & (for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`1/|.p.|-cn)/(1- cn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2>=0 & q<>0.TOP-REAL 2) holds f is continuous proof let cn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=cn, b=(1-cn); reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2; assume that A1: cn<1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2 )|K1 holds f.p=|.p.|* ((p`1/|.p.|-cn)/(1-cn)) and A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2) |K1 holds q`2>=0 & q<>0.TOP-REAL 2; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; b>0 by A1,XREAL_1:149; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 & g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and A6: g3 is continuous by A4,Th5; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A8: dom f=dom g3 by FUNCT_2:def 1; for x being object st x in dom f holds f.x=g3.x proof let x be object; assume A9: x in dom f; then reconsider s=x as Point of (TOP-REAL 2)|K1; x in K1 by A7,A8,A9,PRE_TOPC:8; then reconsider r=x as Point of TOP-REAL 2; A10: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5; A11: g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5; f.r=(|.r.|)* ((r`1/|.r.|-cn)/(1-cn)) by A2,A9; hence thesis by A5,A11,A10; end; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th53: for cn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st -1 =0 & q<>0.TOP-REAL 2) holds f is continuous proof let cn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=cn, b=(1+cn); reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2; assume that A1: -1 =0 & q<>0.TOP-REAL 2; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; 1+cn>0 by A1,XREAL_1:148; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 & g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and A6: g3 is continuous by A4,Th5; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; A8: for x being object st x in dom f holds f.x=g3.x proof let x be object; assume A9: x in dom f; then reconsider s=x as Point of (TOP-REAL 2)|K1; x in dom g3 by A7,A9; then x in K1 by A7,PRE_TOPC:8; then reconsider r=x as Point of TOP-REAL 2; A10: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5; A11: g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5; f.r=(|.r.|)* ((r`1/|.r.|-cn)/(1+cn)) by A2,A9; hence thesis by A5,A11,A10; end; dom f=dom g3 by A7,FUNCT_2:def 1; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th54: for cn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st cn<1 & (for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*( sqrt(1-((p`1/|.p.|- cn)/(1-cn))^2))) & (for q being Point of TOP-REAL 2 st q in the carrier of ( TOP-REAL 2)|K1 holds q`2>=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2) holds f is continuous proof let cn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=cn, b=(1-cn); reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2; assume that A1: cn<1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2 )|K1 holds f.p=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) and A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2) |K1 holds q`2>=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; b>0 by A1,XREAL_1:149; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q =r1 & g1.q=r2 holds g3.q=r2*( sqrt(|.1-((r1/r2-a)/b)^2.|)) and A6: g3 is continuous by A4,Th10; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A8: dom f=dom g3 by FUNCT_2:def 1; for x being object st x in dom f holds f.x=g3.x proof let x be object; A9: 1-cn>0 by A1,XREAL_1:149; assume A10: x in dom f; then x in K1 by A7,A8,PRE_TOPC:8; then reconsider r=x as Point of TOP-REAL 2; A11: |.r.|<>0 by A3,A10,TOPRNS_1:24; |.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1; then A12: ((r`1) -(|.r.|))*((r`1)+|.r.|) =-(r`2)^2; (r`2)^2>=0 by XREAL_1:63; then r`1<= |.r.| by A12,XREAL_1:93; then r`1/|.r.| <= |.r.|/|.r.| by XREAL_1:72; then r`1/|.r.|<=1 by A11,XCMPLX_1:60; then A13: r`1/|.r.|-cn<=(1-cn) by XREAL_1:9; reconsider s=x as Point of (TOP-REAL 2)|K1 by A10; A14: now assume (1-cn)^2=0; then 1-cn+cn=0+cn by XCMPLX_1:6; hence contradiction by A1; end; cn-r`1/|.r.|<=0 by A3,A10,XREAL_1:47; then -(cn- r`1/|.r.|)>=-(1-cn) by A9,XREAL_1:24; then (1-cn)^2>=0 & (r`1/|.r.|-cn)^2<=(1-cn)^2 by A13,SQUARE_1:49,XREAL_1:63 ; then (r`1/|.r.|-cn)^2/(1-cn)^2<=(1-cn)^2/(1-cn)^2 by XREAL_1:72; then (r`1/|.r.|-cn)^2/(1-cn)^2<=1 by A14,XCMPLX_1:60; then ((r`1/|.r.|-cn)/(1-cn))^2<=1 by XCMPLX_1:76; then 1-((r`1/|.r.|-cn)/(1-cn))^2>=0 by XREAL_1:48; then |.1-((r`1/|.r.|-cn)/(1-cn))^2.| =1-((r`1/|.r.|-cn)/(1-cn))^2 by ABSVALUE:def 1; then A15: f.r=(|.r.|)*( sqrt(|.1-((r`1/|.r.|-cn)/(1-cn))^2.|)) by A2,A10; A16: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5; g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5; hence thesis by A5,A15,A16; end; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th55: for cn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st -1 =0 & q`1/|.q.|<=cn & q<>0.TOP-REAL 2) holds f is continuous proof let cn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=cn, b=(1+cn); reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2; assume that A1: -1 =0 & q`1/|.q.|<=cn & q<>0.TOP-REAL 2; A4: 1+cn>0 by A1,XREAL_1:148; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q =r1 & g1.q=r2 holds g3.q=r2*( sqrt(|.1-((r1/r2-a)/b)^2.|)) and A6: g3 is continuous by A4,Th10; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A8: dom f=dom g3 by FUNCT_2:def 1; for x being object st x in dom f holds f.x=g3.x proof let x be object; assume A9: x in dom f; then x in K1 by A7,A8,PRE_TOPC:8; then reconsider r=x as Point of TOP-REAL 2; reconsider s=x as Point of (TOP-REAL 2)|K1 by A9; A10: (1+cn)^2>0 by A4,SQUARE_1:12; A11: |.r.|<>0 by A3,A9,TOPRNS_1:24; |.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1; then A12: ((r`1) -(|.r.|))*((r`1)+|.r.|) =-(r`2)^2; (r`2)^2>=0 by XREAL_1:63; then -(|.r.|)<=r`1 by A12,XREAL_1:93; then r`1/|.r.| >= (-(|.r.|))/|.r.| by XREAL_1:72; then r`1/|.r.|>= -1 by A11,XCMPLX_1:197; then r`1/|.r.|-cn>=-1-cn by XREAL_1:9; then A13: r`1/|.r.|-cn>=-(1+cn); cn-r`1/|.r.|>=0 by A3,A9,XREAL_1:48; then -(cn-r`1/|.r.|)<=-0; then (r`1/|.r.|-cn)^2<=(1+cn)^2 by A4,A13,SQUARE_1:49; then (r`1/|.r.|-cn)^2/(1+cn)^2<=(1+cn)^2/(1+cn)^2 by A4,XREAL_1:72; then (r`1/|.r.|-cn)^2/(1+cn)^2<=1 by A10,XCMPLX_1:60; then ((r`1/|.r.|-cn)/(1+cn))^2<=1 by XCMPLX_1:76; then 1-((r`1/|.r.|-cn)/(1+cn))^2>=0 by XREAL_1:48; then |.1-((r`1/|.r.|-cn)/(1+cn))^2.| =1-((r`1/|.r.|-cn)/(1+cn))^2 by ABSVALUE:def 1; then A14: f.r=(|.r.|)*( sqrt(|.1-((r`1/|.r.|-cn)/(1+cn))^2.|)) by A2,A9; A15: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5; g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5; hence thesis by A5,A14,A15; end; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th56: for cn being Real, K0,B0 being Subset of TOP-REAL 2,f being Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1 =0 & q<>0.TOP-REAL 2} & K0={p: p `1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2} holds f is continuous proof let cn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set sn=sqrt(1-cn^2); set p0=|[cn,sn]|; A1: p0`2=sn by EUCLID:52; p0`1=cn by EUCLID:52; then A2: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1; assume A3: -1 =0 & q<>0.TOP-REAL 2} & K0={p: p`1/|.p.|>=cn & p`2>=0 & p<>0. TOP-REAL 2}; then cn^2<1^2 by SQUARE_1:50; then A4: 1-cn^2>0 by XREAL_1:50; then sn^2=1-cn^2 by SQUARE_1:def 2; then A5: p0`1/|.p0.|=cn by A2,EUCLID:52,SQUARE_1:18; p0`2>0 by A1,A4,SQUARE_1:25; then A6: p0 in K0 by A3,A5,JGRAPH_2:3; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; A7: rng (proj2*((cn-FanMorphN)|K1)) c= the carrier of R^1 by TOPMETR:17; A8: K0 c= B0 proof let x be object; assume x in K0; then ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1/|.p8.|>= cn & p8`2>=0 & p8<>0.TOP-REAL 2 by A3; hence thesis by A3; end; A9: dom ((cn-FanMorphN)|K1) c= dom (proj1*((cn-FanMorphN)|K1)) proof let x be object; assume A10: x in dom ((cn-FanMorphN)|K1); then x in dom (cn-FanMorphN) /\ K1 by RELAT_1:61; then x in dom (cn-FanMorphN) by XBOOLE_0:def 4; then A11: dom proj1 = (the carrier of TOP-REAL 2) & (cn-FanMorphN).x in rng (cn -FanMorphN) by FUNCT_1:3,FUNCT_2:def 1; ((cn-FanMorphN)|K1).x=(cn-FanMorphN).x by A10,FUNCT_1:47; hence thesis by A10,A11,FUNCT_1:11; end; A12: rng (proj1*((cn-FanMorphN)|K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj1*((cn-FanMorphN)|K1)) c= dom ((cn-FanMorphN)|K1) by RELAT_1:25; then dom (proj1*((cn-FanMorphN)|K1)) =dom ((cn-FanMorphN)|K1) by A9, XBOOLE_0:def 10 .=dom (cn-FanMorphN) /\ K1 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g2=proj1*((cn-FanMorphN)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A12,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g2.p=|.p.|* ((p`1/|.p.|-cn)/(1-cn)) proof let p be Point of TOP-REAL 2; A13: dom ((cn-FanMorphN)|K1)=dom (cn-FanMorphN) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A14: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A15: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|>= cn & p3`2>=0 & p3<>0.TOP-REAL 2 by A3,A14; then A16: (cn-FanMorphN).p =|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-( (p`1/|.p.|-cn)/(1-cn))^2))]| by A3,Th51; ((cn-FanMorphN)|K1).p=(cn-FanMorphN).p by A15,A14,FUNCT_1:49; then g2.p=proj1.(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/ |.p.|-cn)/(1-cn))^2))]|) by A15,A13,A14,A16,FUNCT_1:13 .=(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/( 1-cn))^2))]|)`1 by PSCOMP_1:def 5 .=|.p.|* ((p`1/|.p.|-cn)/(1-cn)) by EUCLID:52; hence thesis; end; then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that A17: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f2.p=|.p.|* ((p`1/|.p.|-cn)/(1-cn)); A18: dom ((cn-FanMorphN)|K1) c= dom (proj2*((cn-FanMorphN)|K1)) proof let x be object; assume A19: x in dom ((cn-FanMorphN)|K1); then x in dom (cn-FanMorphN) /\ K1 by RELAT_1:61; then x in dom (cn-FanMorphN) by XBOOLE_0:def 4; then A20: dom proj2 = (the carrier of TOP-REAL 2) & (cn-FanMorphN).x in rng (cn -FanMorphN) by FUNCT_1:3,FUNCT_2:def 1; ((cn-FanMorphN)|K1).x=(cn-FanMorphN).x by A19,FUNCT_1:47; hence thesis by A19,A20,FUNCT_1:11; end; dom (proj2*((cn-FanMorphN)|K1)) c= dom ((cn-FanMorphN)|K1) by RELAT_1:25; then dom (proj2*((cn-FanMorphN)|K1)) =dom ((cn-FanMorphN)|K1) by A18, XBOOLE_0:def 10 .=dom (cn-FanMorphN) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g1=proj2*((cn-FanMorphN)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A7,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g1.p=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) proof let p be Point of TOP-REAL 2; A21: dom ((cn-FanMorphN)|K1)=dom (cn-FanMorphN) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A22: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A23: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|>= cn & p3`2>=0 & p3<>0.TOP-REAL 2 by A3,A22; then A24: (cn-FanMorphN).p=|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-(( p`1/|.p.|-cn)/(1-cn))^2))]| by A3,Th51; ((cn-FanMorphN)|K1).p=(cn-FanMorphN).p by A23,A22,FUNCT_1:49; then g1.p=proj2.(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/ |.p.|-cn)/(1-cn))^2))]|) by A23,A21,A22,A24,FUNCT_1:13 .=(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/( 1-cn))^2))]|)`2 by PSCOMP_1:def 6 .= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52; hence thesis; end; then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that A25: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f1.p=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)); for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2>=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A26: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|>= cn & p3`2>=0 & p3<>0.TOP-REAL 2 by A3,A26; hence thesis; end; then A27: f1 is continuous by A3,A25,Th54; A28: for x,y,s,r being Real st |[x,y]| in K1 & s=f2.(|[x,y]|) & r=f1. (|[x,y]|) holds f.(|[x,y]|)=|[s,r]| proof let x,y,s,r be Real; assume that A29: |[x,y]| in K1 and A30: s=f2.(|[x,y]|) & r=f1.(|[x,y]|); set p99=|[x,y]|; A31: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`1/|.p3.| >=cn & p3`2>=0 & p3<>0.TOP-REAL 2 by A3,A29; A32: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then A33: f1.p99=|.p99.|*( sqrt(1-((p99`1/|.p99.|-cn)/(1-cn))^2)) by A25,A29; ((cn-FanMorphN)|K0).(|[x,y]|)=((cn-FanMorphN)).(|[x,y]|) by A29,FUNCT_1:49 .= |[ |.p99.|* ((p99`1/|.p99.|-cn)/(1-cn)), |.p99.|*( sqrt(1-((p99`1/ |.p99.|-cn)/(1-cn))^2))]| by A3,A31,Th51 .=|[s,r]| by A17,A29,A30,A32,A33; hence thesis by A3; end; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2>=0 & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A34: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|>= cn & p3`2>=0 & p3<>0.TOP-REAL 2 by A3,A34; hence thesis; end; then f2 is continuous by A3,A17,Th52; hence thesis by A6,A8,A27,A28,JGRAPH_2:35; end; theorem Th57: for cn being Real, K0,B0 being Subset of TOP-REAL 2,f being Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1 =0 & q<>0.TOP-REAL 2} & K0={p: p `1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} holds f is continuous proof let cn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set sn=sqrt(1-cn^2); set p0=|[cn,sn]|; A1: p0`2=sn by EUCLID:52; p0`1=cn by EUCLID:52; then A2: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1; assume A3: -1 =0 & q<>0.TOP-REAL 2} & K0={p: p`1/|.p.|<=cn & p`2>=0 & p<>0. TOP-REAL 2}; then cn^2<1^2 by SQUARE_1:50; then A4: 1-cn^2>0 by XREAL_1:50; then sn^2=1-cn^2 by SQUARE_1:def 2; then A5: p0`1/|.p0.|=cn by A2,EUCLID:52,SQUARE_1:18; p0`2>0 by A1,A4,SQUARE_1:25; then A6: p0 in K0 by A3,A5,JGRAPH_2:3; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; A7: rng (proj2*((cn-FanMorphN)|K1)) c= the carrier of R^1 by TOPMETR:17; A8: K0 c= B0 proof let x be object; assume x in K0; then ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1/|.p8.|<= cn & p8`2>=0 & p8<>0.TOP-REAL 2 by A3; hence thesis by A3; end; A9: dom ((cn-FanMorphN)|K1) c= dom (proj1*((cn-FanMorphN)|K1)) proof let x be object; assume A10: x in dom ((cn-FanMorphN)|K1); then x in dom (cn-FanMorphN) /\ K1 by RELAT_1:61; then x in dom (cn-FanMorphN) by XBOOLE_0:def 4; then A11: dom proj1 = (the carrier of TOP-REAL 2) & (cn-FanMorphN).x in rng (cn -FanMorphN) by FUNCT_1:3,FUNCT_2:def 1; ((cn-FanMorphN)|K1).x=(cn-FanMorphN).x by A10,FUNCT_1:47; hence thesis by A10,A11,FUNCT_1:11; end; A12: rng (proj1*((cn-FanMorphN)|K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj1*((cn-FanMorphN)|K1)) c= dom ((cn-FanMorphN)|K1) by RELAT_1:25; then dom (proj1*((cn-FanMorphN)|K1)) =dom ((cn-FanMorphN)|K1) by A9, XBOOLE_0:def 10 .=dom (cn-FanMorphN) /\ K1 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g2=proj1*((cn-FanMorphN)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A12,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g2.p=|.p.|* ((p`1/|.p.|-cn)/(1+cn)) proof let p be Point of TOP-REAL 2; A13: dom ((cn-FanMorphN)|K1)=dom (cn-FanMorphN) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A14: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A15: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|<= cn & p3`2>=0 & p3<>0.TOP-REAL 2 by A3,A14; then A16: (cn-FanMorphN).p =|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-( (p`1/|.p.|-cn)/(1+cn))^2))]| by A3,Th51; ((cn-FanMorphN)|K1).p=(cn-FanMorphN).p by A15,A14,FUNCT_1:49; then g2.p=proj1.(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/ |.p.|-cn)/(1+cn))^2))]|) by A15,A13,A14,A16,FUNCT_1:13 .=(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/( 1+cn))^2))]|)`1 by PSCOMP_1:def 5 .=|.p.|* ((p`1/|.p.|-cn)/(1+cn)) by EUCLID:52; hence thesis; end; then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that A17: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f2.p=|.p.|* ((p`1/|.p.|-cn)/(1+cn)); A18: dom ((cn-FanMorphN)|K1) c= dom (proj2*((cn-FanMorphN)|K1)) proof let x be object; assume A19: x in dom ((cn-FanMorphN)|K1); then x in dom (cn-FanMorphN) /\ K1 by RELAT_1:61; then x in dom (cn-FanMorphN) by XBOOLE_0:def 4; then A20: dom proj2 = (the carrier of TOP-REAL 2) & (cn-FanMorphN).x in rng (cn -FanMorphN) by FUNCT_1:3,FUNCT_2:def 1; ((cn-FanMorphN)|K1).x=(cn-FanMorphN).x by A19,FUNCT_1:47; hence thesis by A19,A20,FUNCT_1:11; end; dom (proj2*((cn-FanMorphN)|K1)) c= dom ((cn-FanMorphN)|K1) by RELAT_1:25; then dom (proj2*((cn-FanMorphN)|K1)) =dom ((cn-FanMorphN)|K1) by A18, XBOOLE_0:def 10 .=dom (cn-FanMorphN) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g1=proj2*((cn-FanMorphN)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A7,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g1.p=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) proof let p be Point of TOP-REAL 2; A21: dom ((cn-FanMorphN)|K1)=dom (cn-FanMorphN) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A22: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A23: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|<= cn & p3`2>=0 & p3<>0.TOP-REAL 2 by A3,A22; then A24: (cn-FanMorphN).p=|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-(( p`1/|.p.|-cn)/(1+cn))^2))]| by A3,Th51; ((cn-FanMorphN)|K1).p=(cn-FanMorphN).p by A23,A22,FUNCT_1:49; then g1.p=proj2.(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/ |.p.|-cn)/(1+cn))^2))]|) by A23,A21,A22,A24,FUNCT_1:13 .=(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/( 1+cn))^2))]|)`2 by PSCOMP_1:def 6 .= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52; hence thesis; end; then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that A25: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f1.p=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)); for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2>=0 & q`1/|.q.|<=cn & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A26: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|<= cn & p3`2>=0 & p3<>0.TOP-REAL 2 by A3,A26; hence thesis; end; then A27: f1 is continuous by A3,A25,Th55; A28: for x,y,s,r being Real st |[x,y]| in K1 & s=f2.(|[x,y]|) & r=f1. (|[x,y]|) holds f.(|[x,y]|)=|[s,r]| proof let x,y,s,r be Real; assume that A29: |[x,y]| in K1 and A30: s=f2.(|[x,y]|) & r=f1.(|[x,y]|); set p99=|[x,y]|; A31: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`1/|.p3.| <=cn & p3`2>=0 & p3<>0.TOP-REAL 2 by A3,A29; A32: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then A33: f1.p99=|.p99.|*( sqrt(1-((p99`1/|.p99.|-cn)/(1+cn))^2)) by A25,A29; ((cn-FanMorphN)|K0).(|[x,y]|)=((cn-FanMorphN)).(|[x,y]|) by A29,FUNCT_1:49 .= |[ |.p99.|* ((p99`1/|.p99.|-cn)/(1+cn)), |.p99.|*( sqrt(1-((p99`1/ |.p99.|-cn)/(1+cn))^2))]| by A3,A31,Th51 .=|[s,r]| by A17,A29,A30,A32,A33; hence thesis by A3; end; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2>=0 & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A34: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|<= cn & p3`2>=0 & p3<>0.TOP-REAL 2 by A3,A34; hence thesis; end; then f2 is continuous by A3,A17,Th53; hence thesis by A6,A8,A27,A28,JGRAPH_2:35; end; theorem Th58: for cn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p`1 >=(cn)*(|.p.|) & p`2>=0} holds K03 is closed proof defpred Q[Point of TOP-REAL 2] means $1`2>=0; let sn be Real, K003 be Subset of TOP-REAL 2; assume A1: K003={p: p`1>=(sn)*(|.p.|) & p`2>=0}; reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; defpred P[Point of TOP-REAL 2] means ($1`1>=sn*|.$1.|); reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1 where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10; K1 is closed & KX is closed by Lm8,JORDAN6:7; hence thesis by A1,A2,TOPS_1:8; end; theorem Th59: for cn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p`1 <=(cn)*(|.p.|) & p`2>=0} holds K03 is closed proof defpred Q[Point of TOP-REAL 2] means $1`2>=0; let sn be Real, K003 be Subset of TOP-REAL 2; assume A1: K003={p: p`1<=(sn)*(|.p.|) & p`2>=0}; reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; defpred P[Point of TOP-REAL 2] means ($1`1<=sn*|.$1.|); reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1 where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10; K1 is closed & KX is closed by Lm10,JORDAN6:7; hence thesis by A1,A2,TOPS_1:8; end; theorem Th60: for cn being Real, K0,B0 being Subset of TOP-REAL 2,f being Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1 =0 & p<>0.TOP-REAL 2} holds f is continuous proof let cn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set sn=sqrt(1-cn^2); set p0=|[cn,sn]|; A1: p0`2=sn by EUCLID:52; assume A2: -1 =0 & p<>0.TOP-REAL 2}; then cn^2<1^2 by SQUARE_1:50; then A3: 1-cn^2>0 by XREAL_1:50; then A4: p0`2>0 by A1,SQUARE_1:25; then p0 in K0 by A2,JGRAPH_2:3; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; p0<>0.TOP-REAL 2 by A1,A3,JGRAPH_2:3,SQUARE_1:25; then not p0 in {0.TOP-REAL 2} by TARSKI:def 1; then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A2,XBOOLE_0:def 5; A5: the carrier of (TOP-REAL 2)|K1 = K1 by PRE_TOPC:8; p0`1=cn by EUCLID:52; then A6: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1; A7: D<>{}; sn^2=1-cn^2 by A3,SQUARE_1:def 2; then A8: p0`1/|.p0.|=cn by A6,EUCLID:52,SQUARE_1:18; then A9: p0 in {p: p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2} by A4,JGRAPH_2:3; A10: {p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} c= K1 proof let x be object; assume x in {p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2}; then ex p st p=x & p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2; hence thesis by A2; end; A11: {p: p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2} c= K1 proof let x be object; assume x in {p: p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2}; then ex p st p=x & p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2; hence thesis by A2; end; then reconsider K00={p: p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2} as non empty Subset of ((TOP-REAL 2)|K1) by A9,PRE_TOPC:8; the carrier of (TOP-REAL 2)|D =D by PRE_TOPC:8; then A12: rng (f|K00) c=D; p0 in {p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} by A4,A8,JGRAPH_2:3; then reconsider K11={p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} as non empty Subset of ((TOP-REAL 2)|K1) by A10,PRE_TOPC:8; the carrier of (TOP-REAL 2)|D =D by PRE_TOPC:8; then A13: rng (f|K11) c=D; the carrier of (TOP-REAL 2)|B0=the carrier of (TOP-REAL 2)|D; then A14: dom f=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1 .=K1 by PRE_TOPC:8; then dom (f|K00)=K00 by A11,RELAT_1:62 .= the carrier of ((TOP-REAL 2)|K1)|K00 by PRE_TOPC:8; then reconsider f1=f|K00 as Function of ((TOP-REAL 2)|K1)|K00,(TOP-REAL 2)|D by A12,FUNCT_2:2 ; dom (f|K11)=K11 by A10,A14,RELAT_1:62 .= the carrier of ((TOP-REAL 2)|K1)|K11 by PRE_TOPC:8; then reconsider f2=f|K11 as Function of ((TOP-REAL 2)|K1)|K11,(TOP-REAL 2)|D by A13,FUNCT_2:2 ; defpred P[Point of TOP-REAL 2] means $1`1/|.$1.|>=cn & $1`2>=0 & $1<>0. TOP-REAL 2; A15: dom f2=the carrier of ((TOP-REAL 2)|K1)|K11 by FUNCT_2:def 1 .=K11 by PRE_TOPC:8; {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; then reconsider K001={p: p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL 2) by A9; A16: the carrier of (TOP-REAL 2)|K1 =K1 by PRE_TOPC:8; defpred P[Point of TOP-REAL 2] means $1`1>=(cn)*(|.$1.|) & $1`2>=0; {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; then reconsider K003={p: p`1>=(cn)*(|.p.|) & p`2>=0} as Subset of (TOP-REAL 2); defpred P[Point of TOP-REAL 2] means $1`1/|.$1.|<=cn & $1`2>=0 & $1<>0. TOP-REAL 2; A17: {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; A18: rng ((cn-FanMorphN)|K001) c= K1 proof let y be object; assume y in rng ((cn-FanMorphN)|K001); then consider x being object such that A19: x in dom ((cn-FanMorphN)|K001) and A20: y=((cn-FanMorphN)|K001).x by FUNCT_1:def 3; x in dom (cn-FanMorphN) by A19,RELAT_1:57; then reconsider q=x as Point of TOP-REAL 2; A21: y=(cn-FanMorphN).q by A19,A20,FUNCT_1:47; dom ((cn-FanMorphN)|K001)=(dom (cn-FanMorphN))/\ K001 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K001 by FUNCT_2:def 1 .=K001 by XBOOLE_1:28; then A22: ex p2 being Point of TOP-REAL 2 st p2=q & p2`1/|.p2.|>= cn & p2`2>=0 & p2<>0.TOP-REAL 2 by A19; then A23: (q`1/|.q.|-cn)>= 0 by XREAL_1:48; |.q.|<>0 by A22,TOPRNS_1:24; then A24: (|.q.|)^2>0^2 by SQUARE_1:12; set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/ (1-cn))^2))]|; A25: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52; A26: 1-cn>0 by A2,XREAL_1:149; 0<=(q`2)^2 by XREAL_1:63; then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`1)^2 <= (|.q.|)^2 by JGRAPH_3:1; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A24,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`1/|.q.| by SQUARE_1:51; then 1-cn>=q`1/|.q.|-cn by XREAL_1:9; then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A26,XREAL_1:72; then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A26,XCMPLX_1:197; then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A26,A23,SQUARE_1:49; then A27: 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then A28: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187; sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>=0 by A27,SQUARE_1:def 2; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)>=0; then A29: sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76; A30: q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; then A31: (q4`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A28,SQUARE_1:def 2; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A25,A31; then A32: q4<>0.TOP-REAL 2 by A24,TOPRNS_1:23; cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q `1/|.q.|-cn)/(1-cn))^2))]| by A2,A22,Th51; hence thesis by A2,A21,A30,A29,A32; end; A33: dom (cn-FanMorphN)=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then dom ((cn-FanMorphN)|K001)=K001 by RELAT_1:62 .= the carrier of (TOP-REAL 2)|K001 by PRE_TOPC:8; then reconsider f3=(cn-FanMorphN)|K001 as Function of (TOP-REAL 2)|K001,(TOP-REAL 2)|K1 by A5,A18,FUNCT_2:2; A34: K003 is closed by Th58; K1 c= D proof let x be object; assume A35: x in K1; then ex p6 being Point of TOP-REAL 2 st p6=x & p6`2>=0 & p6 <>0.TOP-REAL 2 by A2; then not x in {0.TOP-REAL 2} by TARSKI:def 1; hence thesis by A2,A35,XBOOLE_0:def 5; end; then D=K1 \/ D by XBOOLE_1:12; then A36: (TOP-REAL 2)|K1 is SubSpace of (TOP-REAL 2)|D by TOPMETR:4; p0 in {p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} by A4,A8,JGRAPH_2:3; then reconsider K111={p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} as non empty Subset of TOP-REAL 2 by A17; A37: rng ((cn-FanMorphN)|K111) c= K1 proof let y be object; assume y in rng ((cn-FanMorphN)|K111); then consider x being object such that A38: x in dom ((cn-FanMorphN)|K111) and A39: y=((cn-FanMorphN)|K111).x by FUNCT_1:def 3; x in dom (cn-FanMorphN) by A38,RELAT_1:57; then reconsider q=x as Point of TOP-REAL 2; A40: y=(cn-FanMorphN).q by A38,A39,FUNCT_1:47; dom ((cn-FanMorphN)|K111)=(dom (cn-FanMorphN))/\ K111 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K111 by FUNCT_2:def 1 .=K111 by XBOOLE_1:28; then A41: ex p2 being Point of TOP-REAL 2 st p2=q & p2`1/|.p2.|<= cn & p2`2>=0 & p2<>0.TOP-REAL 2 by A38; then A42: (q`1/|.q.|-cn)<=0 by XREAL_1:47; |.q.|<>0 by A41,TOPRNS_1:24; then A43: (|.q.|)^2>0^2 by SQUARE_1:12; set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/ (1+cn))^2))]|; A44: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52; A45: 1+cn>0 by A2,XREAL_1:148; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1 ,XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A43,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`1/|.q.| by SQUARE_1:51; then -1-cn<=q`1/|.q.|-cn by XREAL_1:9; then (-(1+cn))/(1+cn)<=(( q`1/|.q.|-cn))/(1+cn) by A45,XREAL_1:72; then -1<=(( q`1/|.q.|-cn))/(1+cn) by A45,XCMPLX_1:197; then A46: ( (q`1/|.q.|-cn) /(1+cn))^2<=1^2 by A45,A42,SQUARE_1:49; then A47: 1-((q`1/|.q.|-cn)/(1+cn))^2>=0 by XREAL_1:48; 1-(-((q`1/|.q.|-cn)/(1+cn)))^2>=0 by A46,XREAL_1:48; then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187; then sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(q`1/|.q.|-cn)^2/(1+cn)^2)>=0; then A48: sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76; A49: q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52; then A50: (q4`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A47,SQUARE_1:def 2; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A44,A50; then A51: q4<>0.TOP-REAL 2 by A43,TOPRNS_1:23; cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q `1/|.q.|-cn)/(1+cn))^2))]| by A2,A41,Th51; hence thesis by A2,A40,A49,A48,A51; end; dom ((cn-FanMorphN)|K111)=K111 by A33,RELAT_1:62 .= the carrier of (TOP-REAL 2)|K111 by PRE_TOPC:8; then reconsider f4=(cn-FanMorphN)|K111 as Function of (TOP-REAL 2)|K111,(TOP-REAL 2)|K1 by A16,A37,FUNCT_2:2; the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then ((TOP-REAL 2)|K1)|K11=(TOP-REAL 2)|K111 & f2= f4 by A2,FUNCT_1:51 ,GOBOARD9:2; then A52: f2 is continuous by A2,A36,Th57,PRE_TOPC:26; A53: the carrier of ((TOP-REAL 2)|K1)=K0 by PRE_TOPC:8; set T1= ((TOP-REAL 2)|K1)|K00,T2=((TOP-REAL 2)|K1)|K11; A54: [#](((TOP-REAL 2)|K1)|K11)=K11 by PRE_TOPC:def 5; defpred P[Point of TOP-REAL 2] means $1`1<=cn*(|.$1.|) & $1`2>=0; {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; then reconsider K004={p: p`1<=(cn)*(|.p.|) & p`2>=0} as Subset of (TOP-REAL 2); A55: K004 /\ K1 c= K11 proof let x be object; assume A56: x in K004 /\ K1; then x in K004 by XBOOLE_0:def 4; then consider q1 being Point of TOP-REAL 2 such that A57: q1=x and A58: q1`1<=(cn)*(|.q1.|) and q1`2>=0; x in K1 by A56,XBOOLE_0:def 4; then A59: ex q2 being Point of TOP-REAL 2 st q2=x & q2`2>=0 & q2 <>0.TOP-REAL 2 by A2; q1`1/|.q1.|<=(cn)*(|.q1.|)/|.q1.| by A58,XREAL_1:72; then q1`1/|.q1.|<=(cn) by A57,A59,TOPRNS_1:24,XCMPLX_1:89; hence thesis by A57,A59; end; A60: K004 is closed by Th59; the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then ((TOP-REAL 2)|K1)|K00=(TOP-REAL 2)|K001 & f1= f3 by A2,FUNCT_1:51 ,GOBOARD9:2; then A61: f1 is continuous by A2,A36,Th56,PRE_TOPC:26; A62: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5; K11 c= K004 /\ K1 proof let x be object; assume x in K11; then consider p such that A63: p=x and A64: p`1/|.p.|<=cn and A65: p`2>=0 and A66: p<>0.TOP-REAL 2; p`1/|.p.|*|.p.|<=(cn)*(|.p.|) by A64,XREAL_1:64; then p`1<=(cn)*(|.p.|) by A66,TOPRNS_1:24,XCMPLX_1:87; then A67: x in K004 by A63,A65; x in K1 by A2,A63,A65,A66; hence thesis by A67,XBOOLE_0:def 4; end; then K11=K004 /\ [#]((TOP-REAL 2)|K1) by A62,A55,XBOOLE_0:def 10; then A68: K11 is closed by A60,PRE_TOPC:13; A69: K003 /\ K1 c= K00 proof let x be object; assume A70: x in K003 /\ K1; then x in K003 by XBOOLE_0:def 4; then consider q1 being Point of TOP-REAL 2 such that A71: q1=x and A72: q1`1>=(cn)*(|.q1.|) and q1`2>=0; x in K1 by A70,XBOOLE_0:def 4; then A73: ex q2 being Point of TOP-REAL 2 st q2=x & q2`2>=0 & q2 <>0.TOP-REAL 2 by A2; q1`1/|.q1.|>=(cn)*(|.q1.|)/|.q1.| by A72,XREAL_1:72; then q1`1/|.q1.|>=(cn) by A71,A73,TOPRNS_1:24,XCMPLX_1:89; hence thesis by A71,A73; end; K00 c= K003 /\ K1 proof let x be object; assume x in K00; then consider p such that A74: p=x and A75: p`1/|.p.|>=cn and A76: p`2>=0 and A77: p<>0.TOP-REAL 2; p`1/|.p.|*|.p.|>=(cn)*(|.p.|) by A75,XREAL_1:64; then p`1>=(cn)*(|.p.|) by A77,TOPRNS_1:24,XCMPLX_1:87; then A78: x in K003 by A74,A76; x in K1 by A2,A74,A76,A77; hence thesis by A78,XBOOLE_0:def 4; end; then K00=K003 /\ [#]((TOP-REAL 2)|K1) by A62,A69,XBOOLE_0:def 10; then A79: K00 is closed by A34,PRE_TOPC:13; A80: [#](((TOP-REAL 2)|K1)|K00)=K00 by PRE_TOPC:def 5; A81: for p being object st p in ([#]T1)/\([#]T2) holds f1.p = f2.p proof let p be object; assume A82: p in ([#]T1)/\([#]T2); then p in K00 by A80,XBOOLE_0:def 4; hence f1.p=f.p by FUNCT_1:49 .=f2.p by A54,A82,FUNCT_1:49; end; A83: K1 c= K00 \/ K11 proof let x be object; assume x in K1; then consider p such that A84: p=x & p`2>=0 & p<>0.TOP-REAL 2 by A2; per cases; suppose p`1/|.p.|>=cn; then x in K00 by A84; hence thesis by XBOOLE_0:def 3; end; suppose p`1/|.p.| 0.TOP-REAL 2} holds f is continuous proof let cn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set sn=sqrt(1-cn^2); set p0=|[cn,-sn]|; assume A1: -1 0.TOP-REAL 2}; then cn^2<1^2 by SQUARE_1:50; then 1-cn^2>0 by XREAL_1:50; then p0`2=-sn & --sn>0 by EUCLID:52,SQUARE_1:25; then A2: p0`2<0; then p0 in K0 by A1,JGRAPH_2:3; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; not p0 in {0.TOP-REAL 2} by A2,JGRAPH_2:3,TARSKI:def 1; then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A1,XBOOLE_0:def 5; A3: K1 c= D proof let x be object; assume x in K1; then consider p2 being Point of TOP-REAL 2 such that A4: p2=x and p2`2<=0 and A5: p2<>0.TOP-REAL 2 by A1; not p2 in {0.TOP-REAL 2} by A5,TARSKI:def 1; hence thesis by A1,A4,XBOOLE_0:def 5; end; for p being Point of (TOP-REAL 2)|K1,V being Subset of (TOP-REAL 2)|D st f.p in V & V is open holds ex W being Subset of (TOP-REAL 2)|K1 st p in W & W is open & f.:W c= V proof let p be Point of (TOP-REAL 2)|K1,V be Subset of (TOP-REAL 2)|D; assume that A6: f.p in V and A7: V is open; consider V2 being Subset of TOP-REAL 2 such that A8: V2 is open and A9: V2 /\ [#]((TOP-REAL 2)|D)=V by A7,TOPS_2:24; reconsider W2=V2 /\ [#]((TOP-REAL 2)|K1) as Subset of (TOP-REAL 2)| K1; A10: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5; then A11: f.p=(cn-FanMorphN).p by A1,FUNCT_1:49; A12: f.:W2 c= V proof let y be object; assume y in f.:W2; then consider x being object such that A13: x in dom f and A14: x in W2 and A15: y=f.x by FUNCT_1:def 6; f is Function of (TOP-REAL 2)|K1, (TOP-REAL 2)|D; then dom f= K1 by A10,FUNCT_2:def 1; then consider p4 being Point of TOP-REAL 2 such that A16: x=p4 and A17: p4`2<=0 and p4<>0.TOP-REAL 2 by A1,A13; A18: p4 in V2 by A14,A16,XBOOLE_0:def 4; p4 in [#]((TOP-REAL 2)|K1) by A13,A16; then p4 in D by A3,A10; then A19: p4 in [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5; f.p4=(cn-FanMorphN).p4 by A1,A10,A13,A16,FUNCT_1:49 .=p4 by A17,Th49; hence thesis by A9,A15,A16,A18,A19,XBOOLE_0:def 4; end; p in the carrier of (TOP-REAL 2)|K1; then consider q being Point of TOP-REAL 2 such that A20: q=p and A21: q`2<=0 and q <>0.TOP-REAL 2 by A1,A10; (cn-FanMorphN).q=q by A21,Th49; then p in V2 by A6,A9,A11,A20,XBOOLE_0:def 4; then A22: p in W2 by XBOOLE_0:def 4; W2 is open by A8,TOPS_2:24; hence thesis by A22,A12; end; hence thesis by JGRAPH_2:10; end; theorem Th62: for B0 being Subset of TOP-REAL 2, K0 being Subset of (TOP-REAL 2)|B0 st B0=NonZero TOP-REAL 2 & K0={p: p`2>=0 & p<>0.TOP-REAL 2} holds K0 is closed proof set J0 = NonZero TOP-REAL 2; defpred P[Point of TOP-REAL 2] means $1`2>=0; set I1 = {p: P[p] & p<>0.TOP-REAL 2}; let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0; reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; A1: I1 = {p7 where p7 is Point of TOP-REAL 2 : P[p7]} /\ J0 from JGRAPH_3: sch 2; assume B0=J0 & K0=I1; then K1 is closed & K0=K1 /\ [#]((TOP-REAL 2)|B0) by A1,JORDAN6:7 ,PRE_TOPC:def 5; hence thesis by PRE_TOPC:13; end; theorem Th63: for B0 being Subset of TOP-REAL 2, K0 being Subset of (TOP-REAL 2)|B0 st B0=NonZero TOP-REAL 2 & K0={p: p`2<=0 & p<>0.TOP-REAL 2} holds K0 is closed proof set J0 = NonZero TOP-REAL 2; defpred P[Point of TOP-REAL 2] means $1`2<=0; set I1 = {p: P[p] & p<>0.TOP-REAL 2}; let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0; reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; A1: I1 = {p7 where p7 is Point of TOP-REAL 2 : P[p7]} /\ J0 from JGRAPH_3: sch 2; assume B0=J0 & K0=I1; then K1 is closed & K0=K1 /\ [#]((TOP-REAL 2)|B0) by A1,JORDAN6:8 ,PRE_TOPC:def 5; hence thesis by PRE_TOPC:13; end; theorem Th64: for cn being Real, B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2)|B0,f being Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0) st -1 =0 & p<>0.TOP-REAL 2} holds f is continuous proof let cn be Real, B0 be Subset of TOP-REAL 2, K0 be Subset of (TOP-REAL 2)|B0,f be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0); the carrier of (TOP-REAL 2)|B0= B0 by PRE_TOPC:8; then reconsider K1=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1; assume A1: -1 =0 & p<>0.TOP-REAL 2 }; K0 c= B0 proof let x be object; assume x in K0; then A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2>=0 & p8 <>0.TOP-REAL 2 by A1; then not x in {0.TOP-REAL 2} by TARSKI:def 1; hence thesis by A1,A2,XBOOLE_0:def 5; end; then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7; hence thesis by A1,Th60; end; theorem Th65: for cn being Real, B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2)|B0,f being Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0) st -1 0.TOP-REAL 2} holds f is continuous proof let cn be Real, B0 be Subset of TOP-REAL 2, K0 be Subset of (TOP-REAL 2)|B0, f be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0); the carrier of (TOP-REAL 2)|B0= B0 by PRE_TOPC:8; then reconsider K1=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1; assume A1: -1 0.TOP-REAL 2}; K0 c= B0 proof let x be object; assume x in K0; then A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2<=0 & p8 <>0.TOP-REAL 2 by A1; then not x in {0.TOP-REAL 2} by TARSKI:def 1; hence thesis by A1,A2,XBOOLE_0:def 5; end; then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7; hence thesis by A1,Th61; end; theorem Th66: for cn being Real,p being Point of TOP-REAL 2 holds |.(cn -FanMorphN).p.|=|.p.| proof let cn be Real,p be Point of TOP-REAL 2; set f=cn-FanMorphN; set z=f.p; reconsider q=p as Point of TOP-REAL 2; reconsider qz=z as Point of TOP-REAL 2; per cases; suppose A1: q`1/|.q.|>=cn & q`2>0; then A2: (cn-FanMorphN).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-(( q`1/|.q.|-cn)/(1-cn))^2))]| by Th49; then A3: qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; A4: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by A2,EUCLID:52; A5: (q`1/|.q.|-cn)>=0 by A1,XREAL_1:48; A6: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; |.q.|<>0 by A1,JGRAPH_2:3,TOPRNS_1:24; then A7: (|.q.|)^2>0 by SQUARE_1:12; 0<=(q`2)^2 by XREAL_1:63; then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A6,XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A7,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`1/|.q.| by SQUARE_1:51; then A8: 1-cn>=q`1/|.q.|-cn by XREAL_1:9; per cases; suppose A9: 1-cn=0; A10: ((q`1/|.q.|-cn)/(1-cn))=(q`1/|.q.|-cn)*(1-cn)" by XCMPLX_0:def 9 .= (q`1/|.q.|-cn)*0 by A9 .=0; then 1-((q`1/|.q.|-cn)/(1-cn))^2=1; then (cn-FanMorphN).q= |[ |.q.|*0,|.q.|*1]| by A1,A10,Th49,SQUARE_1:18 .=|[0,(|.q.|)]|; then ((cn-FanMorphN).q)`2=(|.q.|) & ((cn-FanMorphN).q)`1=0 by EUCLID:52; then |.(cn-FanMorphN).p.|=sqrt(((|.q.|))^2+0^2) by JGRAPH_3:1 .=|.q.| by SQUARE_1:22; hence thesis; end; suppose A11: 1-cn<>0; per cases by A11; suppose A12: 1-cn>0; -(1-cn)<= -( q`1/|.q.|-cn) by A8,XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A12,XREAL_1:72; then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A12,XCMPLX_1:197; then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A5,A12,SQUARE_1:49; then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then A13: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187; A14: (qz`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2 by A3 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A13,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A4,A14; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; hence thesis by SQUARE_1:22; end; suppose A15: 1-cn<0; 0+(q`1)^2<(q`1)^2+(q`2)^2 by A1,SQUARE_1:12,XREAL_1:8; then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,A6,XREAL_1:74; then (q`1)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60; then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76; then A16: 1 > q`1/|.p.| by SQUARE_1:52; q`1/|.q.|-cn>=0 by A1,XREAL_1:48; hence thesis by A15,A16,XREAL_1:9; end; end; end; suppose A17: q`1/|.q.| 0; then |.q.|<>0 by JGRAPH_2:3,TOPRNS_1:24; then A18: (|.q.|)^2>0 by SQUARE_1:12; A19: (q`1/|.q.|-cn)<0 by A17,XREAL_1:49; A20: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; 0<=(q`2)^2 by XREAL_1:63; then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A20,XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A18,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`1/|.q.| by SQUARE_1:51; then A21: -1-cn<=q`1/|.q.|-cn by XREAL_1:9; A22: (cn-FanMorphN).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-( (q`1/|.q.|-cn)/(1+cn))^2))]| by A17,Th50; then A23: qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52; A24: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by A22,EUCLID:52; per cases; suppose A25: 1+cn=0; ((q`1/|.q.|-cn)/(1+cn))=(q`1/|.q.|-cn)*(1+cn)" by XCMPLX_0:def 9 .= (q`1/|.q.|-cn)*0 by A25 .=0; then ((cn-FanMorphN).q)`2=(|.q.|) & ((cn-FanMorphN).q)`1=0 by A22, EUCLID:52,SQUARE_1:18; then |.(cn-FanMorphN).p.|=sqrt(((|.q.|))^2+0^2) by JGRAPH_3:1 .=|.q.| by SQUARE_1:22; hence thesis; end; suppose A26: 1+cn<>0; per cases by A26; suppose A27: 1+cn>0; then (-(1+cn))/(1+cn)<=(( q`1/|.q.|-cn))/(1+cn) by A21,XREAL_1:72; then -1<=(( q`1/|.q.|-cn))/(1+cn) by A27,XCMPLX_1:197; then ( (q`1/|.q.|-cn) /(1+cn))^2<=1^2 by A19,A27,SQUARE_1:49; then A28: 1-(((q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48; A29: (qz`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2 by A23 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A28,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A24,A29; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; hence thesis by SQUARE_1:22; end; suppose A30: 1+cn<0; 0+(q`1)^2<(q`1)^2+(q`2)^2 by A17,SQUARE_1:12,XREAL_1:8; then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A18,A20,XREAL_1:74; then (q`1)^2/(|.q.|)^2 < 1 by A18,XCMPLX_1:60; then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76; then -1 < q`1/|.p.| by SQUARE_1:52; then A31: q`1/|.q.|-cn>-1-cn by XREAL_1:9; -(1+cn)>-0 by A30,XREAL_1:24; hence thesis by A17,A31,XREAL_1:49; end; end; end; suppose q`2<=0; hence thesis by Th49; end; end; theorem Th67: for cn being Real,x,K0 being set st -1 =0 & p<>0.TOP-REAL 2} holds (cn-FanMorphN).x in K0 proof let cn be Real,x,K0 be set; assume A1: -1 =0 & p<>0.TOP-REAL 2}; then consider p such that A2: p=x and A3: p`2>=0 and A4: p<>0.TOP-REAL 2; A5: now assume |.p.|<=0; then |.p.|=0; hence contradiction by A4,TOPRNS_1:24; end; then A6: (|.p.|)^2>0 by SQUARE_1:12; per cases; suppose A7: p`1/|.p.|<=cn; reconsider p9= (cn-FanMorphN).p as Point of TOP-REAL 2; (cn-FanMorphN).p= |[ |.p.|*((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-(( p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A3,A4,A7,Th51; then A8: p9`2=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52; A9: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; A10: 1+cn>0 by A1,XREAL_1:148; per cases; suppose p`2=0; hence thesis by A1,A2,Th49; end; suppose p`2<>0; then 0+(p`1)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8; then (p`1)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A9,XREAL_1:74; then (p`1)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60; then ((p`1)/|.p.|)^2 < 1 by XCMPLX_1:76; then -1 < p`1/|.p.| by SQUARE_1:52; then -1-cn< p`1/|.p.|-cn by XREAL_1:9; then (-1)*(1+cn)/(1+cn)< (p`1/|.p.|-cn)/(1+cn) by A10,XREAL_1:74; then A11: -1< (p`1/|.p.|-cn)/(1+cn) by A10,XCMPLX_1:89; p`1/|.p.|-cn<=0 by A7,XREAL_1:47; then 1^2> ((p`1/|.p.|-cn)/(1+cn))^2 by A10,A11,SQUARE_1:50; then 1-((p`1/|.p.|-cn)/(1+cn))^2>0 by XREAL_1:50; then sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)>0 by SQUARE_1:25; then |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))>0 by A5,XREAL_1:129; hence thesis by A1,A2,A8,JGRAPH_2:3; end; end; suppose A12: p`1/|.p.|>cn; reconsider p9= (cn-FanMorphN).p as Point of TOP-REAL 2; (cn-FanMorphN).p= |[ |.p.|*((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-(( p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A3,A4,A12,Th51; then A13: p9`2=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52; A14: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; A15: 1-cn>0 by A1,XREAL_1:149; per cases; suppose p`2=0; hence thesis by A1,A2,Th49; end; suppose p`2<>0; then 0+(p`1)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8; then (p`1)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A14,XREAL_1:74; then (p`1)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60; then ((p`1)/|.p.|)^2 < 1 by XCMPLX_1:76; then p`1/|.p.|<1 by SQUARE_1:52; then (p`1/|.p.|-cn)<1-cn by XREAL_1:9; then (p`1/|.p.|-cn)/(1-cn)<(1-cn)/(1-cn) by A15,XREAL_1:74; then A16: (p`1/|.p.|-cn)/(1-cn)<1 by A15,XCMPLX_1:60; -(1-cn)< -0 & p`1/|.p.|-cn>=cn-cn by A12,A15,XREAL_1:9,24; then (-1)*(1-cn)/(1-cn)< (p`1/|.p.|-cn)/(1-cn) by A15,XREAL_1:74; then -1< (p`1/|.p.|-cn)/(1-cn) by A15,XCMPLX_1:89; then 1^2> ((p`1/|.p.|-cn)/(1-cn))^2 by A16,SQUARE_1:50; then 1-((p`1/|.p.|-cn)/(1-cn))^2>0 by XREAL_1:50; then sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)>0 by SQUARE_1:25; then p9`2>0 by A5,A13,XREAL_1:129; hence thesis by A1,A2,JGRAPH_2:3; end; end; end; theorem Th68: for cn being Real,x,K0 being set st -1 0.TOP-REAL 2} holds (cn-FanMorphN).x in K0 proof let cn be Real,x,K0 be set; assume A1: -1 0.TOP-REAL 2}; then ex p st p=x & p`2<=0 & p<>0.TOP-REAL 2; hence thesis by A1,Th49; end; theorem Th69: for cn being Real, D being non empty Subset of TOP-REAL 2 st -1< cn & cn<1 & D`={0.TOP-REAL 2} holds ex h being Function of (TOP-REAL 2)|D,( TOP-REAL 2)|D st h=(cn-FanMorphN)|D & h is continuous proof (|[0,1]|)`2=1 by EUCLID:52; then A1: |[0,1]| in {p where p is Point of TOP-REAL 2: p`2>=0 & p<>0.TOP-REAL 2} by JGRAPH_2:3; set Y1=|[0,-1]|; reconsider B0= {0.TOP-REAL 2} as Subset of TOP-REAL 2; defpred P[Point of TOP-REAL 2] means $1`2>=0; let cn be Real,D be non empty Subset of TOP-REAL 2; assume that A2: -1 0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from InclSub(A5); then reconsider K0={p:p`2>=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL 2)|D by A1; A6: K0=the carrier of ((TOP-REAL 2)|D)|K0 by PRE_TOPC:8; A7: the carrier of ((TOP-REAL 2)|D)=D by PRE_TOPC:8; A8: rng ((cn-FanMorphN)|K0) c= the carrier of ((TOP-REAL 2)|D)|K0 proof let y be object; assume y in rng ((cn-FanMorphN)|K0); then consider x being object such that A9: x in dom ((cn-FanMorphN)|K0) and A10: y=((cn-FanMorphN)|K0).x by FUNCT_1:def 3; x in (dom (cn-FanMorphN)) /\ K0 by A9,RELAT_1:61; then A11: x in K0 by XBOOLE_0:def 4; K0 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1; then reconsider p=x as Point of TOP-REAL 2 by A11; (cn-FanMorphN).p=y by A10,A11,FUNCT_1:49; then y in K0 by A2,A11,Th67; hence thesis by PRE_TOPC:8; end; A12: K0 c= (the carrier of TOP-REAL 2) proof let z be object; assume z in K0; then ex p8 being Point of TOP-REAL 2 st p8=z & p8`2>=0 & p8 <>0.TOP-REAL 2; hence thesis; end; Y1`2=-1 & 0.TOP-REAL 2 <> Y1 by EUCLID:52,JGRAPH_2:3; then A13: Y1 in {p where p is Point of TOP-REAL 2: p`2<=0 & p<>0.TOP-REAL 2}; A14: the carrier of ((TOP-REAL 2)|D) = (NonZero TOP-REAL 2) by A5,PRE_TOPC:8; defpred P[Point of TOP-REAL 2] means $1`2<=0; {p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from InclSub(A5); then reconsider K1={p: p`2<=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL 2)|D by A13; A15: K0 is closed & K1 is closed by A5,Th62,Th63; dom ((cn-FanMorphN)|K0)= dom ((cn-FanMorphN)) /\ K0 by RELAT_1:61 .=((the carrier of TOP-REAL 2)) /\ K0 by FUNCT_2:def 1 .=K0 by A12,XBOOLE_1:28; then reconsider f=(cn-FanMorphN)|K0 as Function of ((TOP-REAL 2)|D)|K0, (( TOP-REAL 2)|D) by A6,A8,FUNCT_2:2,XBOOLE_1:1; A16: K1=the carrier of ((TOP-REAL 2)|D)|K1 by PRE_TOPC:8; A17: rng ((cn-FanMorphN)|K1) c= the carrier of ((TOP-REAL 2)|D)|K1 proof let y be object; assume y in rng ((cn-FanMorphN)|K1); then consider x being object such that A18: x in dom ((cn-FanMorphN)|K1) and A19: y=((cn-FanMorphN)|K1).x by FUNCT_1:def 3; x in (dom (cn-FanMorphN)) /\ K1 by A18,RELAT_1:61; then A20: x in K1 by XBOOLE_0:def 4; K1 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1; then reconsider p=x as Point of TOP-REAL 2 by A20; (cn-FanMorphN).p=y by A19,A20,FUNCT_1:49; then y in K1 by A2,A20,Th68; hence thesis by PRE_TOPC:8; end; A21: K1 c= (the carrier of TOP-REAL 2) proof let z be object; assume z in K1; then ex p8 being Point of TOP-REAL 2 st p8=z & p8`2<=0 & p8 <>0.TOP-REAL 2; hence thesis; end; dom ((cn-FanMorphN)|K1)= dom ((cn-FanMorphN)) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2)) /\ K1 by FUNCT_2:def 1 .=K1 by A21,XBOOLE_1:28; then reconsider g=(cn-FanMorphN)|K1 as Function of ((TOP-REAL 2)|D)|K1, (( TOP-REAL 2)|D) by A16,A17,FUNCT_2:2,XBOOLE_1:1; A22: K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5; A23: D c= K0 \/ K1 proof let x be object; assume A24: x in D; then reconsider px=x as Point of TOP-REAL 2; not x in {0.TOP-REAL 2} by A5,A24,XBOOLE_0:def 5; then px`2>=0 & px<>0.TOP-REAL 2 or px`2<=0 & px<>0.TOP-REAL 2 by TARSKI:def 1; then x in K0 or x in K1; hence thesis by XBOOLE_0:def 3; end; A25: dom f=K0 by A6,FUNCT_2:def 1; A26: K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5; A27: for x be object st x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1))) holds f.x = g.x proof let x be object; assume A28: x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1)) ); then x in K0 by A26,XBOOLE_0:def 4; then f.x=(cn-FanMorphN).x by FUNCT_1:49; hence thesis by A22,A28,FUNCT_1:49; end; D= [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5; then A29: ([#](((TOP-REAL 2)|D)|K0)) \/ ([#](((TOP-REAL 2)|D)|K1)) = [#](( TOP-REAL 2)|D) by A26,A22,A23,XBOOLE_0:def 10; A30: f is continuous & g is continuous by A2,A5,Th64,Th65; then consider h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that A31: h= f+*g and h is continuous by A26,A22,A29,A15,A27,JGRAPH_2:1; A32: dom h=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1; A33: dom g=K1 by A16,FUNCT_2:def 1; K0=[#](((TOP-REAL 2)|D)|K0) & K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5; then A34: f tolerates g by A27,A25,A33,PARTFUN1:def 4; A35: for x being object st x in dom h holds h.x=((cn-FanMorphN)|D).x proof let x be object; assume A36: x in dom h; then reconsider p=x as Point of TOP-REAL 2 by A14,XBOOLE_0:def 5; not x in {0.TOP-REAL 2} by A14,A36,XBOOLE_0:def 5; then A37: x <>0.TOP-REAL 2 by TARSKI:def 1; A38: x in D`` by A32,A36,PRE_TOPC:8; now per cases; case A39: x in K0; A40: (cn-FanMorphN)|D.p=(cn-FanMorphN).p by A38,FUNCT_1:49 .=f.p by A39,FUNCT_1:49; h.p=(g+*f).p by A31,A34,FUNCT_4:34 .=f.p by A25,A39,FUNCT_4:13; hence thesis by A40; end; case not x in K0; then not p`2>=0 by A37; then A41: x in K1 by A37; (cn-FanMorphN)|D.p=(cn-FanMorphN).p by A38,FUNCT_1:49 .=g.p by A41,FUNCT_1:49; hence thesis by A31,A33,A41,FUNCT_4:13; end; end; hence thesis; end; dom (cn-FanMorphN)=(the carrier of (TOP-REAL 2)) by FUNCT_2:def 1; then dom ((cn-FanMorphN)|D)=(the carrier of (TOP-REAL 2))/\ D by RELAT_1:61 .=the carrier of ((TOP-REAL 2)|D) by A4,XBOOLE_1:28; then f+*g=(cn-FanMorphN)|D by A31,A32,A35,FUNCT_1:2; hence thesis by A26,A22,A29,A30,A15,A27,JGRAPH_2:1; end; theorem Th70: for cn being Real st -1 f.(0.TOP-REAL 2) proof let p be Point of (TOP-REAL 2)|D; A5: [#]((TOP-REAL 2)|D)=D by PRE_TOPC:def 5; then reconsider q=p as Point of TOP-REAL 2 by XBOOLE_0:def 5; not p in {0.TOP-REAL 2} by A5,XBOOLE_0:def 5; then A6: not p=0.TOP-REAL 2 by TARSKI:def 1; now per cases; case A7: q`1/|.q.|>=cn & q`2>=0; set q9= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|- cn)/(1-cn))^2))]|; A8: q9`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52; now assume A9: q9=0.TOP-REAL 2; A10: |.q.|<>0^2 by A6,TOPRNS_1:24; then sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)= sqrt(1-0) by A8,A9,JGRAPH_2:3 ,XCMPLX_1:6 .=1 by SQUARE_1:18; hence contradiction by A9,A10,EUCLID:52,JGRAPH_2:3; end; hence thesis by A1,A2,A3,A6,A7,Th51; end; case A11: q`1/|.q.| =0; set q9=|[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|- cn)/(1+cn))^2))]|; A12: q9`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52; now assume A13: q9=0.TOP-REAL 2; A14: |.q.|<>0^2 by A6,TOPRNS_1:24; then sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)= sqrt(1-0) by A12,A13, JGRAPH_2:3,XCMPLX_1:6 .=1 by SQUARE_1:18; hence contradiction by A13,A14,EUCLID:52,JGRAPH_2:3; end; hence thesis by A1,A2,A3,A6,A11,Th51; end; case q`2<0; then f.p=p by Th49; hence thesis by A6,Th49,JGRAPH_2:3; end; end; hence thesis; end; A15: for V being Subset of (TOP-REAL 2) st f.(0.TOP-REAL 2) in V & V is open ex W being Subset of (TOP-REAL 2) st 0.TOP-REAL 2 in W & W is open & f.:W c= V proof reconsider u0=0.TOP-REAL 2 as Point of Euclid 2 by EUCLID:67; let V be Subset of (TOP-REAL 2); reconsider VV = V as Subset of TopSpaceMetr Euclid 2 by Lm11; assume that A16: f.(0.TOP-REAL 2) in V and A17: V is open; VV is open by A17,Lm11,PRE_TOPC:30; then consider r being Real such that A18: r>0 and A19: Ball(u0,r) c= V by A3,A16,TOPMETR:15; reconsider r as Real; the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8; then reconsider W1=Ball(u0,r) as Subset of TOP-REAL 2; A20: W1 is open by GOBOARD6:3; A21: f.:W1 c= W1 proof let z be object; assume z in f.:W1; then consider y being object such that A22: y in dom f and A23: y in W1 and A24: z=f.y by FUNCT_1:def 6; z in rng f by A22,A24,FUNCT_1:def 3; then reconsider qz=z as Point of TOP-REAL 2; reconsider q=y as Point of TOP-REAL 2 by A22; reconsider qy=q as Point of Euclid 2 by EUCLID:67; reconsider pz=qz as Point of Euclid 2 by EUCLID:67; dist(u0,qy) 0.TOP-REAL 2 & q`1/|.q.|>=cn & q`2>=0; then A27: (q`1/|.q.|-cn)>= 0 by XREAL_1:48; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then A28: (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; A29: 1-cn>0 by A2,XREAL_1:149; |.q.|<>0 by A26,TOPRNS_1:24; then (|.q.|)^2>0 by SQUARE_1:12; then (q`1)^2/(|.q.|)^2 <= 1 by A28,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`1/|.q.| by SQUARE_1:51; then 1-cn>=q`1/|.q.|-cn by XREAL_1:9; then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A29,XREAL_1:72; then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A29,XCMPLX_1:197; then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A29,A27,SQUARE_1:49; then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then A30: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187; A31: (cn-FanMorphN).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)) , |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))]| by A1,A2,A26,Th51; then A32: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by A24,EUCLID:52; qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by A24,A31,EUCLID:52 ; then A33: (qz`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A30,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A32,A33; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; then A34: |.qz.|=|.q.| by SQUARE_1:22; |.- q.| 0.TOP-REAL 2 & q`1/|.q.| =0; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then A36: (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; A37: 1+cn>0 by A1,XREAL_1:148; |.q.|<>0 by A35,TOPRNS_1:24; then (|.q.|)^2>0 by SQUARE_1:12; then (q`1)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`1/|.q.| by SQUARE_1:51; then --1>=-q`1/|.q.| by XREAL_1:24; then 1+cn>=-q`1/|.q.|+cn by XREAL_1:7; then A38: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A37,XREAL_1:185; (cn-q`1/|.q.|)>=0 by A35,XREAL_1:48; then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A37; then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A38,SQUARE_1:49; then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48; then A39: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187; A40: (cn-FanMorphN).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))]| by A1,A2,A35,Th51; then A41: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by A24,EUCLID:52; qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by A24,A40,EUCLID:52 ; then A42: (qz`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A39,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A41,A42; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; then A43: |.qz.|=|.q.| by SQUARE_1:22; |.- q.| 0 by A2,XREAL_1:149; now per cases by JGRAPH_2:3; case A7: q`2<=0; then A8: (cn-FanMorphN).q=q by Th49; now per cases by JGRAPH_2:3; case p`2<=0; hence thesis by A5,A8,Th49; end; case A9: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2>=0; set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|. p.|-cn)/(1-cn))^2))]|; A10: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; 0<=(p`2)^2 by XREAL_1:63; then 0+(p`1)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7; then A11: (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A10,XREAL_1:72; A12: |.p.|>0 by A9,Lm1; then (|.p.|)^2>0 by SQUARE_1:12; then (p`1)^2/(|.p.|)^2 <= 1 by A11,XCMPLX_1:60; then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76; then 1>=p`1/|.p.| by SQUARE_1:51; then 1-cn>=p`1/|.p.|-cn by XREAL_1:9; then -(1-cn)<= -( p`1/|.p.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XREAL_1:72; then A13: -1<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XCMPLX_1:197; A14: (p`1/|.p.|-cn)>=0 by A9,XREAL_1:48; A15: cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A2,A9,Th51; (p`1/|.p.|-cn)>= 0 by A9,XREAL_1:48; then ((-(p`1/|.p.|-cn))/(1-cn))^2<=1^2 by A6,A13,SQUARE_1:49; then A16: 1-((-(p`1/|.p.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then sqrt(1-((-(p`1/|.p.|-cn))/(1-cn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(p`1/|.p.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(p`1/|.p.|-cn)^2/(1-cn)^2)>=0; then sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76; then p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) & q`2=0 by A5 ,A7,A8,A15,EUCLID:52; then A17: ( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))=0 by A5,A8,A15,A12,XCMPLX_1:6; 1-(-((p`1/|.p.|-cn))/(1-cn))^2>=0 by A16,XCMPLX_1:187; then 1-((p`1/|.p.|-cn)/(1-cn))^2=0 by A17,SQUARE_1:24; then 1= (p`1/|.p.|-cn)/(1-cn) by A6,A14,SQUARE_1:18,22; then 1 *(1-cn)=(p`1/|.p.|-cn) by A6,XCMPLX_1:87; then 1 *|.p.|=p`1 by A12,XCMPLX_1:87; then p`2=0 by A10,XCMPLX_1:6; hence thesis by A5,A8,Th49; end; case A18: p<>0.TOP-REAL 2 & p`1/|.p.| =0; then A19: |.p.|<>0 by TOPRNS_1:24; then A20: (|.p.|)^2>0 by SQUARE_1:12; set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|. p.|-cn)/(1+cn))^2))]|; A21: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; A22: 1+cn>0 by A1,XREAL_1:148; A23: (p`1/|.p.|-cn)<=0 by A18,XREAL_1:47; then A24: -1<=(-( p`1/|.p.|-cn))/(1+cn) by A22; A25: cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A2,A18,Th51; 0<=(p`2)^2 by XREAL_1:63; then 0+(p`1)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7; then (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A21,XREAL_1:72; then (p`1)^2/(|.p.|)^2 <= 1 by A20,XCMPLX_1:60; then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76; then (-((p`1)/|.p.|))^2 <= 1; then 1>= -p`1/|.p.| by SQUARE_1:51; then (1+cn)>= -p`1/|.p.|+cn by XREAL_1:7; then (-(p`1/|.p.|-cn))/(1+cn)<=1 by A22,XREAL_1:185; then ((-(p`1/|.p.|-cn))/(1+cn))^2<=1^2 by A24,SQUARE_1:49; then A26: 1-((-(p`1/|.p.|-cn))/(1+cn))^2>=0 by XREAL_1:48; then sqrt(1-((-(p`1/|.p.|-cn))/(1+cn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(p`1/|.p.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-((p`1/|.p.|-cn))^2/(1+cn)^2)>=0; then sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76; then p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) & q`2=0 by A5 ,A7,A8,A25,EUCLID:52; then A27: ( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))=0 by A5,A8,A25,A19,XCMPLX_1:6; 1-(-((p`1/|.p.|-cn))/(1+cn))^2>=0 by A26,XCMPLX_1:187; then 1-((p`1/|.p.|-cn)/(1+cn))^2=0 by A27,SQUARE_1:24; then 1=sqrt((-((p`1/|.p.|-cn)/(1+cn)))^2) by SQUARE_1:18; then 1= -((p`1/|.p.|-cn)/(1+cn)) by A22,A23,SQUARE_1:22; then 1= ((-(p`1/|.p.|-cn))/(1+cn)) by XCMPLX_1:187; then 1 *(1+cn)=-(p`1/|.p.|-cn) by A22,XCMPLX_1:87; then 1+cn-cn=-p`1/|.p.|; then 1=(-p`1)/|.p.| by XCMPLX_1:187; then 1 *|.p.|=-p`1 by A18,TOPRNS_1:24,XCMPLX_1:87; then (p`1)^2-(p`1)^2 =(p`2)^2 by A21,XCMPLX_1:26; then p`2=0 by XCMPLX_1:6; hence thesis by A5,A8,Th49; end; end; hence thesis; end; case A28: q`1/|.q.|>=cn & q`2>=0 & q<>0.TOP-REAL 2; then |.q.|<>0 by TOPRNS_1:24; then A29: (|.q.|)^2>0 by SQUARE_1:12; set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|- cn)/(1-cn))^2))]|; A30: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52; A31: cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1 -((q`1/|.q.|-cn)/(1-cn))^2))]| by A1,A2,A28,Th51; A32: q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; now per cases by JGRAPH_2:3; case A33: p`2<=0; then A34: (cn-FanMorphN).p=p by Th49; A35: |.q.|<>0 by A28,TOPRNS_1:24; then A36: (|.q.|)^2>0 by SQUARE_1:12; A37: (q`1/|.q.|-cn)>= 0 by A28,XREAL_1:48; A38: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; A39: (q`1/|.q.|-cn)>=0 by A28,XREAL_1:48; A40: 1-cn>0 by A2,XREAL_1:149; 0<=(q`2)^2 by XREAL_1:63; then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A38,XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`1/|.q.| by SQUARE_1:51; then 1-cn>=q`1/|.q.|-cn by XREAL_1:9; then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A40,XREAL_1:72; then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A40,XCMPLX_1:197; then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A40,A37,SQUARE_1:49; then A41: 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)>=0; then sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76; then p`2=0 by A5,A31,A33,A34,EUCLID:52; then A42: ( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))=0 by A5,A31,A32,A34,A35, XCMPLX_1:6; 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by A41,XCMPLX_1:187; then 1-((q`1/|.q.|-cn)/(1-cn))^2=0 by A42,SQUARE_1:24; then 1= (q`1/|.q.|-cn)/(1-cn) by A40,A39,SQUARE_1:18,22; then 1 *(1-cn)=(q`1/|.q.|-cn) by A40,XCMPLX_1:87; then 1 *|.q.|=q`1 by A28,TOPRNS_1:24,XCMPLX_1:87; then q`2=0 by A38,XCMPLX_1:6; hence thesis by A5,A34,Th49; end; case A43: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2>=0; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A29,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`1/|.q.| by SQUARE_1:51; then 1-cn>=q`1/|.q.|-cn by XREAL_1:9; then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A6,XREAL_1:72; then A44: -1<=(-( q`1/|.q.|-cn))/(1-cn) by A6,XCMPLX_1:197; (q`1/|.q.|-cn)>= 0 by A28,XREAL_1:48; then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A6,A44,SQUARE_1:49; then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then A45: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187; q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; then A46: (q4`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A45,SQUARE_1:def 2; A47: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A47,A46; then A48: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22; then A49: |.q4.|=|.q.| by SQUARE_1:22; 0<=(p`2)^2 by XREAL_1:63; then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`1)^2<=(p`1)^2+(p`2)^2 by JGRAPH_3:1,XREAL_1:7; then A50: (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72; |.p.|<>0 by A43,TOPRNS_1:24; then (|.p.|)^2>0 by SQUARE_1:12; then (p`1)^2/(|.p.|)^2 <= 1 by A50,XCMPLX_1:60; then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76; then 1>=p`1/|.p.| by SQUARE_1:51; then 1-cn>=p`1/|.p.|-cn by XREAL_1:9; then -(1-cn)<= -( p`1/|.p.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XREAL_1:72; then A51: -1<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XCMPLX_1:197; (p`1/|.p.|-cn)>= 0 by A43,XREAL_1:48; then ((-(p`1/|.p.|-cn))/(1-cn))^2<=1^2 by A6,A51,SQUARE_1:49; then 1-((-(p`1/|.p.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then A52: 1-(-((p`1/|.p.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187; set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|. p.|-cn)/(1-cn))^2))]|; A53: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1-cn)) by EUCLID:52; p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52; then A54: (p4`2)^2= (|.p.|)^2*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))^2 .= (|.p.|)^2*(1-((p`1/|.p.|-cn)/(1-cn))^2) by A52,SQUARE_1:def 2; (|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1 .=(|.p.|)^2 by A53,A54; then A55: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22; then A56: |.p4.|=|.p.| by SQUARE_1:22; A57: cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A2,A43,Th51; then ((p`1/|.p.|-cn)/(1-cn)) =|.q.|* ((q`1/|.q.|-cn)/(1-cn))/|.p .| by A5,A31,A30,A43,A53,TOPRNS_1:24,XCMPLX_1:89; then (p`1/|.p.|-cn)/(1-cn)=(q`1/|.q.|-cn)/(1-cn) by A5,A31,A43,A57 ,A48,A55,TOPRNS_1:24,XCMPLX_1:89; then (p`1/|.p.|-cn)/(1-cn)*(1-cn)=q`1/|.q.|-cn by A6,XCMPLX_1:87; then p`1/|.p.|-cn=q`1/|.q.|-cn by A6,XCMPLX_1:87; then p`1/|.p.|*|.p.|=q`1 by A5,A31,A43,A57,A49,A56,TOPRNS_1:24 ,XCMPLX_1:87; then A58: p`1=q`1 by A43,TOPRNS_1:24,XCMPLX_1:87; A59: p=|[p`1,p`2]| by EUCLID:53; |.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1; then p`2=sqrt((q`2)^2) by A5,A31,A43,A57,A49,A56,A58,SQUARE_1:22; then p`2=q`2 by A28,SQUARE_1:22; hence thesis by A58,A59,EUCLID:53; end; case A60: p<>0.TOP-REAL 2 & p`1/|.p.| =0; then p`1/|.p.|-cn<0 by XREAL_1:49; then A61: ((p`1/|.p.|-cn)/(1+cn))<0 by A1,XREAL_1:141,148; set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|. p.|-cn)/(1+cn))^2))]|; A62: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1+cn)) & q`1/|.q.|-cn>=0 by A28, EUCLID:52,XREAL_1:48; A63: 1-cn>0 by A2,XREAL_1:149; cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.p.| -cn)/(1+cn))^2))]| & |.p.|<>0 by A1,A2,A60,Th51,TOPRNS_1:24; hence thesis by A5,A31,A30,A61,A62,A63,XREAL_1:132; end; end; hence thesis; end; case A64: q`1/|.q.| =0 & q<>0.TOP-REAL 2; then A65: |.q.|<>0 by TOPRNS_1:24; then A66: (|.q.|)^2>0 by SQUARE_1:12; set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|- cn)/(1+cn))^2))]|; A67: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52; A68: cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt( 1-((q`1/|.q.|-cn)/(1+cn))^2))]| by A1,A2,A64,Th51; A69: q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52; now per cases by JGRAPH_2:3; case A70: p`2<=0; A71: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; A72: 1+cn>0 by A1,XREAL_1:148; 0<=(q`2)^2 by XREAL_1:63; then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A71,XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then (-((q`1)/|.q.|))^2 <= 1; then 1>= -q`1/|.q.| by SQUARE_1:51; then (1+cn)>= -q`1/|.q.|+cn by XREAL_1:7; then A73: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A72,XREAL_1:185; A74: (q`1/|.q.|-cn)<=0 by A64,XREAL_1:47; then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A72; then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A73,SQUARE_1:49; then A75: 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48; then A76: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187; A77: (cn-FanMorphN).p=p by A70,Th49; sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>=0 by A75,SQUARE_1:def 2; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-((q`1/|.q.|-cn))^2/(1+cn)^2)>=0; then sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76; then p`2=0 by A5,A68,A70,A77,EUCLID:52; then ( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))=0 by A5,A68,A69,A65,A77, XCMPLX_1:6; then 1-((q`1/|.q.|-cn)/(1+cn))^2=0 by A76,SQUARE_1:24; then 1=sqrt((-((q`1/|.q.|-cn)/(1+cn)))^2) by SQUARE_1:18; then 1= -((q`1/|.q.|-cn)/(1+cn)) by A72,A74,SQUARE_1:22; then 1= ((-(q`1/|.q.|-cn))/(1+cn)) by XCMPLX_1:187; then 1 *(1+cn)=-(q`1/|.q.|-cn) by A72,XCMPLX_1:87; then 1+cn-cn=-q`1/|.q.|; then 1=(-q`1)/|.q.| by XCMPLX_1:187; then 1 *|.q.|=-q`1 by A64,TOPRNS_1:24,XCMPLX_1:87; then (q`1)^2-(q`1)^2 =(q`2)^2 by A71,XCMPLX_1:26; then q`2=0 by XCMPLX_1:6; hence thesis by A5,A77,Th49; end; case A78: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2>=0; set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|. p.|-cn)/(1-cn))^2))]|; A79: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1-cn)) & |.q.|<>0 by A64,EUCLID:52 ,TOPRNS_1:24; q`1/|.q.|-cn<0 by A64,XREAL_1:49; then A80: ((q`1/|.q.|-cn)/(1+cn))<0 by A1,XREAL_1:141,148; A81: 1-cn>0 by A2,XREAL_1:149; cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.p.| -cn)/(1-cn))^2))]| & p`1/|.p.|-cn>=0 by A1,A2,A78,Th51, XREAL_1:48; hence thesis by A5,A68,A67,A80,A79,A81,XREAL_1:132; end; case A82: p<>0.TOP-REAL 2 & p`1/|.p.| =0; 0<=(p`2)^2 by XREAL_1:63; then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`1)^2<=(p`1)^2+(p`2)^2 by JGRAPH_3:1,XREAL_1:7; then A83: (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72; A84: 1+cn>0 by A1,XREAL_1:148; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`1/|.q.| by SQUARE_1:51; then -1-cn<=q`1/|.q.|-cn by XREAL_1:9; then -(-1-cn)>= -(q`1/|.q.|-cn) by XREAL_1:24; then A85: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A84,XREAL_1:185; (q`1/|.q.|-cn)<=0 by A64,XREAL_1:47; then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A84; then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A85,SQUARE_1:49; then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48; then A86: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187; q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52; then A87: (q4`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A86,SQUARE_1:def 2; A88: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52; set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|. p.|-cn)/(1+cn))^2))]|; A89: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1+cn)) by EUCLID:52; |.p.|<>0 by A82,TOPRNS_1:24; then (|.p.|)^2>0 by SQUARE_1:12; then (p`1)^2/(|.p.|)^2 <= 1 by A83,XCMPLX_1:60; then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76; then -1<=p`1/|.p.| by SQUARE_1:51; then -1-cn<=p`1/|.p.|-cn by XREAL_1:9; then -(-1-cn)>= -(p`1/|.p.|-cn) by XREAL_1:24; then A90: (-(p`1/|.p.|-cn))/(1+cn)<=1 by A84,XREAL_1:185; (p`1/|.p.|-cn)<=0 by A82,XREAL_1:47; then -1<=(-( p`1/|.p.|-cn))/(1+cn) by A84; then ((-(p`1/|.p.|-cn))/(1+cn))^2<=1^2 by A90,SQUARE_1:49; then 1-((-(p`1/|.p.|-cn))/(1+cn))^2>=0 by XREAL_1:48; then A91: 1-(-((p`1/|.p.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187; p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52; then A92: (p4`2)^2= (|.p.|)^2*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))^2 .= (|.p.|)^2*(1-((p`1/|.p.|-cn)/(1+cn))^2) by A91,SQUARE_1:def 2; (|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1 .=(|.p.|)^2 by A89,A92; then A93: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22; then A94: |.p4.|=|.p.| by SQUARE_1:22; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A88,A87; then A95: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22; then A96: |.q4.|=|.q.| by SQUARE_1:22; A97: cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A2,A82,Th51; then ((p`1/|.p.|-cn)/(1+cn))=|.q.|* ((q`1/|.q.|-cn)/(1+cn))/|.p.| by A5,A68,A67,A82,A89,TOPRNS_1:24,XCMPLX_1:89; then (p`1/|.p.|-cn)/(1+cn)=(q`1/|.q.|-cn)/(1+cn) by A5,A68,A82,A97 ,A95,A93,TOPRNS_1:24,XCMPLX_1:89; then (p`1/|.p.|-cn)/(1+cn)*(1+cn)=q`1/|.q.|-cn by A84,XCMPLX_1:87; then p`1/|.p.|-cn=q`1/|.q.|-cn by A84,XCMPLX_1:87; then p`1/|.p.|*|.p.|=q`1 by A5,A68,A82,A97,A96,A94,TOPRNS_1:24 ,XCMPLX_1:87; then A98: p`1=q`1 by A82,TOPRNS_1:24,XCMPLX_1:87; A99: p=|[p`1,p`2]| by EUCLID:53; |.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1; then p`2=sqrt((q`2)^2) by A5,A68,A82,A97,A96,A94,A98,SQUARE_1:22; then p`2=q`2 by A64,SQUARE_1:22; hence thesis by A98,A99,EUCLID:53; end; end; hence thesis; end; end; hence thesis; end; hence thesis by FUNCT_1:def 4; end; theorem Th72: for cn being Real st -1 =0 & q`2>=0 & q<>0.TOP-REAL 2; --(1+cn)>0 by A1,XREAL_1:148; then A6: -(-1-cn)>0; A7: 1-cn>=0 by A2,XREAL_1:149; then q`1/|.q.|*(1-cn)>=0 by A5; then -1-cn<= q`1/|.q.|*(1-cn) by A6; then A8: -1-cn+cn<= q`1/|.q.|*(1-cn)+cn by XREAL_1:7; set px=|[ |.q.|*(q`1/|.q.|*(1-cn)+cn), (|.q.|)*sqrt(1-(q`1/|.q.|*(1- cn)+cn)^2)]|; A9: px`1 = |.q.|*(q`1/|.q.|*(1-cn)+cn) by EUCLID:52; |.q.|<>0 by A5,TOPRNS_1:24; then A10: |.q.|^2>0 by SQUARE_1:12; A11: dom (cn-FanMorphN)=the carrier of TOP-REAL 2 by FUNCT_2:def 1; A12: 1-cn>0 by A2,XREAL_1:149; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A10,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then q`1/|.q.|<=1 by SQUARE_1:51; then q`1/|.q.|*(1-cn) <=1 *(1-cn) by A12,XREAL_1:64; then q`1/|.q.|*(1-cn)+cn-cn <=1-cn; then (q`1/|.q.|*(1-cn)+cn) <=1 by XREAL_1:9; then 1^2>=(q`1/|.q.|*(1-cn)+cn)^2 by A8,SQUARE_1:49; then A13: 1-(q`1/|.q.|*(1-cn)+cn)^2>=0 by XREAL_1:48; then A14: sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2)>=0 by SQUARE_1:def 2; A15: px`2 = (|.q.|)*sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2) by EUCLID:52; then |.px.|^2=((|.q.|)*sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2))^2 +(|.q.|*(q `1/|.q.|*(1-cn)+cn))^2 by A9,JGRAPH_3:1 .=(|.q.|)^2*(sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2))^2 +(|.q.|)^2*((q`1 /|.q.|*(1-cn)+cn))^2; then A16: |.px.|^2=(|.q.|)^2*(1-(q`1/|.q.|*(1-cn)+cn)^2) +(|.q.|)^2*((q`1 /|.q.|*(1-cn)+cn))^2 by A13,SQUARE_1:def 2 .= (|.q.|)^2; then A17: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22 .=|.q.| by SQUARE_1:22; then A18: px<>0.TOP-REAL 2 by A5,TOPRNS_1:23,24; (q`1/|.q.|*(1-cn)+cn)>=0+cn by A5,A7,XREAL_1:7; then px`1/|.px.| >=cn by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89; then A19: (cn-FanMorphN).px =|[ |.px.|* ((px`1/|.px.|-cn)/(1-cn) ), |.px .|*( sqrt(1-((px`1/|.px.|-cn)/(1-cn))^2))]| by A1,A2,A15,A14,A18,Th51; A20: |.px.|*( sqrt((q`2/|.q.|)^2))=|.q.|*(q`2/|.q.|) by A5,A17,SQUARE_1:22 .=q`2 by A5,TOPRNS_1:24,XCMPLX_1:87; A21: |.px.|* ((px`1/|.px.|-cn)/(1-cn)) =|.q.|* (( ((q`1/|.q.|*(1-cn) +cn))-cn)/(1-cn)) by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89 .=|.q.|* ( q`1/|.q.|) by A12,XCMPLX_1:89 .= q`1 by A5,TOPRNS_1:24,XCMPLX_1:87; then |.px.|*( sqrt(1-((px`1/|.px.|-cn)/(1-cn))^2)) = |.px.|*( sqrt(1 -(q`1/|.px.|)^2)) by A5,A17,TOPRNS_1:24,XCMPLX_1:89 .= |.px.|*( sqrt(1-(q`1)^2/(|.px.|)^2)) by XCMPLX_1:76 .= |.px.|*( sqrt( (|.px.|)^2/(|.px.|)^2-(q`1)^2/(|.px.|)^2)) by A10 ,A16,XCMPLX_1:60 .= |.px.|*( sqrt( ((|.px.|)^2-(q`1)^2)/(|.px.|)^2)) by XCMPLX_1:120 .= |.px.|*( sqrt( ((q`1)^2+(q`2)^2-(q`1)^2)/(|.px.|)^2)) by A16, JGRAPH_3:1 .= |.px.|*( sqrt((q`2/|.q.|)^2)) by A17,XCMPLX_1:76; hence ex x being set st x in dom (cn-FanMorphN) & y=(cn-FanMorphN).x by A19,A21,A20,A11,EUCLID:53; end; case A22: q`1/|.q.|<0 & q`2>=0 & q<>0.TOP-REAL 2; A23: 1+cn>=0 by A1,XREAL_1:148; (1-cn)>0 by A2,XREAL_1:149; then A24: 1-cn+cn>= q`1/|.q.|*(1+cn)+cn by A22,A23,XREAL_1:7; A25: 1+cn>0 by A1,XREAL_1:148; |.q.|<>0 by A22,TOPRNS_1:24; then A26: |.q.|^2>0 by SQUARE_1:12; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A26,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then q`1/|.q.|>=-1 by SQUARE_1:51; then q`1/|.q.|*(1+cn) >=(-1)*(1+cn) by A25,XREAL_1:64; then q`1/|.q.|*(1+cn)+cn-cn >=-1-cn; then (q`1/|.q.|*(1+cn)+cn) >=-1 by XREAL_1:9; then 1^2>=(q`1/|.q.|*(1+cn)+cn)^2 by A24,SQUARE_1:49; then A27: 1-(q`1/|.q.|*(1+cn)+cn)^2>=0 by XREAL_1:48; then A28: sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2)>=0 by SQUARE_1:def 2; A29: dom (cn-FanMorphN)=the carrier of TOP-REAL 2 by FUNCT_2:def 1; set px=|[ |.q.|*(q`1/|.q.|*(1+cn)+cn), (|.q.|)*sqrt(1-(q`1/|.q.|*(1+ cn)+cn)^2)]|; A30: px`1 = |.q.|*(q`1/|.q.|*(1+cn)+cn) by EUCLID:52; A31: px`2 = (|.q.|)*sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2) by EUCLID:52; then |.px.|^2=((|.q.|)*sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2))^2 +(|.q.|*(q `1/|.q.|*(1+cn)+cn))^2 by A30,JGRAPH_3:1 .=(|.q.|)^2*(sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2))^2 +(|.q.|)^2*((q`1 /|.q.|*(1+cn)+cn))^2; then A32: |.px.|^2=(|.q.|)^2*(1-(q`1/|.q.|*(1+cn)+cn)^2) +(|.q.|)^2*((q`1 /|.q.|*(1+cn)+cn))^2 by A27,SQUARE_1:def 2 .= (|.q.|)^2; then A33: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22 .=|.q.| by SQUARE_1:22; then A34: px<>0.TOP-REAL 2 by A22,TOPRNS_1:23,24; (q`1/|.q.|*(1+cn)+cn)<=0+cn by A22,A23,XREAL_1:7; then px`1/|.px.| <=cn by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89; then A35: (cn-FanMorphN).px =|[ |.px.|* ((px`1/|.px.|-cn)/(1+cn) ), |.px .|*( sqrt(1-((px`1/|.px.|-cn)/(1+cn))^2))]| by A1,A2,A31,A28,A34,Th51; A36: |.px.|*( sqrt((q`2/|.q.|)^2))=|.q.|*(q`2/|.q.|) by A22,A33, SQUARE_1:22 .=q`2 by A22,TOPRNS_1:24,XCMPLX_1:87; A37: |.px.|* ((px`1/|.px.|-cn)/(1+cn)) =|.q.|* (( ((q`1/|.q.|*(1+cn) +cn))-cn)/(1+cn)) by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89 .=|.q.|* ( q`1/|.q.|) by A25,XCMPLX_1:89 .= q`1 by A22,TOPRNS_1:24,XCMPLX_1:87; then |.px.|*( sqrt(1-((px`1/|.px.|-cn)/(1+cn))^2)) = |.px.|*( sqrt(1 -(q`1/|.px.|)^2)) by A22,A33,TOPRNS_1:24,XCMPLX_1:89 .= |.px.|*( sqrt(1-(q`1)^2/(|.px.|)^2)) by XCMPLX_1:76 .= |.px.|*( sqrt( (|.px.|)^2/(|.px.|)^2-(q`1)^2/(|.px.|)^2)) by A26 ,A32,XCMPLX_1:60 .= |.px.|*( sqrt( ((|.px.|)^2-(q`1)^2)/(|.px.|)^2)) by XCMPLX_1:120 .= |.px.|*( sqrt( ((q`1)^2+(q`2)^2-(q`1)^2)/(|.px.|)^2)) by A32, JGRAPH_3:1 .= |.px.|*( sqrt((q`2/|.q.|)^2)) by A33,XCMPLX_1:76; hence ex x being set st x in dom (cn-FanMorphN) & y=(cn-FanMorphN).x by A35,A37,A36,A29,EUCLID:53; end; end; hence thesis by A3,FUNCT_1:def 3; end; hence thesis by A3,XBOOLE_0:def 10; end; hence thesis; end; theorem Th73: for cn being Real,p2 being Point of TOP-REAL 2 st -1 0 & q`1/|.q.|>=cn holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds p`2>0 & p`1>=0 proof let cn be Real,q be Point of TOP-REAL 2; assume that A1: cn<1 and A2: q`2>0 and A3: q`1/|.q.|>=cn; A4: (q`1/|.q.|-cn)>= 0 by A3,XREAL_1:48; let p be Point of TOP-REAL 2; set qz=p; A5: 1-cn>0 by A1,XREAL_1:149; A6: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24; then A7: (|.q.|)^2>0 by SQUARE_1:12; (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1 ,SQUARE_1:12,XREAL_1:8; then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,XREAL_1:74; then (q`1)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60; then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76; then 1>q`1/|.q.| by SQUARE_1:52; then 1-cn>q`1/|.q.|-cn by XREAL_1:9; then -(1-cn)< -( q`1/|.q.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<(-( q`1/|.q.|-cn))/(1-cn) by A5,XREAL_1:74; then -1<(-( q`1/|.q.|-cn))/(1-cn) by A5,XCMPLX_1:197; then ((-(q`1/|.q.|-cn))/(1-cn))^2<1^2 by A5,A4,SQUARE_1:50; then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>0 by XREAL_1:50; then sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>0 by SQUARE_1:25; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)> 0 by XCMPLX_1:76; then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)> 0; then A8: sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)> 0 by XCMPLX_1:76; assume p=(cn-FanMorphN).q; then A9: p=|[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1- cn))^2))]| by A2,A3,Th49; then qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; hence thesis by A9,A6,A5,A4,A8,EUCLID:52,XREAL_1:129; end; theorem Th76: for cn being Real,q being Point of TOP-REAL 2 st -1 0 & q`1/|.q.| 0 & p`1<0 proof let cn be Real,q be Point of TOP-REAL 2; assume that A1: -1 0 and A3: q`1/|.q.| 0 by A1,XREAL_1:148; let p be Point of TOP-REAL 2; set qz=p; assume p=(cn-FanMorphN).q; then p=|[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+ cn))^2))]| by A2,A3,Th50; then A5: qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) & qz`1= |.q.|* ((q`1/ |.q.|- cn)/(1+cn)) by EUCLID:52; A6: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24; then A7: (|.q.|)^2>0 by SQUARE_1:12; A8: (q`1/|.q.|-cn)< 0 by A3,XREAL_1:49; then -( q`1/|.q.|-cn)>0 by XREAL_1:58; then (-(1+cn))/(1+cn)<(-( q`1/|.q.|-cn))/(1+cn) by A4,XREAL_1:74; then A9: -1<(-( q`1/|.q.|-cn))/(1+cn) by A4,XCMPLX_1:197; (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1 ,SQUARE_1:12,XREAL_1:8; then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,XREAL_1:74; then (q`1)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60; then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76; then -1 -(q`1/|.q.|-cn) by XREAL_1:24; then (-(q`1/|.q.|-cn))/(1+cn)<1 by A4,XREAL_1:191; then ((-(q`1/|.q.|-cn))/(1+cn))^2<1^2 by A9,SQUARE_1:50; then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>0 by XREAL_1:50; then sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>0 by SQUARE_1:25; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)> 0 by XCMPLX_1:76; then sqrt(1-(q`1/|.q.|-cn)^2/(1+cn)^2)> 0; then A10: sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)> 0 by XCMPLX_1:76; ((q`1/|.q.|-cn)/(1+cn))<0 by A1,A8,XREAL_1:141,148; hence thesis by A6,A5,A10,XREAL_1:129,132; end; theorem Th77: for cn being Real,q1,q2 being Point of TOP-REAL 2 st cn<1 & q1`2 >0 & q1`1/|.q1.|>=cn & q2`2>0 & q2`1/|.q2.|>=cn & q1`1/|.q1.|0 and A3: q1`1/|.q1.|>=cn and A4: q2`2>0 and A5: q2`1/|.q2.|>=cn and A6: q1`1/|.q1.| 0 by A1,A6,XREAL_1:9,149; let p1,p2 be Point of TOP-REAL 2; assume that A8: p1=(cn-FanMorphN).q1 and A9: p2=(cn-FanMorphN).q2; A10: |.p2.|=|.q2.| by A9,Th66; p2=|[ |.q2.|* ((q2`1/|.q2.|-cn)/(1-cn)), |.q2.|*( sqrt(1-((q2`1/|.q2.|- cn)/(1-cn))^2))]| by A4,A5,A9,Th49; then A11: p2`1= |.q2.|* ((q2`1/|.q2.|-cn)/(1-cn)) by EUCLID:52; |.q2.|>0 by A4,Lm1,JGRAPH_2:3; then A12: p2`1/|.p2.|= ((q2`1/|.q2.|-cn)/(1-cn)) by A11,A10,XCMPLX_1:89; p1=|[ |.q1.|* ((q1`1/|.q1.|-cn)/(1-cn)), |.q1.|*( sqrt(1-((q1`1/|.q1.|- cn)/(1-cn))^2))]| by A2,A3,A8,Th49; then A13: p1`1= |.q1.|* ((q1`1/|.q1.|-cn)/(1-cn)) by EUCLID:52; A14: |.p1.|=|.q1.| by A8,Th66; |.q1.|>0 by A2,Lm1,JGRAPH_2:3; then p1`1/|.p1.|= ((q1`1/|.q1.|-cn)/(1-cn)) by A13,A14,XCMPLX_1:89; hence thesis by A12,A7,XREAL_1:74; end; theorem Th78: for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1 0 & q1`1/|.q1.| 0 & q2`1/|.q2.| 0 and A3: q1`1/|.q1.| 0 and A5: q2`1/|.q2.| 0 by A1,A6,XREAL_1:9,148; let p1,p2 be Point of TOP-REAL 2; assume that A8: p1=(cn-FanMorphN).q1 and A9: p2=(cn-FanMorphN).q2; A10: |.p2.|=|.q2.| by A9,Th66; p2=|[ |.q2.|* ((q2`1/|.q2.|-cn)/(1+cn)), |.q2.|*( sqrt(1-((q2`1/|.q2.|- cn)/(1+cn))^2))]| by A4,A5,A9,Th50; then A11: p2`1= |.q2.|* ((q2`1/|.q2.|-cn)/(1+cn)) by EUCLID:52; |.q2.|>0 by A4,Lm1,JGRAPH_2:3; then A12: p2`1/|.p2.|= ((q2`1/|.q2.|-cn)/(1+cn)) by A11,A10,XCMPLX_1:89; p1=|[ |.q1.|* ((q1`1/|.q1.|-cn)/(1+cn)), |.q1.|*( sqrt(1-((q1`1/|.q1.|- cn)/(1+cn))^2))]| by A2,A3,A8,Th50; then A13: p1`1= |.q1.|* ((q1`1/|.q1.|-cn)/(1+cn)) by EUCLID:52; A14: |.p1.|=|.q1.| by A8,Th66; |.q1.|>0 by A2,Lm1,JGRAPH_2:3; then p1`1/|.p1.|= ((q1`1/|.q1.|-cn)/(1+cn)) by A13,A14,XCMPLX_1:89; hence thesis by A12,A7,XREAL_1:74; end; theorem for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1 0 & q2`2>0 & q1`1/|.q1.| 0 and A4: q2`2>0 and A5: q1`1/|.q1.| =cn & q2`1/|.q2.|>=cn; hence thesis by A2,A3,A4,A5,A6,A7,Th77; end; suppose q1`1/|.q1.|>=cn & q2`1/|.q2.| =cn; then p2`1>=0 by A2,A4,A7,Th75; then A9: p2`1/|.p2.|>=0; p1`1<0 by A1,A3,A6,A8,Th76; hence thesis by A9,Lm1,JGRAPH_2:3,XREAL_1:141; end; suppose q1`1/|.q1.| 0 & q`1/|.q.|=cn holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds p`2>0 & p`1=0 proof let cn be Real,q be Point of TOP-REAL 2; assume that A1: q`2>0 and A2: q`1/|.q.|=cn; A3: |.q.|<>0 & sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>0 by A1,A2,JGRAPH_2:3 ,SQUARE_1:25,TOPRNS_1:24; let p be Point of TOP-REAL 2; assume p=(cn-FanMorphN).q; then A4: p=|[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1- cn))^2))]| by A1,A2,Th49; then p`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; hence thesis by A2,A4,A3,EUCLID:52,XREAL_1:129; end; theorem for cn being Real holds 0.TOP-REAL 2=(cn-FanMorphN).(0.TOP-REAL 2) by Th49,JGRAPH_2:3; begin :: Fan Morphism for East definition let s be Real, q be Point of TOP-REAL 2; func FanE(s,q) -> Point of TOP-REAL 2 equals :Def6: |.q.|*|[sqrt(1-((q`2/|.q .|-s)/(1-s))^2), (q`2/|.q.|-s)/(1-s)]| if q`2/|.q.|>=s & q`1>0, |.q.|*|[sqrt(1- ((q`2/|.q.|-s)/(1+s))^2), (q`2/|.q.|-s)/(1+s)]| if q`2/|.q.| ~~0 otherwise q; correctness; end; definition let s be Real; func s-FanMorphE -> Function of TOP-REAL 2, TOP-REAL 2 means :Def7: for q being Point of TOP-REAL 2 holds it.q=FanE(s,q); existence proof deffunc F(Point of TOP-REAL 2)=FanE(s,$1); thus ex IT being Function of TOP-REAL 2, TOP-REAL 2 st for q being Point of TOP-REAL 2 holds IT.q=F(q)from FUNCT_2:sch 4; end; uniqueness proof deffunc F(Point of TOP-REAL 2)=FanE(s,$1); thus for a,b being Function of TOP-REAL 2, TOP-REAL 2 st (for q being Point of TOP-REAL 2 holds a.q=F(q)) & (for q being Point of TOP-REAL 2 holds b. q=F(q)) holds a=b from BINOP_2:sch 1; end; end; theorem Th82: for sn being Real holds (q`2/|.q.|>=sn & q`1>0 implies sn -FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|- sn)/(1-sn))]|)& (q`1<=0 implies sn-FanMorphE.q=q) proof let sn be Real; hereby assume q`2/|.q.|>=sn & q`1>0; then FanE(sn,q)= |.q.|*|[sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2), (q`2/|.q.|-sn)/ (1-sn)]| by Def6 .= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn )/(1-sn))]| by EUCLID:58; hence sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(1-sn))]| by Def7; end; assume A1: q`1<=0; sn-FanMorphE.q=FanE(sn,q) by Def7; hence thesis by A1,Def6; end; theorem Th83: for sn being Real holds (q`2/|.q.|<=sn & q`1>0 implies sn -FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|*((q`2/|.q.|- sn)/(1+sn))]|) proof let sn be Real; assume that A1: q`2/|.q.|<=sn and A2: q`1>0; now per cases by A1,XXREAL_0:1; case q`2/|.q.|~~=sn & q`1>=0 & q<>0.TOP-REAL 2 implies sn-FanMorphE.q = |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1- sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(1-sn))]|) & (q`2/|.q.|<=sn & q`1>=0 & q<>0. TOP-REAL 2 implies sn-FanMorphE.q = |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2) ), |.q.|*((q`2/|.q.|-sn)/(1+sn))]|) proof let sn be Real; assume that A1: -1 =sn & q`1>=0 & q<>0.TOP-REAL 2; per cases; suppose A4: q`1>0; then FanE(sn,q)= |.q.|*|[sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2), (q`2/|.q.|-sn )/(1-sn)]| by A3,Def6 .= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|- sn)/(1-sn))]| by EUCLID:58; hence thesis by A4,Def7,Th83; end; suppose A5: q`1<=0; then A6: sn-FanMorphE.q=q by Th82; A7: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; A8: 1-sn>0 by A2,XREAL_1:149; A9: q`1=0 by A3,A5; |.q.|<>0 by A3,TOPRNS_1:24; then |.q.|^2>0 by SQUARE_1:12; then (q`2)^2/|.q.|^2=1^2 by A7,A9,XCMPLX_1:60; then ((q`2)/|.q.|)^2=1^2 by XCMPLX_1:76; then A10: sqrt(((q`2)/|.q.|)^2)=1 by SQUARE_1:22; A11: now assume q`2<0; then -((q`2)/|.q.|)=1 by A10,SQUARE_1:23; hence contradiction by A1,A3; end; sqrt((|.q.|)^2)=|.q.| by SQUARE_1:22; then A12: |.q.|=q`2 by A7,A9,A11,SQUARE_1:22; then 1=q`2/|.q.| by A3,TOPRNS_1:24,XCMPLX_1:60; then (q`2/|.q.|-sn)/(1-sn)=1 by A8,XCMPLX_1:60; hence thesis by A2,A6,A9,A12,EUCLID:53,SQUARE_1:17,TOPRNS_1:24 ,XCMPLX_1:60; end; end; suppose A13: q`2/|.q.|<=sn & q`1>=0 & q<>0.TOP-REAL 2; per cases; suppose q`1>0; hence thesis by Th82,Th83; end; suppose A14: q`1<=0; then A15: q`1=0 by A13; A16: 1+sn>0 by A1,XREAL_1:148; A17: |.q.|<>0 by A13,TOPRNS_1:24; 1>q`2/|.q.| by A2,A13,XXREAL_0:2; then 1 *(|.q.|)>q`2/|.q.|*(|.q.|) by A17,XREAL_1:68; then A18: (|.q.|)^2 =(q`1)^2+(q`2)^2 & (|.q.|)>q`2 by A13,JGRAPH_3:1,TOPRNS_1:24 ,XCMPLX_1:87; then A19: |.q.|=-q`2 by A15,SQUARE_1:40; A20: q`2= -(|.q.|) by A15,A18,SQUARE_1:40; then -1=q`2/|.q.| by A13,TOPRNS_1:24,XCMPLX_1:197; then (q`2/|.q.|-sn)/(1+sn) =(-(1+sn))/(1+sn) .=-1 by A16,XCMPLX_1:197; then |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-sn )/(1+sn))]| =q by A15,A19,EUCLID:53,SQUARE_1:17; hence thesis by A1,A14,A17,A20,Th82,XCMPLX_1:197; end; end; suppose q`1<0 or q=0.TOP-REAL 2; hence thesis; end; end; theorem Th85: for sn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st sn<1 & (for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`2/|.p.|-sn)/(1- sn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1>=0 & q<>0.TOP-REAL 2) holds f is continuous proof let sn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=sn, b=(1-sn); reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3; assume that A1: sn<1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2 )|K1 holds f.p=|.p.|* ((p`2/|.p.|-sn)/(1-sn)) and A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2) |K1 holds q`1>=0 & q<>0.TOP-REAL 2; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; b>0 by A1,XREAL_1:149; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 & g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and A6: g3 is continuous by A4,Th5; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A8: dom f=dom g3 by FUNCT_2:def 1; for x being object st x in dom f holds f.x=g3.x proof let x be object; assume A9: x in dom f; then reconsider s=x as Point of (TOP-REAL 2)|K1; x in K1 by A7,A8,A9,PRE_TOPC:8; then reconsider r=x as Point of (TOP-REAL 2); A10: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6; A11: g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5; f.r=(|.r.|)* ((r`2/|.r.|-sn)/(1-sn)) by A2,A9; hence thesis by A5,A11,A10; end; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th86: for sn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st -1 =0 & q<>0.TOP-REAL 2) holds f is continuous proof let sn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=sn, b=(1+sn); reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3; assume that A1: -1 =0 & q<>0.TOP-REAL 2; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; 1+sn>0 by A1,XREAL_1:148; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 & g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and A6: g3 is continuous by A4,Th5; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; A8: for x being object st x in dom f holds f.x=g3.x proof let x be object; assume A9: x in dom f; then reconsider s=x as Point of (TOP-REAL 2)|K1; x in dom g3 by A7,A9; then x in K1 by A7,PRE_TOPC:8; then reconsider r=x as Point of (TOP-REAL 2); A10: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6; A11: g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5; f.r=(|.r.|)* ((r`2/|.r.|-sn)/(1+sn)) by A2,A9; hence thesis by A5,A11,A10; end; dom f=dom g3 by A7,FUNCT_2:def 1; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th87: for sn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st sn<1 & (for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*(sqrt(1-((p`2/|.p.|- sn)/(1-sn))^2))) & (for q being Point of TOP-REAL 2 st q in the carrier of ( TOP-REAL 2)|K1 holds q`1>=0 & q`2/|.q.|>=sn & q<>0.TOP-REAL 2) holds f is continuous proof let sn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=sn, b=(1-sn); reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3; assume that A1: sn<1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2 )|K1 holds f.p=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) and A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2) |K1 holds q`1>=0 & q`2/|.q.|>=sn & q<>0.TOP-REAL 2; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; b>0 by A1,XREAL_1:149; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q =r1 & g1.q=r2 holds g3.q=r2*(sqrt(|.1-((r1/r2-a)/b)^2.|)) and A6: g3 is continuous by A4,Th10; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A8: dom f=dom g3 by FUNCT_2:def 1; for x being object st x in dom f holds f.x=g3.x proof let x be object; A9: 1-sn>0 by A1,XREAL_1:149; assume A10: x in dom f; then x in K1 by A7,A8,PRE_TOPC:8; then reconsider r=x as Point of (TOP-REAL 2); A11: |.r.|<>0 by A3,A10,TOPRNS_1:24; |.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1; then A12: ((r`2) -(|.r.|))*((r`2)+|.r.|) =-(r`1)^2; (r`1)^2>=0 by XREAL_1:63; then r`2<= |.r.| by A12,XREAL_1:93; then r`2/|.r.| <= |.r.|/|.r.| by XREAL_1:72; then r`2/|.r.|<=1 by A11,XCMPLX_1:60; then A13: r`2/|.r.|-sn<=(1-sn) by XREAL_1:9; reconsider s=x as Point of (TOP-REAL 2)|K1 by A10; A14: now assume (1-sn)^2=0; then 1-sn+sn=0+sn by XCMPLX_1:6; hence contradiction by A1; end; sn-r`2/|.r.|<=0 by A3,A10,XREAL_1:47; then -(sn- r`2/|.r.|)>=-(1-sn) by A9,XREAL_1:24; then (1-sn)^2>=0 & (r`2/|.r.|-sn)^2<=(1-sn)^2 by A13,SQUARE_1:49,XREAL_1:63 ; then (r`2/|.r.|-sn)^2/(1-sn)^2<=(1-sn)^2/(1-sn)^2 by XREAL_1:72; then (r`2/|.r.|-sn)^2/(1-sn)^2<=1 by A14,XCMPLX_1:60; then ((r`2/|.r.|-sn)/(1-sn))^2<=1 by XCMPLX_1:76; then 1-((r`2/|.r.|-sn)/(1-sn))^2>=0 by XREAL_1:48; then |.1-((r`2/|.r.|-sn)/(1-sn))^2.| =1-((r`2/|.r.|-sn)/(1-sn))^2 by ABSVALUE:def 1; then A15: f.r=(|.r.|)*(sqrt(|.1-((r`2/|.r.|-sn)/(1-sn))^2.|)) by A2,A10; A16: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6; g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5; hence thesis by A5,A15,A16; end; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th88: for sn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st -1 =0 & q`2/|.q.|<=sn & q<>0.TOP-REAL 2) holds f is continuous proof let sn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=sn, b=(1+sn); reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3; assume that A1: -1 =0 & q`2/|.q.|<=sn & q<>0.TOP-REAL 2; A4: 1+sn>0 by A1,XREAL_1:148; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q =r1 & g1.q=r2 holds g3.q=r2*(sqrt(|.1-((r1/r2-a)/b)^2.|)) and A6: g3 is continuous by A4,Th10; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A8: dom f=dom g3 by FUNCT_2:def 1; for x being object st x in dom f holds f.x=g3.x proof let x be object; assume A9: x in dom f; then x in K1 by A7,A8,PRE_TOPC:8; then reconsider r=x as Point of (TOP-REAL 2); reconsider s=x as Point of (TOP-REAL 2)|K1 by A9; A10: (1+sn)^2>0 by A4,SQUARE_1:12; A11: |.r.|<>0 by A3,A9,TOPRNS_1:24; |.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1; then A12: ((r`2) -(|.r.|))*((r`2)+|.r.|) =-(r`1)^2; (r`1)^2>=0 by XREAL_1:63; then -(|.r.|)<=r`2 by A12,XREAL_1:93; then r`2/|.r.| >= (-(|.r.|))/|.r.| by XREAL_1:72; then r`2/|.r.|>= -1 by A11,XCMPLX_1:197; then r`2/|.r.|-sn>=-1-sn by XREAL_1:9; then A13: r`2/|.r.|-sn>=-(1+sn); sn-r`2/|.r.|>=0 by A3,A9,XREAL_1:48; then -(sn-r`2/|.r.|)<=-0; then (r`2/|.r.|-sn)^2<=(1+sn)^2 by A4,A13,SQUARE_1:49; then (r`2/|.r.|-sn)^2/(1+sn)^2<=(1+sn)^2/(1+sn)^2 by A4,XREAL_1:72; then (r`2/|.r.|-sn)^2/(1+sn)^2<=1 by A10,XCMPLX_1:60; then ((r`2/|.r.|-sn)/(1+sn))^2<=1 by XCMPLX_1:76; then 1-((r`2/|.r.|-sn)/(1+sn))^2>=0 by XREAL_1:48; then |.1-((r`2/|.r.|-sn)/(1+sn))^2.| =1-((r`2/|.r.|-sn)/(1+sn))^2 by ABSVALUE:def 1; then A14: f.r=(|.r.|)*(sqrt(|.1-((r`2/|.r.|-sn)/(1+sn))^2.|)) by A2,A9; A15: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6; g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5; hence thesis by A5,A14,A15; end; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th89: for sn being Real, K0,B0 being Subset of TOP-REAL 2,f being Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1 =0 & q<>0.TOP-REAL 2} & K0={p: p `2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2} holds f is continuous proof let sn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set cn=sqrt(1-sn^2); set p0=|[cn,sn]|; A1: p0`1=cn by EUCLID:52; p0`2=sn by EUCLID:52; then A2: |.p0.|=sqrt((cn)^2+sn^2) by A1,JGRAPH_3:1; assume A3: -1 =0 & q<>0.TOP-REAL 2} & K0={p: p`2/|.p.|>=sn & p`1>=0 & p<>0. TOP-REAL 2}; then sn^2<1^2 by SQUARE_1:50; then A4: 1-sn^2>0 by XREAL_1:50; then A5: --cn>0 by SQUARE_1:25; cn^2=1-sn^2 by A4,SQUARE_1:def 2; then p0`2/|.p0.|=sn by A2,EUCLID:52,SQUARE_1:18; then A6: p0 in K0 by A3,A1,A5,JGRAPH_2:3; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; A7: rng (proj1*((sn-FanMorphE)|K1)) c= the carrier of R^1 by TOPMETR:17; A8: K0 c= B0 proof let x be object; assume x in K0; then ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2/|.p8.|>= sn & p8`1>=0 & p8<>0.TOP-REAL 2 by A3; hence thesis by A3; end; A9: dom ((sn-FanMorphE)|K1) c= dom (proj2*((sn-FanMorphE)|K1)) proof let x be object; assume A10: x in dom ((sn-FanMorphE)|K1); then x in dom (sn-FanMorphE) /\ K1 by RELAT_1:61; then x in dom (sn-FanMorphE) by XBOOLE_0:def 4; then A11: dom proj2 = (the carrier of TOP-REAL 2) & (sn-FanMorphE).x in rng (sn -FanMorphE) by FUNCT_1:3,FUNCT_2:def 1; ((sn-FanMorphE)|K1).x=(sn-FanMorphE).x by A10,FUNCT_1:47; hence thesis by A10,A11,FUNCT_1:11; end; A12: rng (proj2*((sn-FanMorphE)|K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj2*((sn-FanMorphE)|K1)) c= dom ((sn-FanMorphE)|K1) by RELAT_1:25; then dom (proj2*((sn-FanMorphE)|K1)) =dom ((sn-FanMorphE)|K1) by A9, XBOOLE_0:def 10 .=dom (sn-FanMorphE) /\ K1 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g2=proj2*((sn-FanMorphE)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A12,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g2.p=|.p.|* ((p`2/|.p.|-sn)/(1-sn)) proof let p be Point of TOP-REAL 2; A13: dom ((sn-FanMorphE)|K1)=dom (sn-FanMorphE) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A14: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A15: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|>= sn & p3`1>=0 & p3<>0.TOP-REAL 2 by A3,A14; then A16: (sn-FanMorphE).p =|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2/|.p.|-sn)/(1-sn))]| by A3,Th84; ((sn-FanMorphE)|K1).p=(sn-FanMorphE).p by A15,A14,FUNCT_1:49; then g2.p=proj2.(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2 /|.p.|-sn)/(1-sn))]|) by A15,A13,A14,A16,FUNCT_1:13 .=(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2/|.p.|-sn) /(1-sn))]|)`2 by PSCOMP_1:def 6 .=|.p.|* ((p`2/|.p.|-sn)/(1-sn)) by EUCLID:52; hence thesis; end; then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that A17: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f2.p=|.p.|* ((p`2/|.p.|-sn)/(1-sn)); A18: dom ((sn-FanMorphE)|K1) c= dom (proj1*((sn-FanMorphE)|K1)) proof let x be object; assume A19: x in dom ((sn-FanMorphE)|K1); then x in dom (sn-FanMorphE) /\ K1 by RELAT_1:61; then x in dom (sn-FanMorphE) by XBOOLE_0:def 4; then A20: dom proj1 = (the carrier of TOP-REAL 2) & (sn-FanMorphE).x in rng (sn -FanMorphE) by FUNCT_1:3,FUNCT_2:def 1; ((sn-FanMorphE)|K1).x=(sn-FanMorphE).x by A19,FUNCT_1:47; hence thesis by A19,A20,FUNCT_1:11; end; dom (proj1*((sn-FanMorphE)|K1)) c= dom ((sn-FanMorphE)|K1) by RELAT_1:25; then dom (proj1*((sn-FanMorphE)|K1)) =dom ((sn-FanMorphE)|K1) by A18, XBOOLE_0:def 10 .=dom (sn-FanMorphE) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g1=proj1*((sn-FanMorphE)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A7,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g1.p=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) proof let p be Point of TOP-REAL 2; A21: dom ((sn-FanMorphE)|K1)=dom (sn-FanMorphE) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A22: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A23: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|>= sn & p3`1>=0 & p3<>0.TOP-REAL 2 by A3,A22; then A24: (sn-FanMorphE).p=|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2/|.p.|-sn)/(1-sn))]| by A3,Th84; ((sn-FanMorphE)|K1).p=(sn-FanMorphE).p by A23,A22,FUNCT_1:49; then g1.p=proj1.(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2 /|.p.|-sn)/(1-sn))]|) by A23,A21,A22,A24,FUNCT_1:13 .=(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2/|.p.|-sn) /(1-sn)) ]|)`1 by PSCOMP_1:def 5 .= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) by EUCLID:52; hence thesis; end; then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that A25: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f1.p=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)); for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1>=0 & q`2/|.q.|>=sn & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A26: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|>= sn & p3`1>=0 & p3<>0.TOP-REAL 2 by A3,A26; hence thesis; end; then A27: f1 is continuous by A3,A25,Th87; A28: for x,y,r,s being Real st |[x,y]| in K1 & r=f1.(|[x,y]|) & s=f2. (|[x,y]|) holds f.(|[x,y]|)=|[r,s]| proof let x,y,r,s be Real; assume that A29: |[x,y]| in K1 and A30: r=f1.(|[x,y]|) & s=f2.(|[x,y]|); set p99=|[x,y]|; A31: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`2/|.p3.| >=sn & p3`1>=0 & p3<>0.TOP-REAL 2 by A3,A29; A32: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then A33: f1.p99=|.p99.|*(sqrt(1-((p99`2/|.p99.|-sn)/(1-sn))^2)) by A25,A29; ((sn-FanMorphE)|K0).(|[x,y]|)=((sn-FanMorphE)).(|[x,y]|) by A29,FUNCT_1:49 .= |[|.p99.|*(sqrt(1-((p99`2/|.p99.|-sn)/(1-sn))^2)), |.p99.|* ((p99`2 /|.p99.|-sn)/(1-sn))]| by A3,A31,Th84 .=|[r,s]| by A17,A29,A30,A32,A33; hence thesis by A3; end; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1>=0 & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A34: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|>= sn & p3`1>=0 & p3<>0.TOP-REAL 2 by A3,A34; hence thesis; end; then f2 is continuous by A3,A17,Th85; hence thesis by A6,A8,A27,A28,JGRAPH_2:35; end; theorem Th90: for sn being Real, K0,B0 being Subset of TOP-REAL 2,f being Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1 =0 & q<>0.TOP-REAL 2} & K0={p: p `2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} holds f is continuous proof let sn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set cn=sqrt(1-sn^2); set p0=|[cn,sn]|; A1: p0`1=cn by EUCLID:52; p0`2=sn by EUCLID:52; then A2: |.p0.|=sqrt((cn)^2+sn^2) by A1,JGRAPH_3:1; assume A3: -1 =0 & q<>0.TOP-REAL 2} & K0={p: p`2/|.p.|<=sn & p`1>=0 & p<>0. TOP-REAL 2}; then sn^2<1^2 by SQUARE_1:50; then A4: 1-sn^2>0 by XREAL_1:50; then A5: --cn>0 by SQUARE_1:25; cn^2=1-sn^2 by A4,SQUARE_1:def 2; then p0`2/|.p0.|=sn by A2,EUCLID:52,SQUARE_1:18; then A6: p0 in K0 by A3,A1,A5,JGRAPH_2:3; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; A7: rng (proj1*((sn-FanMorphE)|K1)) c= the carrier of R^1 by TOPMETR:17; A8: K0 c= B0 proof let x be object; assume x in K0; then ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2/|.p8.|<= sn & p8`1>=0 & p8<>0.TOP-REAL 2 by A3; hence thesis by A3; end; A9: dom ((sn-FanMorphE)|K1) c= dom (proj2*((sn-FanMorphE)|K1)) proof let x be object; assume A10: x in dom ((sn-FanMorphE)|K1); then x in dom (sn-FanMorphE) /\ K1 by RELAT_1:61; then x in dom (sn-FanMorphE) by XBOOLE_0:def 4; then A11: dom proj2 = (the carrier of TOP-REAL 2) & (sn-FanMorphE).x in rng (sn -FanMorphE) by FUNCT_1:3,FUNCT_2:def 1; ((sn-FanMorphE)|K1).x=(sn-FanMorphE).x by A10,FUNCT_1:47; hence thesis by A10,A11,FUNCT_1:11; end; A12: rng (proj2*((sn-FanMorphE)|K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj2*((sn-FanMorphE)|K1)) c= dom ((sn-FanMorphE)|K1) by RELAT_1:25; then dom (proj2*((sn-FanMorphE)|K1)) =dom ((sn-FanMorphE)|K1) by A9, XBOOLE_0:def 10 .=dom (sn-FanMorphE) /\ K1 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g2=proj2*((sn-FanMorphE)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A12,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g2.p=|.p.|* ((p`2/|.p.|-sn)/(1+sn)) proof let p be Point of TOP-REAL 2; A13: dom ((sn-FanMorphE)|K1)=dom (sn-FanMorphE) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A14: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A15: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|<= sn & p3`1>=0 & p3<>0.TOP-REAL 2 by A3,A14; then A16: (sn-FanMorphE).p =|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A3,Th84; ((sn-FanMorphE)|K1).p=(sn-FanMorphE).p by A15,A14,FUNCT_1:49; then g2.p=proj2.(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2 /|.p.|-sn)/(1+sn))]|) by A15,A13,A14,A16,FUNCT_1:13 .=(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2/|.p.|-sn) /(1+sn))]|)`2 by PSCOMP_1:def 6 .=|.p.|* ((p`2/|.p.|-sn)/(1+sn)) by EUCLID:52; hence thesis; end; then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that A17: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f2.p=|.p.|* ((p`2/|.p.|-sn)/(1+sn)); A18: dom ((sn-FanMorphE)|K1) c= dom (proj1*((sn-FanMorphE)|K1)) proof let x be object; assume A19: x in dom ((sn-FanMorphE)|K1); then x in dom (sn-FanMorphE) /\ K1 by RELAT_1:61; then x in dom (sn-FanMorphE) by XBOOLE_0:def 4; then A20: dom proj1 = (the carrier of TOP-REAL 2) & (sn-FanMorphE).x in rng (sn -FanMorphE) by FUNCT_1:3,FUNCT_2:def 1; ((sn-FanMorphE)|K1).x=(sn-FanMorphE).x by A19,FUNCT_1:47; hence thesis by A19,A20,FUNCT_1:11; end; dom (proj1*((sn-FanMorphE)|K1)) c= dom ((sn-FanMorphE)|K1) by RELAT_1:25; then dom (proj1*((sn-FanMorphE)|K1)) =dom ((sn-FanMorphE)|K1) by A18, XBOOLE_0:def 10 .=dom (sn-FanMorphE) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g1=proj1*((sn-FanMorphE)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A7,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g1.p=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) proof let p be Point of TOP-REAL 2; A21: dom ((sn-FanMorphE)|K1)=dom (sn-FanMorphE) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A22: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A23: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|<= sn & p3`1>=0 & p3<>0.TOP-REAL 2 by A3,A22; then A24: (sn-FanMorphE).p=|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A3,Th84; ((sn-FanMorphE)|K1).p=(sn-FanMorphE).p by A23,A22,FUNCT_1:49; then g1.p=proj1.(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2 /|.p.|-sn)/(1+sn))]|) by A23,A21,A22,A24,FUNCT_1:13 .=(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2/|.p.|-sn) /(1+sn)) ]|)`1 by PSCOMP_1:def 5 .= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) by EUCLID:52; hence thesis; end; then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that A25: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f1.p=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)); for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1>=0 & q`2/|.q.|<=sn & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A26: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|<= sn & p3`1>=0 & p3<>0.TOP-REAL 2 by A3,A26; hence thesis; end; then A27: f1 is continuous by A3,A25,Th88; A28: for x,y,r,s being Real st |[x,y]| in K1 & r=f1.(|[x,y]|) & s=f2. (|[x,y]|) holds f.(|[x,y]|)=|[r,s]| proof let x,y,r,s be Real; assume that A29: |[x,y]| in K1 and A30: r=f1.(|[x,y]|) & s=f2.(|[x,y]|); set p99=|[x,y]|; A31: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`2/|.p3.| <=sn & p3`1>=0 & p3<>0.TOP-REAL 2 by A3,A29; A32: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then A33: f1.p99=|.p99.|*(sqrt(1-((p99`2/|.p99.|-sn)/(1+sn))^2)) by A25,A29; ((sn-FanMorphE)|K0).(|[x,y]|)=((sn-FanMorphE)).(|[x,y]|) by A29,FUNCT_1:49 .= |[|.p99.|*(sqrt(1-((p99`2/|.p99.|-sn)/(1+sn))^2)), |.p99.|* ((p99`2 /|.p99.|-sn)/(1+sn))]| by A3,A31,Th84 .=|[r,s]| by A17,A29,A30,A32,A33; hence thesis by A3; end; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1>=0 & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A34: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|<= sn & p3`1>=0 & p3<>0.TOP-REAL 2 by A3,A34; hence thesis; end; then f2 is continuous by A3,A17,Th86; hence thesis by A6,A8,A27,A28,JGRAPH_2:35; end; theorem Th91: for sn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p`2 >=(sn)*(|.p.|) & p`1>=0} holds K03 is closed proof defpred Q[Point of TOP-REAL 2] means $1`1>=0; let sn be Real, K003 be Subset of TOP-REAL 2; assume A1: K003={p: p`2>=(sn)*(|.p.|) & p`1>=0}; reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; defpred P[Point of TOP-REAL 2] means ($1`2>=sn*|.$1.|); reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1 where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10; K1 is closed & KX is closed by Lm7,JORDAN6:4; hence thesis by A1,A2,TOPS_1:8; end; theorem Th92: for sn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p`2 <=(sn)*(|.p.|) & p`1>=0} holds K03 is closed proof defpred Q[Point of TOP-REAL 2] means $1`1>=0; let sn be Real, K003 be Subset of TOP-REAL 2; assume A1: K003={p: p`2<=(sn)*(|.p.|) & p`1>=0}; reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; defpred P[Point of TOP-REAL 2] means ($1`2<=sn*|.$1.|); reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1 where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10; K1 is closed & KX is closed by Lm9,JORDAN6:4; hence thesis by A1,A2,TOPS_1:8; end; theorem Th93: for sn being Real, K0,B0 being Subset of TOP-REAL 2,f being Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1 =0 & p<>0.TOP-REAL 2} holds f is continuous proof let sn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set cn=sqrt(1-sn^2); set p0=|[cn,sn]|; A1: p0`1=cn by EUCLID:52; defpred P[Point of TOP-REAL 2] means $1`2/|.$1.|>=sn & $1`1>=0 & $1<>0. TOP-REAL 2; p0`2=sn by EUCLID:52; then A2: |.p0.|=sqrt((cn)^2+sn^2) by A1,JGRAPH_3:1; assume A3: -1 =0 & p<>0.TOP-REAL 2}; then sn^2<1^2 by SQUARE_1:50; then A4: 1-sn^2>0 by XREAL_1:50; then A5: p0`1>0 by A1,SQUARE_1:25; then p0 in K0 by A3,JGRAPH_2:3; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; cn^2=1-sn^2 by A4,SQUARE_1:def 2; then A6: p0`2/|.p0.|=sn by A2,EUCLID:52,SQUARE_1:18; then A7: p0 in {p: p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2} by A5,JGRAPH_2:3; {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; then reconsider K001={p: p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL 2) by A7; A8: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; defpred P[Point of TOP-REAL 2] means $1`2>=(sn)*(|.$1.|) & $1`1>=0; {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; then reconsider K003={p: p`2>=(sn)*(|.p.|) & p`1>=0} as Subset of (TOP-REAL 2); defpred P[Point of TOP-REAL 2] means $1`2/|.$1.|<=sn & $1`1>=0 & $1<>0. TOP-REAL 2; A9: {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; A10: --cn>0 by A4,SQUARE_1:25; then p0 in {p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} by A1,A6,JGRAPH_2:3; then reconsider K111={p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL 2) by A9; A11: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5; A12: rng ((sn-FanMorphE)|K001) c= K1 proof let y be object; assume y in rng ((sn-FanMorphE)|K001); then consider x being object such that A13: x in dom ((sn-FanMorphE)|K001) and A14: y=((sn-FanMorphE)|K001).x by FUNCT_1:def 3; x in dom (sn-FanMorphE) by A13,RELAT_1:57; then reconsider q=x as Point of TOP-REAL 2; A15: y=(sn-FanMorphE).q by A13,A14,FUNCT_1:47; dom ((sn-FanMorphE)|K001)=(dom (sn-FanMorphE))/\ K001 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K001 by FUNCT_2:def 1 .=K001 by XBOOLE_1:28; then A16: ex p2 being Point of TOP-REAL 2 st p2=q & p2`2/|.p2.|>= sn & p2`1>=0 & p2<>0.TOP-REAL 2 by A13; then A17: (q`2/|.q.|-sn)>= 0 by XREAL_1:48; |.q.|<>0 by A16,TOPRNS_1:24; then A18: (|.q.|)^2>0^2 by SQUARE_1:12; set q4= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|- sn)/(1-sn))]|; A19: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52; A20: 1-sn>0 by A3,XREAL_1:149; 0<=(q`1)^2 by XREAL_1:63; then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`2)^2 <= (|.q.|)^2 by JGRAPH_3:1; then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`2)^2/(|.q.|)^2 <= 1 by A18,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`2/|.q.| by SQUARE_1:51; then 1-sn>=q`2/|.q.|-sn by XREAL_1:9; then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24; then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A20,XREAL_1:72; then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A20,XCMPLX_1:197; then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A20,A17,SQUARE_1:49; then A21: 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48; then A22: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187; sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>=0 by A21,SQUARE_1:def 2; then sqrt(1-(-(q`2/|.q.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(q`2/|.q.|-sn)^2/(1-sn)^2)>=0; then A23: sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76; A24: q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52; then A25: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2 .= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A22,SQUARE_1:def 2; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A19,A25; then A26: q4<>0.TOP-REAL 2 by A18,TOPRNS_1:23; sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(1-sn))]| by A3,A16,Th84; hence thesis by A3,A15,A24,A23,A26; end; A27: {p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} c= K1 proof let x be object; assume x in {p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2}; then ex p st p=x & p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2; hence thesis by A3; end; p0<>0.TOP-REAL 2 by A1,A4,JGRAPH_2:3,SQUARE_1:25; then not p0 in {0.TOP-REAL 2} by TARSKI:def 1; then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A3,XBOOLE_0:def 5; K1 c= D proof let x be object; assume A28: x in K1; then ex p6 being Point of TOP-REAL 2 st p6=x & p6`1>=0 & p6 <>0.TOP-REAL 2 by A3; then not x in {0.TOP-REAL 2} by TARSKI:def 1; hence thesis by A3,A28,XBOOLE_0:def 5; end; then D=K1 \/ D by XBOOLE_1:12; then A29: (TOP-REAL 2)|K1 is SubSpace of (TOP-REAL 2)|D by TOPMETR:4; A30: {p: p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2} c= K1 proof let x be object; assume x in {p: p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2}; then ex p st p=x & p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2; hence thesis by A3; end; then reconsider K00={p: p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2} as non empty Subset of ((TOP-REAL 2)|K1) by A7,PRE_TOPC:8; A31: K003 is closed by Th91; A32: rng ((sn-FanMorphE)|K111) c= K1 proof let y be object; assume y in rng ((sn-FanMorphE)|K111); then consider x being object such that A33: x in dom ((sn-FanMorphE)|K111) and A34: y=((sn-FanMorphE)|K111).x by FUNCT_1:def 3; x in dom (sn-FanMorphE) by A33,RELAT_1:57; then reconsider q=x as Point of TOP-REAL 2; A35: y=(sn-FanMorphE).q by A33,A34,FUNCT_1:47; dom ((sn-FanMorphE)|K111)=(dom (sn-FanMorphE))/\ K111 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K111 by FUNCT_2:def 1 .=K111 by XBOOLE_1:28; then A36: ex p2 being Point of TOP-REAL 2 st p2=q & p2`2/|.p2.|<= sn & p2`1>=0 & p2<>0.TOP-REAL 2 by A33; then A37: (q`2/|.q.|-sn)<=0 by XREAL_1:47; |.q.|<>0 by A36,TOPRNS_1:24; then A38: (|.q.|)^2>0^2 by SQUARE_1:12; set q4= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|- sn)/(1+sn))]|; A39: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52; A40: 1+sn>0 by A3,XREAL_1:148; 0<=(q`1)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1 ,XREAL_1:7; then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`2)^2/(|.q.|)^2 <= 1 by A38,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`2/|.q.| by SQUARE_1:51; then -1-sn<=q`2/|.q.|-sn by XREAL_1:9; then (-(1+sn))/(1+sn)<=(( q`2/|.q.|-sn))/(1+sn) by A40,XREAL_1:72; then -1<=(( q`2/|.q.|-sn))/(1+sn) by A40,XCMPLX_1:197; then A41: ( (q`2/|.q.|-sn) /(1+sn))^2<=1^2 by A40,A37,SQUARE_1:49; then A42: 1-((q`2/|.q.|-sn)/(1+sn))^2>=0 by XREAL_1:48; 1-(-((q`2/|.q.|-sn)/(1+sn)))^2>=0 by A41,XREAL_1:48; then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187; then sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(q`2/|.q.|-sn)^2/(1+sn)^2)>=0; then A43: sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76; A44: q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52; then A45: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2 .= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A42,SQUARE_1:def 2; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A39,A45; then A46: q4<>0.TOP-REAL 2 by A38,TOPRNS_1:23; sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(1+sn))]| by A3,A36,Th84; hence thesis by A3,A35,A44,A43,A46; end; the carrier of (TOP-REAL 2)|D=D by PRE_TOPC:8; then A47: rng (f|K00) c=D; the carrier of (TOP-REAL 2)|B0=the carrier of (TOP-REAL 2)|D; then A48: dom f=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1 .=K1 by PRE_TOPC:8; then dom (f|K00)=K00 by A30,RELAT_1:62 .= the carrier of ((TOP-REAL 2)|K1)|K00 by PRE_TOPC:8; then reconsider f1=f|K00 as Function of ((TOP-REAL 2)|K1)|K00,(TOP-REAL 2)|D by A47,FUNCT_2:2 ; A49: the carrier of ((TOP-REAL 2)|K1)=K0 by PRE_TOPC:8; p0 in {p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} by A1,A10,A6,JGRAPH_2:3; then reconsider K11={p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} as non empty Subset of ((TOP-REAL 2)|K1) by A27,PRE_TOPC:8; A50: the carrier of (TOP-REAL 2)|K1 =K1 by PRE_TOPC:8; A51: dom (sn-FanMorphE)=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then dom ((sn-FanMorphE)|K001)=K001 by RELAT_1:62 .= the carrier of (TOP-REAL 2)|K001 by PRE_TOPC:8; then reconsider f3=(sn-FanMorphE)|K001 as Function of (TOP-REAL 2)|K001,(TOP-REAL 2)|K1 by A8,A12,FUNCT_2:2; A52: D<>{}; dom ((sn-FanMorphE)|K111)=K111 by A51,RELAT_1:62 .= the carrier of (TOP-REAL 2)|K111 by PRE_TOPC:8; then reconsider f4=(sn-FanMorphE)|K111 as Function of (TOP-REAL 2)|K111,(TOP-REAL 2)|K1 by A50,A32,FUNCT_2:2; the carrier of (TOP-REAL 2)|D =D by PRE_TOPC:8; then A53: rng (f|K11) c=D; dom (f|K11)=K11 by A27,A48,RELAT_1:62 .= the carrier of ((TOP-REAL 2)|K1)|K11 by PRE_TOPC:8; then reconsider f2=f|K11 as Function of ((TOP-REAL 2)|K1)|K11,(TOP-REAL 2)|D by A53,FUNCT_2:2 ; the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then ((TOP-REAL 2)|K1)|K11=(TOP-REAL 2)|K111 & f2= f4 by A3,FUNCT_1:51 ,GOBOARD9:2; then A54: f2 is continuous by A3,A29,Th90,PRE_TOPC:26; the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then ((TOP-REAL 2)|K1)|K00=(TOP-REAL 2)|K001 & f1= f3 by A3,FUNCT_1:51 ,GOBOARD9:2; then A55: f1 is continuous by A3,A29,Th89,PRE_TOPC:26; A56: dom f2=the carrier of ((TOP-REAL 2)|K1)|K11 by FUNCT_2:def 1 .=K11 by PRE_TOPC:8; defpred P[Point of TOP-REAL 2] means $1`2<=(sn)*(|.$1.|) & $1`1>=0; {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; then reconsider K004={p: p`2<=(sn)*(|.p.|) & p`1>=0} as Subset of (TOP-REAL 2); A57: K004 /\ K1 c= K11 proof let x be object; assume A58: x in K004 /\ K1; then x in K004 by XBOOLE_0:def 4; then consider q1 being Point of TOP-REAL 2 such that A59: q1=x and A60: q1`2<=(sn)*(|.q1.|) and q1`1>=0; x in K1 by A58,XBOOLE_0:def 4; then A61: ex q2 being Point of TOP-REAL 2 st q2=x & q2`1>=0 & q2 <>0.TOP-REAL 2 by A3; q1`2/|.q1.|<=(sn)*(|.q1.|)/|.q1.| by A60,XREAL_1:72; then q1`2/|.q1.|<=(sn) by A59,A61,TOPRNS_1:24,XCMPLX_1:89; hence thesis by A59,A61; end; A62: K004 is closed by Th92; K11 c= K004 /\ K1 proof let x be object; assume x in K11; then consider p such that A63: p=x and A64: p`2/|.p.|<=sn and A65: p`1>=0 and A66: p<>0.TOP-REAL 2; p`2/|.p.|*|.p.|<=(sn)*(|.p.|) by A64,XREAL_1:64; then p`2<=(sn)*(|.p.|) by A66,TOPRNS_1:24,XCMPLX_1:87; then A67: x in K004 by A63,A65; x in K1 by A3,A63,A65,A66; hence thesis by A67,XBOOLE_0:def 4; end; then K11=K004 /\ [#]((TOP-REAL 2)|K1) by A11,A57,XBOOLE_0:def 10; then A68: K11 is closed by A62,PRE_TOPC:13; A69: K003 /\ K1 c= K00 proof let x be object; assume A70: x in K003 /\ K1; then x in K003 by XBOOLE_0:def 4; then consider q1 being Point of TOP-REAL 2 such that A71: q1=x and A72: q1`2>=(sn)*(|.q1.|) and q1`1>=0; x in K1 by A70,XBOOLE_0:def 4; then A73: ex q2 being Point of TOP-REAL 2 st q2=x & q2`1>=0 & q2 <>0.TOP-REAL 2 by A3; q1`2/|.q1.|>=(sn)*(|.q1.|)/|.q1.| by A72,XREAL_1:72; then q1`2/|.q1.|>=(sn) by A71,A73,TOPRNS_1:24,XCMPLX_1:89; hence thesis by A71,A73; end; K00 c= K003 /\ K1 proof let x be object; assume x in K00; then consider p such that A74: p=x and A75: p`2/|.p.|>=sn and A76: p`1>=0 and A77: p<>0.TOP-REAL 2; p`2/|.p.|*|.p.|>=(sn)*(|.p.|) by A75,XREAL_1:64; then p`2>=(sn)*(|.p.|) by A77,TOPRNS_1:24,XCMPLX_1:87; then A78: x in K003 by A74,A76; x in K1 by A3,A74,A76,A77; hence thesis by A78,XBOOLE_0:def 4; end; then K00=K003 /\ [#]((TOP-REAL 2)|K1) by A11,A69,XBOOLE_0:def 10; then A79: K00 is closed by A31,PRE_TOPC:13; set T1= ((TOP-REAL 2)|K1)|K00,T2=((TOP-REAL 2)|K1)|K11; A80: [#](((TOP-REAL 2)|K1)|K11)=K11 by PRE_TOPC:def 5; A81: [#](((TOP-REAL 2)|K1)|K00)=K00 by PRE_TOPC:def 5; A82: for p being object st p in ([#]T1)/\([#]T2) holds f1.p = f2.p proof let p be object; assume A83: p in ([#]T1)/\([#]T2); then p in K00 by A81,XBOOLE_0:def 4; hence f1.p=f.p by FUNCT_1:49 .=f2.p by A80,A83,FUNCT_1:49; end; A84: K1 c= K00 \/ K11 proof let x be object; assume x in K1; then consider p such that A85: p=x & p`1>=0 & p<>0.TOP-REAL 2 by A3; per cases; suppose p`2/|.p.|>=sn; then x in K00 by A85; hence thesis by XBOOLE_0:def 3; end; suppose p`2/|.p.| 0.TOP-REAL 2} holds f is continuous proof let sn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set cn=sqrt(1-sn^2); set p0=|[-cn,-sn]|; assume A1: -1 0.TOP-REAL 2}; then sn^2<1^2 by SQUARE_1:50; then 1-sn^2>0 by XREAL_1:50; then --cn>0 by SQUARE_1:25; then A2: p0`1=-cn & -cn<0 by EUCLID:52; then p0 in K0 by A1,JGRAPH_2:3; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; not p0 in {0.TOP-REAL 2} by A2,JGRAPH_2:3,TARSKI:def 1; then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A1,XBOOLE_0:def 5; A3: K1 c= D proof let x be object; assume x in K1; then consider p2 being Point of TOP-REAL 2 such that A4: p2=x and p2`1<=0 and A5: p2<>0.TOP-REAL 2 by A1; not p2 in {0.TOP-REAL 2} by A5,TARSKI:def 1; hence thesis by A1,A4,XBOOLE_0:def 5; end; for p being Point of (TOP-REAL 2)|K1,V being Subset of (TOP-REAL 2)|D st f.p in V & V is open holds ex W being Subset of (TOP-REAL 2)|K1 st p in W & W is open & f.:W c= V proof let p be Point of (TOP-REAL 2)|K1,V be Subset of (TOP-REAL 2)|D; assume that A6: f.p in V and A7: V is open; consider V2 being Subset of TOP-REAL 2 such that A8: V2 is open and A9: V2 /\ [#]((TOP-REAL 2)|D)=V by A7,TOPS_2:24; reconsider W2=V2 /\ [#]((TOP-REAL 2)|K1) as Subset of (TOP-REAL 2)| K1; A10: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5; then A11: f.p=(sn-FanMorphE).p by A1,FUNCT_1:49; A12: f.:W2 c= V proof let y be object; assume y in f.:W2; then consider x being object such that A13: x in dom f and A14: x in W2 and A15: y=f.x by FUNCT_1:def 6; f is Function of (TOP-REAL 2)|K1, (TOP-REAL 2)|D; then dom f= K1 by A10,FUNCT_2:def 1; then consider p4 being Point of TOP-REAL 2 such that A16: x=p4 and A17: p4`1<=0 and p4<>0.TOP-REAL 2 by A1,A13; A18: p4 in V2 by A14,A16,XBOOLE_0:def 4; p4 in [#]((TOP-REAL 2)|K1) by A13,A16; then p4 in D by A3,A10; then A19: p4 in [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5; f.p4=(sn-FanMorphE).p4 by A1,A10,A13,A16,FUNCT_1:49 .=p4 by A17,Th82; hence thesis by A9,A15,A16,A18,A19,XBOOLE_0:def 4; end; p in the carrier of (TOP-REAL 2)|K1; then consider q being Point of TOP-REAL 2 such that A20: q=p and A21: q`1<=0 and q <>0.TOP-REAL 2 by A1,A10; (sn-FanMorphE).q=q by A21,Th82; then p in V2 by A6,A9,A11,A20,XBOOLE_0:def 4; then A22: p in W2 by XBOOLE_0:def 4; W2 is open by A8,TOPS_2:24; hence thesis by A22,A12; end; hence thesis by JGRAPH_2:10; end; theorem Th95: for sn being Real, B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2)|B0,f being Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2 )|B0) st -1 =0 & p<>0.TOP-REAL 2} holds f is continuous proof let sn be Real, B0 be Subset of TOP-REAL 2, K0 be Subset of (TOP-REAL 2)|B0,f be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0); the carrier of (TOP-REAL 2)|B0= B0 by PRE_TOPC:8; then reconsider K1=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1; assume A1: -1 =0 & p<>0.TOP-REAL 2 }; K0 c= B0 proof let x be object; assume x in K0; then A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1>=0 & p8 <>0.TOP-REAL 2 by A1; then not x in {0.TOP-REAL 2} by TARSKI:def 1; hence thesis by A1,A2,XBOOLE_0:def 5; end; then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7; hence thesis by A1,Th93; end; theorem Th96: for sn being Real, B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2)|B0,f being Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2 )|B0) st -1 0.TOP-REAL 2} holds f is continuous proof let sn be Real, B0 be Subset of TOP-REAL 2, K0 be Subset of (TOP-REAL 2)|B0,f be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0); the carrier of (TOP-REAL 2)|B0= B0 by PRE_TOPC:8; then reconsider K1=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1; assume A1: -1 0.TOP-REAL 2}; K0 c= B0 proof let x be object; assume x in K0; then A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1<=0 & p8 <>0.TOP-REAL 2 by A1; then not x in {0.TOP-REAL 2} by TARSKI:def 1; hence thesis by A1,A2,XBOOLE_0:def 5; end; then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7; hence thesis by A1,Th94; end; theorem Th97: for sn being Real,p being Point of TOP-REAL 2 holds |.(sn -FanMorphE).p.|=|.p.| proof let sn be Real,p be Point of TOP-REAL 2; set f=sn-FanMorphE; set z=f.p; reconsider q=p as Point of TOP-REAL 2; reconsider qz=z as Point of TOP-REAL 2; per cases; suppose A1: q`2/|.q.|>=sn & q`1>0; then A2: (sn-FanMorphE).q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(1-sn))]| by Th82; then A3: qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52; A4: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by A2,EUCLID:52; A5: (q`2/|.q.|-sn)>=0 by A1,XREAL_1:48; A6: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; |.q.|<>0 by A1,JGRAPH_2:3,TOPRNS_1:24; then A7: (|.q.|)^2>0 by SQUARE_1:12; 0<=(q`1)^2 by XREAL_1:63; then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A6,XREAL_1:72; then (q`2)^2/(|.q.|)^2 <= 1 by A7,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`2/|.q.| by SQUARE_1:51; then A8: 1-sn>=q`2/|.q.|-sn by XREAL_1:9; per cases; suppose A9: 1-sn=0; A10: ((q`2/|.q.|-sn)/(1-sn))=(q`2/|.q.|-sn)*(1-sn)" by XCMPLX_0:def 9 .= (q`2/|.q.|-sn)*0 by A9 .=0; then 1-((q`2/|.q.|-sn)/(1-sn))^2=1; then (sn-FanMorphE).q= |[|.q.|*1,|.q.|*0]| by A1,A10,Th82,SQUARE_1:18 .=|[(|.q.|),0]|; then ((sn-FanMorphE).q)`1=(|.q.|) & ((sn-FanMorphE).q)`2=0 by EUCLID:52; then |.(sn-FanMorphE).p.|=sqrt(((|.q.|))^2+0^2) by JGRAPH_3:1 .=|.q.| by SQUARE_1:22; hence thesis; end; suppose A11: 1-sn<>0; per cases by A11; suppose A12: 1-sn>0; -(1-sn)<= -( q`2/|.q.|-sn) by A8,XREAL_1:24; then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A12,XREAL_1:72; then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A12,XCMPLX_1:197; then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A5,A12,SQUARE_1:49; then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48; then A13: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187; A14: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2 by A3 .= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A13,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A4,A14; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; hence thesis by SQUARE_1:22; end; suppose A15: 1-sn<0; 0+(q`2)^2<(q`1)^2+(q`2)^2 by A1,SQUARE_1:12,XREAL_1:8; then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,A6,XREAL_1:74; then (q`2)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60; then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76; then A16: 1 > q`2/|.p.| by SQUARE_1:52; q`2/|.q.|-sn>=0 by A1,XREAL_1:48; hence thesis by A15,A16,XREAL_1:9; end; end; end; suppose A17: q`2/|.q.| 0; then |.q.|<>0 by JGRAPH_2:3,TOPRNS_1:24; then A18: (|.q.|)^2>0 by SQUARE_1:12; A19: (q`2/|.q.|-sn)<0 by A17,XREAL_1:49; A20: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; 0<=(q`1)^2 by XREAL_1:63; then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A20,XREAL_1:72; then (q`2)^2/(|.q.|)^2 <= 1 by A18,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`2/|.q.| by SQUARE_1:51; then A21: -1-sn<=q`2/|.q.|-sn by XREAL_1:9; A22: (sn-FanMorphE).q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.| * ((q`2/|.q.|-sn)/(1+sn))]| by A17,Th83; then A23: qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52; A24: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by A22,EUCLID:52; per cases; suppose A25: 1+sn=0; ((q`2/|.q.|-sn)/(1+sn))=(q`2/|.q.|-sn)*(1+sn)" by XCMPLX_0:def 9 .= (q`2/|.q.|-sn)*0 by A25 .=0; then ((sn-FanMorphE).q)`1=(|.q.|) & ((sn-FanMorphE).q)`2=0 by A22, EUCLID:52,SQUARE_1:18; then |.(sn-FanMorphE).p.|=sqrt(((|.q.|))^2+0^2) by JGRAPH_3:1 .=|.q.| by SQUARE_1:22; hence thesis; end; suppose A26: 1+sn<>0; per cases by A26; suppose A27: 1+sn>0; then (-(1+sn))/(1+sn)<=(( q`2/|.q.|-sn))/(1+sn) by A21,XREAL_1:72; then -1<=(( q`2/|.q.|-sn))/(1+sn) by A27,XCMPLX_1:197; then ( (q`2/|.q.|-sn) /(1+sn))^2<=1^2 by A19,A27,SQUARE_1:49; then A28: 1-(((q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48; A29: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2 by A23 .= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A28,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A24,A29; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; hence thesis by SQUARE_1:22; end; suppose A30: 1+sn<0; 0+(q`2)^2<(q`1)^2+(q`2)^2 by A17,SQUARE_1:12,XREAL_1:8; then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A18,A20,XREAL_1:74; then (q`2)^2/(|.q.|)^2 < 1 by A18,XCMPLX_1:60; then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76; then -1 < q`2/|.p.| by SQUARE_1:52; then A31: q`2/|.q.|-sn>-1-sn by XREAL_1:9; -(1+sn)>-0 by A30,XREAL_1:24; hence thesis by A17,A31,XREAL_1:49; end; end; end; suppose q`1<=0; hence thesis by Th82; end; end; theorem Th98: for sn being Real,x,K0 being set st -1 =0 & p<>0.TOP-REAL 2} holds (sn-FanMorphE).x in K0 proof let sn be Real,x,K0 be set; assume A1: -1 =0 & p<>0.TOP-REAL 2}; then consider p such that A2: p=x and A3: p`1>=0 and A4: p<>0.TOP-REAL 2; A5: now assume |.p.|<=0; then |.p.|=0; hence contradiction by A4,TOPRNS_1:24; end; then A6: (|.p.|)^2>0 by SQUARE_1:12; per cases; suppose A7: p`2/|.p.|<=sn; reconsider p9= (sn-FanMorphE).p as Point of TOP-REAL 2; (sn-FanMorphE).p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.| *((p`2/|.p.|-sn)/(1+sn))]| by A1,A3,A4,A7,Th84; then A8: p9`1=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) by EUCLID:52; A9: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; A10: 1+sn>0 by A1,XREAL_1:148; per cases; suppose p`1=0; hence thesis by A1,A2,Th82; end; suppose p`1<>0; then 0+(p`2)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8; then (p`2)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A9,XREAL_1:74; then (p`2)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60; then ((p`2)/|.p.|)^2 < 1 by XCMPLX_1:76; then -1 < p`2/|.p.| by SQUARE_1:52; then -1-sn< p`2/|.p.|-sn by XREAL_1:9; then (-1)*(1+sn)/(1+sn)< (p`2/|.p.|-sn)/(1+sn) by A10,XREAL_1:74; then A11: -1< (p`2/|.p.|-sn)/(1+sn) by A10,XCMPLX_1:89; p`2/|.p.|-sn<=0 by A7,XREAL_1:47; then 1^2> ((p`2/|.p.|-sn)/(1+sn))^2 by A10,A11,SQUARE_1:50; then 1-((p`2/|.p.|-sn)/(1+sn))^2>0 by XREAL_1:50; then sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)>0 by SQUARE_1:25; then |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))>0 by A5,XREAL_1:129; hence thesis by A1,A2,A8,JGRAPH_2:3; end; end; suppose A12: p`2/|.p.|>sn; reconsider p9= (sn-FanMorphE).p as Point of TOP-REAL 2; (sn-FanMorphE).p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.| *((p`2/|.p.|-sn)/(1-sn))]| by A1,A3,A4,A12,Th84; then A13: p9`1=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) by EUCLID:52; A14: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; A15: 1-sn>0 by A1,XREAL_1:149; per cases; suppose p`1=0; hence thesis by A1,A2,Th82; end; suppose p`1<>0; then 0+(p`2)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8; then (p`2)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A14,XREAL_1:74; then (p`2)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60; then ((p`2)/|.p.|)^2 < 1 by XCMPLX_1:76; then p`2/|.p.|<1 by SQUARE_1:52; then (p`2/|.p.|-sn)<1-sn by XREAL_1:9; then (p`2/|.p.|-sn)/(1-sn)<(1-sn)/(1-sn) by A15,XREAL_1:74; then A16: (p`2/|.p.|-sn)/(1-sn)<1 by A15,XCMPLX_1:60; -(1-sn)< -0 & p`2/|.p.|-sn>=sn-sn by A12,A15,XREAL_1:9,24; then (-1)*(1-sn)/(1-sn)< (p`2/|.p.|-sn)/(1-sn) by A15,XREAL_1:74; then -1< (p`2/|.p.|-sn)/(1-sn) by A15,XCMPLX_1:89; then 1^2> ((p`2/|.p.|-sn)/(1-sn))^2 by A16,SQUARE_1:50; then 1-((p`2/|.p.|-sn)/(1-sn))^2>0 by XREAL_1:50; then sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)>0 by SQUARE_1:25; then p9`1>0 by A5,A13,XREAL_1:129; hence thesis by A1,A2,JGRAPH_2:3; end; end; end; theorem Th99: for sn being Real,x,K0 being set st -1 0.TOP-REAL 2} holds (sn-FanMorphE).x in K0 proof let sn be Real,x,K0 be set; assume A1: -1 0.TOP-REAL 2}; then ex p st p=x & p`1<=0 & p<>0.TOP-REAL 2; hence thesis by A1,Th82; end; theorem Th100: for sn being Real, D being non empty Subset of TOP-REAL 2 st -1 =0 & p<>0.TOP-REAL 2} by JGRAPH_2:3; set Y1=|[0,1]|; defpred P[Point of TOP-REAL 2] means $1`1>=0; reconsider B0= {0.TOP-REAL 2} as Subset of TOP-REAL 2; let sn be Real,D be non empty Subset of TOP-REAL 2; assume that A2: -1 0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from InclSub(A5); then reconsider K0={p:p`1>=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL 2)|D by A1; A6: K0=the carrier of ((TOP-REAL 2)|D)|K0 by PRE_TOPC:8; A7: the carrier of ((TOP-REAL 2)|D)=D by PRE_TOPC:8; A8: rng ((sn-FanMorphE)|K0) c= the carrier of ((TOP-REAL 2)|D)|K0 proof let y be object; assume y in rng ((sn-FanMorphE)|K0); then consider x being object such that A9: x in dom ((sn-FanMorphE)|K0) and A10: y=((sn-FanMorphE)|K0).x by FUNCT_1:def 3; x in (dom (sn-FanMorphE)) /\ K0 by A9,RELAT_1:61; then A11: x in K0 by XBOOLE_0:def 4; K0 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1; then reconsider p=x as Point of TOP-REAL 2 by A11; (sn-FanMorphE).p=y by A10,A11,FUNCT_1:49; then y in K0 by A2,A11,Th98; hence thesis by PRE_TOPC:8; end; A12: K0 c= (the carrier of TOP-REAL 2) proof let z be object; assume z in K0; then ex p8 being Point of TOP-REAL 2 st p8=z & p8`1>=0 & p8 <>0.TOP-REAL 2; hence thesis; end; Y1`1=0 & Y1`2=1 by EUCLID:52; then A13: Y1 in {p where p is Point of TOP-REAL 2: p`1<=0 & p<>0.TOP-REAL 2} by JGRAPH_2:3; A14: the carrier of ((TOP-REAL 2)|D)= (NonZero TOP-REAL 2) by A5,PRE_TOPC:8; defpred P[Point of TOP-REAL 2] means $1`1<=0; {p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from InclSub(A5); then reconsider K1={p: p`1<=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL 2)|D by A13; A15: K0 is closed & K1 is closed by A5,Th29,Th31; dom ((sn-FanMorphE)|K0)= dom ((sn-FanMorphE)) /\ K0 by RELAT_1:61 .=((the carrier of TOP-REAL 2)) /\ K0 by FUNCT_2:def 1 .=K0 by A12,XBOOLE_1:28; then reconsider f=(sn-FanMorphE)|K0 as Function of ((TOP-REAL 2)|D)|K0, (( TOP-REAL 2)|D) by A6,A8,FUNCT_2:2,XBOOLE_1:1; A16: K1=the carrier of ((TOP-REAL 2)|D)|K1 by PRE_TOPC:8; A17: rng ((sn-FanMorphE)|K1) c= the carrier of ((TOP-REAL 2)|D)|K1 proof let y be object; assume y in rng ((sn-FanMorphE)|K1); then consider x being object such that A18: x in dom ((sn-FanMorphE)|K1) and A19: y=((sn-FanMorphE)|K1).x by FUNCT_1:def 3; x in (dom (sn-FanMorphE)) /\ K1 by A18,RELAT_1:61; then A20: x in K1 by XBOOLE_0:def 4; K1 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1; then reconsider p=x as Point of TOP-REAL 2 by A20; (sn-FanMorphE).p=y by A19,A20,FUNCT_1:49; then y in K1 by A2,A20,Th99; hence thesis by PRE_TOPC:8; end; A21: K1 c= (the carrier of TOP-REAL 2) proof let z be object; assume z in K1; then ex p8 being Point of TOP-REAL 2 st p8=z & p8`1<=0 & p8 <>0.TOP-REAL 2; hence thesis; end; dom ((sn-FanMorphE)|K1)= dom ((sn-FanMorphE)) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2)) /\ K1 by FUNCT_2:def 1 .=K1 by A21,XBOOLE_1:28; then reconsider g=(sn-FanMorphE)|K1 as Function of ((TOP-REAL 2)|D)|K1, (( TOP-REAL 2)|D) by A16,A17,FUNCT_2:2,XBOOLE_1:1; A22: K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5; A23: D c= K0 \/ K1 proof let x be object; assume A24: x in D; then reconsider px=x as Point of TOP-REAL 2; not x in {0.TOP-REAL 2} by A5,A24,XBOOLE_0:def 5; then px`1>=0 & px<>0.TOP-REAL 2 or px`1<=0 & px<>0.TOP-REAL 2 by TARSKI:def 1; then x in K0 or x in K1; hence thesis by XBOOLE_0:def 3; end; A25: dom f=K0 by A6,FUNCT_2:def 1; A26: K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5; A27: for x be object st x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1))) holds f.x = g.x proof let x be object; assume A28: x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1)) ); then x in K0 by A26,XBOOLE_0:def 4; then f.x=(sn-FanMorphE).x by FUNCT_1:49; hence thesis by A22,A28,FUNCT_1:49; end; D= [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5; then A29: ([#](((TOP-REAL 2)|D)|K0)) \/ ([#](((TOP-REAL 2)|D)|K1)) = [#](( TOP-REAL 2)|D) by A26,A22,A23,XBOOLE_0:def 10; A30: f is continuous & g is continuous by A2,A5,Th95,Th96; then consider h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that A31: h= f+*g and h is continuous by A26,A22,A29,A15,A27,JGRAPH_2:1; A32: dom h=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1; A33: dom g=K1 by A16,FUNCT_2:def 1; K0=[#](((TOP-REAL 2)|D)|K0) & K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5; then A34: f tolerates g by A27,A25,A33,PARTFUN1:def 4; A35: for x being object st x in dom h holds h.x=((sn-FanMorphE)|D).x proof let x be object; assume A36: x in dom h; then reconsider p=x as Point of TOP-REAL 2 by A14,XBOOLE_0:def 5; not x in {0.TOP-REAL 2} by A14,A36,XBOOLE_0:def 5; then A37: x <>0.TOP-REAL 2 by TARSKI:def 1; A38: x in D`` by A32,A36,PRE_TOPC:8; now per cases; case A39: x in K0; A40: (sn-FanMorphE)|D.p=(sn-FanMorphE).p by A38,FUNCT_1:49 .=f.p by A39,FUNCT_1:49; h.p=(g+*f).p by A31,A34,FUNCT_4:34 .=f.p by A25,A39,FUNCT_4:13; hence thesis by A40; end; case not x in K0; then not p`1>=0 by A37; then A41: x in K1 by A37; (sn-FanMorphE)|D.p=(sn-FanMorphE).p by A38,FUNCT_1:49 .=g.p by A41,FUNCT_1:49; hence thesis by A31,A33,A41,FUNCT_4:13; end; end; hence thesis; end; dom (sn-FanMorphE)=(the carrier of (TOP-REAL 2)) by FUNCT_2:def 1; then dom ((sn-FanMorphE)|D)=(the carrier of (TOP-REAL 2))/\ D by RELAT_1:61 .=the carrier of ((TOP-REAL 2)|D) by A4,XBOOLE_1:28; then f+*g=(sn-FanMorphE)|D by A31,A32,A35,FUNCT_1:2; hence thesis by A26,A22,A29,A30,A15,A27,JGRAPH_2:1; end; theorem Th101: for sn being Real st -1 f.(0.TOP-REAL 2) proof let p be Point of (TOP-REAL 2)|D; A5: [#]((TOP-REAL 2)|D)=D by PRE_TOPC:def 5; then reconsider q=p as Point of TOP-REAL 2 by XBOOLE_0:def 5; not p in {0.TOP-REAL 2} by A5,XBOOLE_0:def 5; then A6: not p=0.TOP-REAL 2 by TARSKI:def 1; now per cases; case A7: q`2/|.q.|>=sn & q`1>=0; set q9= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q .|-sn)/(1-sn))]|; A8: q9`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52; now assume A9: q9=0.TOP-REAL 2; A10: |.q.|<>0 by A6,TOPRNS_1:24; then sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)=sqrt(1-0^2) by A8,A9, JGRAPH_2:3,XCMPLX_1:6 .=1 by SQUARE_1:18; hence contradiction by A9,A10,EUCLID:52,JGRAPH_2:3; end; hence thesis by A1,A2,A3,A6,A7,Th84; end; case A11: q`2/|.q.| =0; set q9=|[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q .|-sn)/(1+sn))]|; A12: q9`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52; now assume A13: q9=0.TOP-REAL 2; A14: |.q.|<>0 by A6,TOPRNS_1:24; then sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)=sqrt(1-0^2) by A12,A13, JGRAPH_2:3,XCMPLX_1:6 .=1 by SQUARE_1:18; hence contradiction by A13,A14,EUCLID:52,JGRAPH_2:3; end; hence thesis by A1,A2,A3,A6,A11,Th84; end; case q`1<0; then f.p=p by Th82; hence thesis by A6,Th82,JGRAPH_2:3; end; end; hence thesis; end; A15: for V being Subset of (TOP-REAL 2) st f.(0.TOP-REAL 2) in V & V is open ex W being Subset of (TOP-REAL 2) st 0.TOP-REAL 2 in W & W is open & f.:W c= V proof reconsider u0=0.TOP-REAL 2 as Point of Euclid 2 by EUCLID:67; let V be Subset of (TOP-REAL 2); reconsider VV = V as Subset of TopSpaceMetr Euclid 2 by Lm11; assume that A16: f.(0.TOP-REAL 2) in V and A17: V is open; VV is open by A17,Lm11,PRE_TOPC:30; then consider r being Real such that A18: r>0 and A19: Ball(u0,r) c= V by A3,A16,TOPMETR:15; reconsider r as Real; the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8; then reconsider W1=Ball(u0,r) as Subset of TOP-REAL 2; A20: W1 is open by GOBOARD6:3; A21: f.:W1 c= W1 proof let z be object; assume z in f.:W1; then consider y being object such that A22: y in dom f and A23: y in W1 and A24: z=f.y by FUNCT_1:def 6; z in rng f by A22,A24,FUNCT_1:def 3; then reconsider qz=z as Point of TOP-REAL 2; reconsider q=y as Point of TOP-REAL 2 by A22; reconsider qy=q as Point of Euclid 2 by EUCLID:67; reconsider pz=qz as Point of Euclid 2 by EUCLID:67; dist(u0,qy) 0.TOP-REAL 2 & q`2/|.q.|>=sn & q`1>=0; then A27: (q`2/|.q.|-sn)>= 0 by XREAL_1:48; 0<=(q`1)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then A28: (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; A29: 1-sn>0 by A2,XREAL_1:149; |.q.|<>0 by A26,TOPRNS_1:24; then (|.q.|)^2>0 by SQUARE_1:12; then (q`2)^2/(|.q.|)^2 <= 1 by A28,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`2/|.q.| by SQUARE_1:51; then 1-sn>=q`2/|.q.|-sn by XREAL_1:9; then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24; then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A29,XREAL_1:72; then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A29,XCMPLX_1:197; then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A29,A27,SQUARE_1:49; then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48; then A30: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187; A31: (sn-FanMorphE).q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1- sn))^2)) , |.q.|* ((q`2/|.q.|-sn)/(1-sn))]| by A1,A2,A26,Th84; then A32: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by A24,EUCLID:52; qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by A24,A31,EUCLID:52; then A33: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2 .= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A30,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A32,A33; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; then A34: |.qz.|=|.q.| by SQUARE_1:22; |.- q.| 0.TOP-REAL 2 & q`2/|.q.| =0; 0<=(q`1)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then A36: (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; A37: 1+sn>0 by A1,XREAL_1:148; |.q.|<>0 by A35,TOPRNS_1:24; then (|.q.|)^2>0 by SQUARE_1:12; then (q`2)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`2/|.q.| by SQUARE_1:51; then --1>=-q`2/|.q.| by XREAL_1:24; then 1+sn>=-q`2/|.q.|+sn by XREAL_1:7; then A38: (-(q`2/|.q.|-sn))/(1+sn)<=1 by A37,XREAL_1:185; (sn-q`2/|.q.|)>=0 by A35,XREAL_1:48; then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A37; then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A38,SQUARE_1:49; then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48; then A39: 1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187; A40: (sn-FanMorphE).q= |[|.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+ sn))^2) ) , |.q.|* ((q`2/|.q.|-sn)/(1+sn))]| by A1,A2,A35,Th84; then A41: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by A24,EUCLID:52; qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by A24,A40,EUCLID:52; then A42: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2 .= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A39,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A41,A42; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; then A43: |.qz.|=|.q.| by SQUARE_1:22; |.- q.| 0 by A2,XREAL_1:149; now per cases by JGRAPH_2:3; case A7: q`1<=0; then A8: (sn-FanMorphE).q=q by Th82; now per cases by JGRAPH_2:3; case p`1<=0; hence thesis by A5,A8,Th82; end; case A9: p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1>=0; then A10: (p`2/|.p.|-sn)>=0 by XREAL_1:48; set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2 /|.p.|-sn)/(1-sn))]|; A11: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; A12: |.p.|<>0 by A9,TOPRNS_1:24; then A13: (|.p.|)^2>0 by SQUARE_1:12; 0<=(p`1)^2 by XREAL_1:63; then 0+(p`2)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7; then (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A11,XREAL_1:72; then (p`2)^2/(|.p.|)^2 <= 1 by A13,XCMPLX_1:60; then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76; then 1>=p`2/|.p.| by SQUARE_1:51; then 1-sn>=p`2/|.p.|-sn by XREAL_1:9; then -(1-sn)<= -( p`2/|.p.|-sn) by XREAL_1:24; then (-(1-sn))/(1-sn)<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XREAL_1:72; then A14: -1<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XCMPLX_1:197; A15: sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) , |.p.|* ((p`2/|.p.|-sn)/(1-sn))]| by A1,A2,A9,Th84; (p`2/|.p.|-sn)>= 0 by A9,XREAL_1:48; then ((-(p`2/|.p.|-sn))/(1-sn))^2<=1^2 by A6,A14,SQUARE_1:49; then A16: 1-((-(p`2/|.p.|-sn))/(1-sn))^2>=0 by XREAL_1:48; then sqrt(1-((-(p`2/|.p.|-sn))/(1-sn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(p`2/|.p.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(p`2/|.p.|-sn)^2/(1-sn)^2)>=0; then sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76; then p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) & q`1=0 by A5,A7,A8 ,A15,EUCLID:52; then A17: (sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))=0 by A5,A8,A15,A12,XCMPLX_1:6; 1-(-((p`2/|.p.|-sn))/(1-sn))^2>=0 by A16,XCMPLX_1:187; then 1-((p`2/|.p.|-sn)/(1-sn))^2=0 by A17,SQUARE_1:24; then 1= (p`2/|.p.|-sn)/(1-sn) by A6,A10,SQUARE_1:18,22; then 1 *(1-sn)=(p`2/|.p.|-sn) by A6,XCMPLX_1:87; then 1 *|.p.|=p`2 by A9,TOPRNS_1:24,XCMPLX_1:87; then p`1=0 by A11,XCMPLX_1:6; hence thesis by A5,A8,Th82; end; case A18: p<>0.TOP-REAL 2 & p`2/|.p.| =0; then A19: |.p.|<>0 by TOPRNS_1:24; then A20: (|.p.|)^2>0 by SQUARE_1:12; set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2 /|.p.|-sn)/(1+sn))]|; A21: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; A22: 1+sn>0 by A1,XREAL_1:148; A23: (p`2/|.p.|-sn)<=0 by A18,XREAL_1:47; then A24: -1<=(-( p`2/|.p.|-sn))/(1+sn) by A22; A25: sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) , |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A1,A2,A18,Th84; 0<=(p`1)^2 by XREAL_1:63; then 0+(p`2)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7; then (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A21,XREAL_1:72; then (p`2)^2/(|.p.|)^2 <= 1 by A20,XCMPLX_1:60; then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76; then (-((p`2)/|.p.|))^2 <= 1; then 1>= -p`2/|.p.| by SQUARE_1:51; then (1+sn)>= -p`2/|.p.|+sn by XREAL_1:7; then (-(p`2/|.p.|-sn))/(1+sn)<=1 by A22,XREAL_1:185; then ((-(p`2/|.p.|-sn))/(1+sn))^2<=1^2 by A24,SQUARE_1:49; then A26: 1-((-(p`2/|.p.|-sn))/(1+sn))^2>=0 by XREAL_1:48; then sqrt(1-((-(p`2/|.p.|-sn))/(1+sn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(p`2/|.p.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76; then sqrt(1-((p`2/|.p.|-sn))^2/(1+sn)^2)>=0; then sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76; then p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) & q`1=0 by A5,A7,A8 ,A25,EUCLID:52; then A27: (sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))=0 by A5,A8,A25,A19,XCMPLX_1:6; 1-(-((p`2/|.p.|-sn))/(1+sn))^2>=0 by A26,XCMPLX_1:187; then 1-((p`2/|.p.|-sn)/(1+sn))^2=0 by A27,SQUARE_1:24; then 1=sqrt((-((p`2/|.p.|-sn)/(1+sn)))^2) by SQUARE_1:18; then 1= -((p`2/|.p.|-sn)/(1+sn)) by A22,A23,SQUARE_1:22; then 1= ((-(p`2/|.p.|-sn))/(1+sn)) by XCMPLX_1:187; then 1 *(1+sn)=-(p`2/|.p.|-sn) by A22,XCMPLX_1:87; then 1+sn-sn=-p`2/|.p.|; then 1=(-p`2)/|.p.| by XCMPLX_1:187; then 1 *|.p.|=-p`2 by A18,TOPRNS_1:24,XCMPLX_1:87; then (p`2)^2-(p`2)^2 =(p`1)^2 by A21,XCMPLX_1:26; then p`1=0 by XCMPLX_1:6; hence thesis by A5,A8,Th82; end; end; hence thesis; end; case A28: q`2/|.q.|>=sn & q`1>=0 & q<>0.TOP-REAL 2; then |.q.|<>0 by TOPRNS_1:24; then A29: (|.q.|)^2>0 by SQUARE_1:12; set q4= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q .|-sn)/(1-sn))]|; A30: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52; A31: sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2 )), |. q.|* ((q`2/|.q.|-sn)/(1-sn))]| by A1,A2,A28,Th84; A32: q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52; now per cases by JGRAPH_2:3; case A33: p`1<=0; then A34: (sn-FanMorphE).p=p by Th82; A35: |.q.|<>0 by A28,TOPRNS_1:24; then A36: (|.q.|)^2>0 by SQUARE_1:12; A37: (q`2/|.q.|-sn)>= 0 by A28,XREAL_1:48; A38: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; A39: (q`2/|.q.|-sn)>=0 by A28,XREAL_1:48; A40: 1-sn>0 by A2,XREAL_1:149; 0<=(q`1)^2 by XREAL_1:63; then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A38,XREAL_1:72; then (q`2)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`2/|.q.| by SQUARE_1:51; then 1-sn>=q`2/|.q.|-sn by XREAL_1:9; then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24; then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A40,XREAL_1:72; then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A40,XCMPLX_1:197; then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A40,A37,SQUARE_1:49; then A41: 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48; then sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(q`2/|.q.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(q`2/|.q.|-sn)^2/(1-sn)^2)>=0; then sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76; then p`1=0 by A5,A31,A33,A34,EUCLID:52; then A42: (sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))=0 by A5,A31,A32,A34,A35, XCMPLX_1:6; 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by A41,XCMPLX_1:187; then 1-((q`2/|.q.|-sn)/(1-sn))^2=0 by A42,SQUARE_1:24; then 1= (q`2/|.q.|-sn)/(1-sn) by A40,A39,SQUARE_1:18,22; then 1 *(1-sn)=(q`2/|.q.|-sn) by A40,XCMPLX_1:87; then 1 *|.q.|=q`2 by A28,TOPRNS_1:24,XCMPLX_1:87; then q`1=0 by A38,XCMPLX_1:6; hence thesis by A5,A34,Th82; end; case A43: p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1>=0; 0<=(q`1)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`2)^2/(|.q.|)^2 <= 1 by A29,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`2/|.q.| by SQUARE_1:51; then 1-sn>=q`2/|.q.|-sn by XREAL_1:9; then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24; then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A6,XREAL_1:72; then A44: -1<=(-( q`2/|.q.|-sn))/(1-sn) by A6,XCMPLX_1:197; (q`2/|.q.|-sn)>= 0 by A28,XREAL_1:48; then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A6,A44,SQUARE_1:49; then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48; then A45: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187; q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52; then A46: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2 .= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A45,SQUARE_1:def 2; A47: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A47,A46; then A48: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22; then A49: |.q4.|=|.q.| by SQUARE_1:22; 0<=(p`1)^2 by XREAL_1:63; then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`2)^2<=(p`1)^2+(p`2)^2 by JGRAPH_3:1,XREAL_1:7; then A50: (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72; |.p.|<>0 by A43,TOPRNS_1:24; then (|.p.|)^2>0 by SQUARE_1:12; then (p`2)^2/(|.p.|)^2 <= 1 by A50,XCMPLX_1:60; then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76; then 1>=p`2/|.p.| by SQUARE_1:51; then 1-sn>=p`2/|.p.|-sn by XREAL_1:9; then -(1-sn)<= -( p`2/|.p.|-sn) by XREAL_1:24; then (-(1-sn))/(1-sn)<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XREAL_1:72; then A51: -1<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XCMPLX_1:197; (p`2/|.p.|-sn)>= 0 by A43,XREAL_1:48; then ((-(p`2/|.p.|-sn))/(1-sn))^2<=1^2 by A6,A51,SQUARE_1:49; then 1-((-(p`2/|.p.|-sn))/(1-sn))^2>=0 by XREAL_1:48; then A52: 1-(-((p`2/|.p.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187; set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2 /|.p.|-sn)/(1-sn))]|; A53: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1-sn)) by EUCLID:52; p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) by EUCLID:52; then A54: (p4`1)^2= (|.p.|)^2*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))^2 .= (|.p.|)^2*(1-((p`2/|.p.|-sn)/(1-sn))^2) by A52,SQUARE_1:def 2; (|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1 .=(|.p.|)^2 by A53,A54; then A55: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22; then A56: |.p4.|=|.p.| by SQUARE_1:22; A57: sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2) ), |.p.|* ((p`2/|.p.|-sn)/(1-sn))]| by A1,A2,A43,Th84; then ((p`2/|.p.|-sn)/(1-sn)) =|.q.|* ((q`2/|.q.|-sn)/(1-sn))/|.p .| by A5,A31,A30,A43,A53,TOPRNS_1:24,XCMPLX_1:89; then (p`2/|.p.|-sn)/(1-sn)=(q`2/|.q.|-sn)/(1-sn) by A5,A31,A43,A57 ,A48,A55,TOPRNS_1:24,XCMPLX_1:89; then (p`2/|.p.|-sn)/(1-sn)*(1-sn)=q`2/|.q.|-sn by A6,XCMPLX_1:87; then p`2/|.p.|-sn=q`2/|.q.|-sn by A6,XCMPLX_1:87; then p`2/|.p.|*|.p.|=q`2 by A5,A31,A43,A57,A49,A56,TOPRNS_1:24 ,XCMPLX_1:87; then A58: p`2=q`2 by A43,TOPRNS_1:24,XCMPLX_1:87; A59: p=|[p`1,p`2]| by EUCLID:53; |.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1; then p`1=sqrt((q`1)^2) by A5,A31,A43,A57,A49,A56,A58,SQUARE_1:22; then p`1=q`1 by A28,SQUARE_1:22; hence thesis by A58,A59,EUCLID:53; end; case A60: p<>0.TOP-REAL 2 & p`2/|.p.| =0; then p`2/|.p.|-sn<0 by XREAL_1:49; then A61: ((p`2/|.p.|-sn)/(1+sn))<0 by A1,XREAL_1:141,148; set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2 /|.p.|-sn)/(1+sn))]|; A62: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1+sn)) & q`2/|.q.|-sn>=0 by A28, EUCLID:52,XREAL_1:48; A63: 1-sn>0 by A2,XREAL_1:149; sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2) ), |.p.|* ((p`2/ |.p.|-sn)/(1+sn))]| & |.p.|<>0 by A1,A2,A60,Th84,TOPRNS_1:24; hence thesis by A5,A31,A30,A61,A62,A63,XREAL_1:132; end; end; hence thesis; end; case A64: q`2/|.q.| =0 & q<>0.TOP-REAL 2; then A65: |.q.|<>0 by TOPRNS_1:24; then A66: (|.q.|)^2>0 by SQUARE_1:12; set q4= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q .|-sn)/(1+sn))]|; A67: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52; A68: sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2 )), |.q.|* ((q`2/|.q.|-sn)/(1+sn))]| by A1,A2,A64,Th84; A69: q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52; now per cases by JGRAPH_2:3; case A70: p`1<=0; A71: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; A72: 1+sn>0 by A1,XREAL_1:148; 0<=(q`1)^2 by XREAL_1:63; then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A71,XREAL_1:72; then (q`2)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then (-((q`2)/|.q.|))^2 <= 1; then 1>= -q`2/|.q.| by SQUARE_1:51; then (1+sn)>= -q`2/|.q.|+sn by XREAL_1:7; then A73: (-(q`2/|.q.|-sn))/(1+sn)<=1 by A72,XREAL_1:185; A74: (q`2/|.q.|-sn)<=0 by A64,XREAL_1:47; then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A72; then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A73,SQUARE_1:49; then A75: 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48; then A76: 1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187; A77: (sn-FanMorphE).p=p by A70,Th82; sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>=0 by A75,SQUARE_1:def 2; then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76; then sqrt(1-((q`2/|.q.|-sn))^2/(1+sn)^2)>=0; then sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76; then p`1=0 by A5,A68,A70,A77,EUCLID:52; then (sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))=0 by A5,A68,A69,A65,A77, XCMPLX_1:6; then 1-((q`2/|.q.|-sn)/(1+sn))^2=0 by A76,SQUARE_1:24; then 1=sqrt((-((q`2/|.q.|-sn)/(1+sn)))^2) by SQUARE_1:18; then 1= -((q`2/|.q.|-sn)/(1+sn)) by A72,A74,SQUARE_1:22; then 1= ((-(q`2/|.q.|-sn))/(1+sn)) by XCMPLX_1:187; then 1 *(1+sn)=-(q`2/|.q.|-sn) by A72,XCMPLX_1:87; then 1+sn-sn=-q`2/|.q.|; then 1=(-q`2)/|.q.| by XCMPLX_1:187; then 1 *|.q.|=-q`2 by A64,TOPRNS_1:24,XCMPLX_1:87; then (q`2)^2-(q`2)^2 =(q`1)^2 by A71,XCMPLX_1:26; then q`1=0 by XCMPLX_1:6; hence thesis by A5,A77,Th82; end; case A78: p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1>=0; set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2 /|.p.|-sn)/(1-sn))]|; A79: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1-sn)) & |.q.|<>0 by A64,EUCLID:52 ,TOPRNS_1:24; q`2/|.q.|-sn<0 by A64,XREAL_1:49; then A80: ((q`2/|.q.|-sn)/(1+sn))<0 by A1,XREAL_1:141,148; A81: 1-sn>0 by A2,XREAL_1:149; sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2) ), |.p.|* ((p`2/ |.p.|-sn)/(1-sn))]| & p`2/|.p.|-sn>=0 by A1,A2,A78,Th84, XREAL_1:48; hence thesis by A5,A68,A67,A80,A79,A81,XREAL_1:132; end; case A82: p<>0.TOP-REAL 2 & p`2/|.p.| =0; 0<=(p`1)^2 by XREAL_1:63; then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`2)^2<=(p`1)^2+(p`2)^2 by JGRAPH_3:1,XREAL_1:7; then A83: (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72; A84: 1+sn>0 by A1,XREAL_1:148; 0<=(q`1)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`2)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`2/|.q.| by SQUARE_1:51; then -1-sn<=q`2/|.q.|-sn by XREAL_1:9; then -(-1-sn)>= -(q`2/|.q.|-sn) by XREAL_1:24; then A85: (-(q`2/|.q.|-sn))/(1+sn)<=1 by A84,XREAL_1:185; (q`2/|.q.|-sn)<=0 by A64,XREAL_1:47; then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A84; then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A85,SQUARE_1:49; then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48; then A86: 1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187; q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52; then A87: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2 .= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A86,SQUARE_1:def 2; A88: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52; set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2 /|.p.|-sn)/(1+sn))]|; A89: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1+sn)) by EUCLID:52; |.p.|<>0 by A82,TOPRNS_1:24; then (|.p.|)^2>0 by SQUARE_1:12; then (p`2)^2/(|.p.|)^2 <= 1 by A83,XCMPLX_1:60; then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76; then -1<=p`2/|.p.| by SQUARE_1:51; then -1-sn<=p`2/|.p.|-sn by XREAL_1:9; then -(-1-sn)>= -(p`2/|.p.|-sn) by XREAL_1:24; then A90: (-(p`2/|.p.|-sn))/(1+sn)<=1 by A84,XREAL_1:185; (p`2/|.p.|-sn)<=0 by A82,XREAL_1:47; then -1<=(-( p`2/|.p.|-sn))/(1+sn) by A84; then ((-(p`2/|.p.|-sn))/(1+sn))^2<=1^2 by A90,SQUARE_1:49; then 1-((-(p`2/|.p.|-sn))/(1+sn))^2>=0 by XREAL_1:48; then A91: 1-(-((p`2/|.p.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187; p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) by EUCLID:52; then A92: (p4`1)^2= (|.p.|)^2*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))^2 .= (|.p.|)^2*(1-((p`2/|.p.|-sn)/(1+sn))^2) by A91,SQUARE_1:def 2; (|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1 .=(|.p.|)^2 by A89,A92; then A93: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22; then A94: |.p4.|=|.p.| by SQUARE_1:22; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A88,A87; then A95: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22; then A96: |.q4.|=|.q.| by SQUARE_1:22; A97: sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2) ), |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A1,A2,A82,Th84; then ((p`2/|.p.|-sn)/(1+sn)) =|.q.|* ((q`2/|.q.|-sn)/(1+sn))/|.p .| by A5,A68,A67,A82,A89,TOPRNS_1:24,XCMPLX_1:89; then (p`2/|.p.|-sn)/(1+sn)=(q`2/|.q.|-sn)/(1+sn) by A5,A68,A82,A97 ,A95,A93,TOPRNS_1:24,XCMPLX_1:89; then (p`2/|.p.|-sn)/(1+sn)*(1+sn)=q`2/|.q.|-sn by A84,XCMPLX_1:87; then p`2/|.p.|-sn=q`2/|.q.|-sn by A84,XCMPLX_1:87; then p`2/|.p.|*|.p.|=q`2 by A5,A68,A82,A97,A96,A94,TOPRNS_1:24 ,XCMPLX_1:87; then A98: p`2=q`2 by A82,TOPRNS_1:24,XCMPLX_1:87; A99: p=|[p`1,p`2]| by EUCLID:53; |.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1; then p`1=sqrt((q`1)^2) by A5,A68,A82,A97,A96,A94,A98,SQUARE_1:22; then p`1=q`1 by A64,SQUARE_1:22; hence thesis by A98,A99,EUCLID:53; end; end; hence thesis; end; end; hence thesis; end; hence thesis by FUNCT_1:def 4; end; theorem Th103: for sn being Real st -1 =0 & q`1>=0 & q<>0.TOP-REAL 2; --(1+sn)>0 by A1,XREAL_1:148; then A6: -(-1-sn)>0; A7: 1-sn>=0 by A2,XREAL_1:149; then q`2/|.q.|*(1-sn)>=0 by A5; then -1-sn<= q`2/|.q.|*(1-sn) by A6; then A8: -1-sn+sn<= q`2/|.q.|*(1-sn)+sn by XREAL_1:7; set px=|[ (|.q.|)*sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2), |.q.|*(q`2/|.q.|* (1-sn)+sn)]|; A9: px`2 = |.q.|*(q`2/|.q.|*(1-sn)+sn) by EUCLID:52; |.q.|<>0 by A5,TOPRNS_1:24; then A10: |.q.|^2>0 by SQUARE_1:12; A11: dom (sn-FanMorphE)=the carrier of TOP-REAL 2 by FUNCT_2:def 1; A12: 1-sn>0 by A2,XREAL_1:149; 0<=(q`1)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`2)^2/(|.q.|)^2 <= 1 by A10,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then q`2/|.q.|<=1 by SQUARE_1:51; then q`2/|.q.|*(1-sn) <= 1 *(1-sn) by A12,XREAL_1:64; then q`2/|.q.|*(1-sn)+sn-sn <=1-sn; then (q`2/|.q.|*(1-sn)+sn) <=1 by XREAL_1:9; then 1^2>=(q`2/|.q.|*(1-sn)+sn)^2 by A8,SQUARE_1:49; then A13: 1-(q`2/|.q.|*(1-sn)+sn)^2>=0 by XREAL_1:48; then A14: sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2)>=0 by SQUARE_1:def 2; A15: px`1 = (|.q.|)*sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2) by EUCLID:52; then |.px.|^2=((|.q.|)*sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2))^2 +(|.q.|*(q `2/|.q.|*(1-sn)+sn))^2 by A9,JGRAPH_3:1 .=(|.q.|)^2*(sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2))^2 +(|.q.|)^2*((q`2 /|.q.|*(1-sn)+sn))^2; then A16: |.px.|^2=(|.q.|)^2*(1-(q`2/|.q.|*(1-sn)+sn)^2) +(|.q.|)^2*((q`2 /|.q.|*(1-sn)+sn))^2 by A13,SQUARE_1:def 2 .= (|.q.|)^2; then A17: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22 .=|.q.| by SQUARE_1:22; then A18: px<>0.TOP-REAL 2 by A5,TOPRNS_1:23,24; (q`2/|.q.|*(1-sn)+sn)>=0+sn by A5,A7,XREAL_1:7; then px`2/|.px.| >=sn by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89; then A19: (sn-FanMorphE).px =|[ |.px.|*(sqrt(1-((px`2/|.px.|-sn) /(1-sn)) ^2)), |.px.|* ((px`2/|.px.|-sn)/(1-sn))]| by A1,A2,A15,A14,A18,Th84; A20: |.px.|*(sqrt((q`1/|.q.|)^2))=|.q.|*(q`1/|.q.|) by A5,A17,SQUARE_1:22 .=q`1 by A5,TOPRNS_1:24,XCMPLX_1:87; A21: |.px.|* ((px`2/|.px.|-sn)/(1-sn)) =|.q.|* (( ((q`2/|.q.|*(1-sn) +sn))-sn)/(1-sn)) by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89 .=|.q.|* ( q`2/|.q.|) by A12,XCMPLX_1:89 .= q`2 by A5,TOPRNS_1:24,XCMPLX_1:87; then |.px.|*(sqrt(1-((px`2/|.px.|-sn)/(1-sn))^2)) = |.px.|*(sqrt(1-( q`2/|.px.|)^2)) by A5,A17,TOPRNS_1:24,XCMPLX_1:89 .= |.px.|*(sqrt(1-(q`2)^2/(|.px.|)^2)) by XCMPLX_1:76 .= |.px.|*(sqrt( (|.px.|)^2/(|.px.|)^2-(q`2)^2/(|.px.|)^2)) by A10 ,A16,XCMPLX_1:60 .= |.px.|*(sqrt( ((|.px.|)^2-(q`2)^2)/(|.px.|)^2)) by XCMPLX_1:120 .= |.px.|*(sqrt( ((q`1)^2+(q`2)^2-(q`2)^2)/(|.px.|)^2)) by A16, JGRAPH_3:1 .= |.px.|*(sqrt((q`1/|.q.|)^2)) by A17,XCMPLX_1:76; hence ex x being set st x in dom (sn-FanMorphE) & y=(sn-FanMorphE).x by A19,A21,A20,A11,EUCLID:53; end; suppose A22: q`2/|.q.|<0 & q`1>=0 & q<>0.TOP-REAL 2; A23: 1+sn>=0 by A1,XREAL_1:148; (1-sn)>0 by A2,XREAL_1:149; then A24: 1-sn+sn>= q`2/|.q.|*(1+sn)+sn by A22,A23,XREAL_1:7; A25: 1+sn>0 by A1,XREAL_1:148; |.q.|<>0 by A22,TOPRNS_1:24; then A26: |.q.|^2>0 by SQUARE_1:12; 0<=(q`1)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`2)^2/(|.q.|)^2 <= 1 by A26,XCMPLX_1:60; then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76; then q`2/|.q.|>=-1 by SQUARE_1:51; then q`2/|.q.|*(1+sn) >=(-1)*(1+sn) by A25,XREAL_1:64; then q`2/|.q.|*(1+sn)+sn-sn >=-1-sn; then (q`2/|.q.|*(1+sn)+sn) >=-1 by XREAL_1:9; then 1^2>=(q`2/|.q.|*(1+sn)+sn)^2 by A24,SQUARE_1:49; then A27: 1-(q`2/|.q.|*(1+sn)+sn)^2>=0 by XREAL_1:48; then A28: sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2)>=0 by SQUARE_1:def 2; A29: dom (sn-FanMorphE)=the carrier of TOP-REAL 2 by FUNCT_2:def 1; set px=|[ (|.q.|)*sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2), |.q.|*(q`2/|.q.|* (1+sn)+sn)]|; A30: px`2 = |.q.|*(q`2/|.q.|*(1+sn)+sn) by EUCLID:52; A31: px`1 = (|.q.|)*sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2) by EUCLID:52; then |.px.|^2=((|.q.|)*sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2))^2 +(|.q.|*(q `2/|.q.|*(1+sn)+sn))^2 by A30,JGRAPH_3:1 .=(|.q.|)^2*(sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2))^2 +(|.q.|)^2*((q`2 /|.q.|*(1+sn)+sn))^2; then A32: |.px.|^2=(|.q.|)^2*(1-(q`2/|.q.|*(1+sn)+sn)^2) +(|.q.|)^2*((q`2 /|.q.|*(1+sn)+sn))^2 by A27,SQUARE_1:def 2 .= (|.q.|)^2; then A33: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22 .=|.q.| by SQUARE_1:22; then A34: px<>0.TOP-REAL 2 by A22,TOPRNS_1:23,24; (q`2/|.q.|*(1+sn)+sn)<=0+sn by A22,A23,XREAL_1:7; then px`2/|.px.| <=sn by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89; then A35: (sn-FanMorphE).px =|[ |.px.|*(sqrt(1-((px`2/|.px.|-sn) /(1+sn)) ^2)), |.px.|* ((px`2/|.px.|-sn)/(1+sn))]| by A1,A2,A31,A28,A34,Th84; A36: |.px.|*(sqrt((q`1/|.q.|)^2)) =|.q.|*(q`1/|.q.|) by A22,A33, SQUARE_1:22 .=q`1 by A22,TOPRNS_1:24,XCMPLX_1:87; A37: |.px.|* ((px`2/|.px.|-sn)/(1+sn)) =|.q.|* (( ((q`2/|.q.|*(1+sn) +sn))-sn)/(1+sn)) by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89 .=|.q.|* ( q`2/|.q.|) by A25,XCMPLX_1:89 .= q`2 by A22,TOPRNS_1:24,XCMPLX_1:87; then |.px.|*(sqrt(1-((px`2/|.px.|-sn)/(1+sn))^2)) = |.px.|*(sqrt(1-( q`2/|.px.|)^2)) by A22,A33,TOPRNS_1:24,XCMPLX_1:89 .= |.px.|*(sqrt(1-(q`2)^2/(|.px.|)^2)) by XCMPLX_1:76 .= |.px.|*(sqrt( (|.px.|)^2/(|.px.|)^2-(q`2)^2/(|.px.|)^2)) by A26 ,A32,XCMPLX_1:60 .= |.px.|*(sqrt( ((|.px.|)^2-(q`2)^2)/(|.px.|)^2)) by XCMPLX_1:120 .= |.px.|*(sqrt( ((q`1)^2+(q`2)^2-(q`2)^2)/(|.px.|)^2)) by A32, JGRAPH_3:1 .= |.px.|*(sqrt((q`1/|.q.|)^2)) by A33,XCMPLX_1:76; hence ex x being set st x in dom (sn-FanMorphE) & y=(sn-FanMorphE).x by A35,A37,A36,A29,EUCLID:53; end; end; hence thesis by A3,FUNCT_1:def 3; end; hence thesis by A3,XBOOLE_0:def 10; end; hence thesis; end; theorem Th104: for sn being Real,p2 being Point of TOP-REAL 2 st -1 0 & q`2/|.q.|>=sn holds for p being Point of TOP-REAL 2 st p=(sn-FanMorphE).q holds p`1>0 & p`2>=0 proof let sn be Real,q be Point of TOP-REAL 2; assume that A1: sn<1 and A2: q`1>0 and A3: q`2/|.q.|>=sn; A4: (q`2/|.q.|-sn)>= 0 by A3,XREAL_1:48; let p be Point of TOP-REAL 2; set qz=p; A5: 1-sn>0 by A1,XREAL_1:149; A6: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24; then A7: (|.q.|)^2>0 by SQUARE_1:12; (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1 ,SQUARE_1:12,XREAL_1:8; then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,XREAL_1:74; then (q`2)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60; then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76; then 1>q`2/|.q.| by SQUARE_1:52; then 1-sn>q`2/|.q.|-sn by XREAL_1:9; then -(1-sn)< -( q`2/|.q.|-sn) by XREAL_1:24; then (-(1-sn))/(1-sn)<(-( q`2/|.q.|-sn))/(1-sn) by A5,XREAL_1:74; then -1<(-( q`2/|.q.|-sn))/(1-sn) by A5,XCMPLX_1:197; then ((-(q`2/|.q.|-sn))/(1-sn))^2<1^2 by A5,A4,SQUARE_1:50; then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>0 by XREAL_1:50; then sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>0 by SQUARE_1:25; then sqrt(1-(-(q`2/|.q.|-sn))^2/(1-sn)^2)> 0 by XCMPLX_1:76; then sqrt(1-(q`2/|.q.|-sn)^2/(1-sn)^2)> 0; then A8: sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)> 0 by XCMPLX_1:76; assume p=(sn-FanMorphE).q; then A9: p=|[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/( 1-sn))]| by A2,A3,Th82; then qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52; hence thesis by A9,A6,A5,A4,A8,EUCLID:52,XREAL_1:129; end; theorem Th107: for sn being Real,q being Point of TOP-REAL 2 st -1 0 & q`2/|.q.| 0 & p`2<0 proof let sn be Real,q be Point of TOP-REAL 2; assume that A1: -1 0 and A3: q`2/|.q.| 0 by A1,XREAL_1:148; let p be Point of TOP-REAL 2; set qz=p; assume p=(sn-FanMorphE).q; then p=|[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-sn)/( 1+sn))]| by A2,A3,Th83; then A5: qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) & qz`2= |.q.|* ((q`2/|. q.|- sn)/(1+sn)) by EUCLID:52; A6: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24; then A7: (|.q.|)^2>0 by SQUARE_1:12; A8: (q`2/|.q.|-sn)< 0 by A3,XREAL_1:49; then -( q`2/|.q.|-sn)>0 by XREAL_1:58; then (-(1+sn))/(1+sn)<(-( q`2/|.q.|-sn))/(1+sn) by A4,XREAL_1:74; then A9: -1<(-( q`2/|.q.|-sn))/(1+sn) by A4,XCMPLX_1:197; (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1 ,SQUARE_1:12,XREAL_1:8; then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,XREAL_1:74; then (q`2)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60; then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76; then -1 -(q`2/|.q.|-sn) by XREAL_1:24; then (-(q`2/|.q.|-sn))/(1+sn)<1 by A4,XREAL_1:191; then ((-(q`2/|.q.|-sn))/(1+sn))^2<1^2 by A9,SQUARE_1:50; then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>0 by XREAL_1:50; then sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>0 by SQUARE_1:25; then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)> 0 by XCMPLX_1:76; then sqrt(1-(q`2/|.q.|-sn)^2/(1+sn)^2)> 0; then A10: sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)> 0 by XCMPLX_1:76; ((q`2/|.q.|-sn)/(1+sn))<0 by A1,A8,XREAL_1:141,148; hence thesis by A6,A5,A10,XREAL_1:129,132; end; theorem Th108: for sn being Real,q1,q2 being Point of TOP-REAL 2 st sn<1 & q1 `1>0 & q1`2/|.q1.|>=sn & q2`1>0 & q2`2/|.q2.|>=sn & q1`2/|.q1.|0 and A3: q1`2/|.q1.|>=sn and A4: q2`1>0 and A5: q2`2/|.q2.|>=sn and A6: q1`2/|.q1.| 0 by A1,A6,XREAL_1:9,149; let p1,p2 be Point of TOP-REAL 2; assume that A8: p1=(sn-FanMorphE).q1 and A9: p2=(sn-FanMorphE).q2; A10: |.p2.|=|.q2.| by A9,Th97; p2=|[ |.q2.|*(sqrt(1-((q2`2/|.q2.|-sn)/(1-sn))^2)), |.q2.|* ((q2`2/|.q2 .|-sn)/(1-sn))]| by A4,A5,A9,Th82; then A11: p2`2= |.q2.|* ((q2`2/|.q2.|-sn)/(1-sn)) by EUCLID:52; |.q2.|>0 by A4,Lm1,JGRAPH_2:3; then A12: p2`2/|.p2.|= ((q2`2/|.q2.|-sn)/(1-sn)) by A11,A10,XCMPLX_1:89; p1=|[ |.q1.|*(sqrt(1-((q1`2/|.q1.|-sn)/(1-sn))^2)), |.q1.|* ((q1`2/|.q1 .|-sn)/(1-sn))]| by A2,A3,A8,Th82; then A13: p1`2= |.q1.|* ((q1`2/|.q1.|-sn)/(1-sn)) by EUCLID:52; A14: |.p1.|=|.q1.| by A8,Th97; |.q1.|>0 by A2,Lm1,JGRAPH_2:3; then p1`2/|.p1.|= ((q1`2/|.q1.|-sn)/(1-sn)) by A13,A14,XCMPLX_1:89; hence thesis by A12,A7,XREAL_1:74; end; theorem Th109: for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1 0 & q1`2/|.q1.| 0 & q2`2/|.q2.| 0 and A3: q1`2/|.q1.| 0 and A5: q2`2/|.q2.| 0 by A1,A6,XREAL_1:9,148; let p1,p2 be Point of TOP-REAL 2; assume that A8: p1=(sn-FanMorphE).q1 and A9: p2=(sn-FanMorphE).q2; A10: |.p2.|=|.q2.| by A9,Th97; p2=|[ |.q2.|*(sqrt(1-((q2`2/|.q2.|-sn)/(1+sn))^2)), |.q2.|* ((q2`2/|.q2 .|-sn)/(1+sn))]| by A4,A5,A9,Th83; then A11: p2`2= |.q2.|* ((q2`2/|.q2.|-sn)/(1+sn)) by EUCLID:52; |.q2.|>0 by A4,Lm1,JGRAPH_2:3; then A12: p2`2/|.p2.|= (q2`2/|.q2.|-sn)/(1+sn) by A11,A10,XCMPLX_1:89; p1=|[ |.q1.|*(sqrt(1-((q1`2/|.q1.|-sn)/(1+sn))^2)), |.q1.|* ((q1`2/|.q1 .|-sn)/(1+sn))]| by A2,A3,A8,Th83; then A13: p1`2= |.q1.|* ((q1`2/|.q1.|-sn)/(1+sn)) by EUCLID:52; A14: |.p1.|=|.q1.| by A8,Th97; |.q1.|>0 by A2,Lm1,JGRAPH_2:3; then p1`2/|.p1.|= (q1`2/|.q1.|-sn)/(1+sn) by A13,A14,XCMPLX_1:89; hence thesis by A12,A7,XREAL_1:74; end; theorem for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1 0 & q2`1>0 & q1`2/|.q1.| 0 and A4: q2`1>0 and A5: q1`2/|.q1.| =sn & q2`2/|.q2.|>=sn; hence thesis by A2,A3,A4,A5,A6,A7,Th108; end; suppose q1`2/|.q1.|>=sn & q2`2/|.q2.| =sn; then p2`2>=0 by A2,A4,A7,Th106; then A9: p2`2/|.p2.|>=0; p1`2<0 by A1,A3,A6,A8,Th107; hence thesis by A9,Lm1,JGRAPH_2:3,XREAL_1:141; end; suppose q1`2/|.q1.| 0 & q`2/|.q.|=sn holds for p being Point of TOP-REAL 2 st p=(sn-FanMorphE).q holds p`1>0 & p`2=0 proof let sn be Real,q be Point of TOP-REAL 2; assume that A1: q`1>0 and A2: q`2/|.q.|=sn; A3: |.q.|<>0 & sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>0 by A1,A2,JGRAPH_2:3 ,SQUARE_1:25,TOPRNS_1:24; let p be Point of TOP-REAL 2; assume p=(sn-FanMorphE).q; then A4: p=|[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/( 1-sn))]| by A1,A2,Th82; then p`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52; hence thesis by A2,A4,A3,EUCLID:52,XREAL_1:129; end; theorem for sn being Real holds 0.TOP-REAL 2=(sn-FanMorphE).(0.TOP-REAL 2) by Th82,JGRAPH_2:3; begin :: Fan Morphism for South definition let s be Real, q be Point of TOP-REAL 2; func FanS(s,q) -> Point of TOP-REAL 2 equals :Def8: |.q.|*|[(q`1/|.q.|-s)/(1 -s), -sqrt(1-((q`1/|.q.|-s)/(1-s))^2)]| if q`1/|.q.|>=s & q`2<0, |.q.|*|[(q`1/ |.q.|-s)/(1+s), -sqrt(1-((q`1/|.q.|-s)/(1+s))^2)]| if q`1/|.q.| ~~Function of TOP-REAL 2, TOP-REAL 2 means :Def9: for q being Point of TOP-REAL 2 holds it.q=FanS(c,q); existence proof deffunc F(Point of TOP-REAL 2)=FanS(c,$1); thus ex IT being Function of TOP-REAL 2, TOP-REAL 2 st for q being Point of TOP-REAL 2 holds IT.q=F(q)from FUNCT_2:sch 4; end; uniqueness proof deffunc F(Point of TOP-REAL 2)=FanS(c,$1); thus for a,b being Function of TOP-REAL 2, TOP-REAL 2 st (for q being Point of TOP-REAL 2 holds a.q=F(q)) & (for q being Point of TOP-REAL 2 holds b. q=F(q)) holds a=b from BINOP_2:sch 1; end; end; theorem Th113: for cn being Real holds (q`1/|.q.|>=cn & q`2<0 implies cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(-sqrt(1-((q`1/|.q.|- cn)/(1-cn))^2))]|)& (q`2>=0 implies cn-FanMorphS.q=q) proof let cn be Real; hereby assume q`1/|.q.|>=cn & q`2<0; then FanS(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1-cn), -sqrt(1-((q`1/|.q.|-cn)/(1- cn))^2)]| by Def8 .= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(-sqrt(1-((q`1/|.q.|-cn)/( 1-cn))^2))]| by EUCLID:58; hence cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(-sqrt(1-((q `1/|.q.|-cn)/(1-cn))^2))]| by Def9; end; assume A1: q`2>=0; cn-FanMorphS.q=FanS(cn,q) by Def9; hence thesis by A1,Def8; end; theorem Th114: for cn being Real holds (q`1/|.q.|<=cn & q`2<0 implies cn -FanMorphS.q= |[ |.q.|*((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn) /(1+cn))^2))]|) proof let cn be Real; assume that A1: q`1/|.q.|<=cn and A2: q`2<0; per cases by A1,XXREAL_0:1; suppose q`1/|.q.|~~=cn & q`2<=0 & q<>0.TOP-REAL 2 implies cn-FanMorphS.q = |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))]|) & (q`1/|.q.|<=cn & q`2<=0 & q<> 0.TOP-REAL 2 implies cn-FanMorphS.q = |[ |.q.|*((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))]|) proof let cn be Real; assume that A1: -1 =cn & q`2<=0 & q<>0.TOP-REAL 2; per cases; suppose A4: q`2<0; then FanS(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1-cn), -sqrt(1-((q`1/|.q.|-cn)/( 1-cn))^2)]| by A3,Def8 .= |[|.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn) /(1-cn))^2))]| by EUCLID:58; hence thesis by A4,Def9,Th114; end; suppose A5: q`2>=0; then A6: cn-FanMorphS.q=q by Th113; A7: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; A8: 1-cn>0 by A2,XREAL_1:149; A9: q`2=0 by A3,A5; |.q.|<>0 by A3,TOPRNS_1:24; then |.q.|^2>0 by SQUARE_1:12; then (q`1)^2/|.q.|^2=1^2 by A7,A9,XCMPLX_1:60; then ((q`1)/|.q.|)^2=1^2 by XCMPLX_1:76; then A10: sqrt(((q`1)/|.q.|)^2)=1 by SQUARE_1:22; A11: now assume q`1<0; then -((q`1)/|.q.|)=1 by A10,SQUARE_1:23; hence contradiction by A1,A3; end; sqrt((|.q.|)^2)=|.q.| by SQUARE_1:22; then A12: |.q.|=q`1 by A7,A9,A11,SQUARE_1:22; then 1=q`1/|.q.| by A3,TOPRNS_1:24,XCMPLX_1:60; then (q`1/|.q.|-cn)/(1-cn)=1 by A8,XCMPLX_1:60; hence thesis by A2,A6,A9,A12,EUCLID:53,SQUARE_1:17,TOPRNS_1:24 ,XCMPLX_1:60; end; end; suppose A13: q`1/|.q.|<=cn & q`2<=0 & q<>0.TOP-REAL 2; per cases; suppose q`2<0; hence thesis by Th113,Th114; end; suppose A14: q`2>=0; then A15: q`2=0 by A13; A16: 1+cn>0 by A1,XREAL_1:148; A17: |.q.|<>0 by A13,TOPRNS_1:24; 1>q`1/|.q.| by A2,A13,XXREAL_0:2; then 1 *(|.q.|)>q`1/|.q.|*(|.q.|) by A17,XREAL_1:68; then A18: (|.q.|)^2 =(q`1)^2+(q`2)^2 & (|.q.|)>q`1 by A13,JGRAPH_3:1,TOPRNS_1:24 ,XCMPLX_1:87; then A19: |.q.|=-q`1 by A15,SQUARE_1:40; A20: q`1= -(|.q.|) by A15,A18,SQUARE_1:40; then -1=q`1/|.q.| by A13,TOPRNS_1:24,XCMPLX_1:197; then (q`1/|.q.|-cn)/(1+cn) =(-(1+cn))/(1+cn) .=-1 by A16,XCMPLX_1:197; then |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*(-sqrt(1-((q`1/|.q.|-cn)/( 1+cn))^2))]| =q by A15,A19,EUCLID:53,SQUARE_1:17; hence thesis by A1,A14,A17,A20,Th113,XCMPLX_1:197; end; end; suppose q`2>0 or q=0.TOP-REAL 2; hence thesis; end; end; theorem Th116: for cn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st cn<1 & (for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`1/|.p.|-cn)/(1- cn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<=0 & q<>0.TOP-REAL 2) holds f is continuous proof let cn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=cn, b=(1-cn); reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2; assume that A1: cn<1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2 )|K1 holds f.p=|.p.|* ((p`1/|.p.|-cn)/(1-cn)) and A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2) |K1 holds q`2<=0 & q<>0.TOP-REAL 2; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; b>0 by A1,XREAL_1:149; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 & g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and A6: g3 is continuous by A4,Th5; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A8: dom f=dom g3 by FUNCT_2:def 1; for x being object st x in dom f holds f.x=g3.x proof let x be object; assume A9: x in dom f; then reconsider s=x as Point of (TOP-REAL 2)|K1; x in K1 by A7,A8,A9,PRE_TOPC:8; then reconsider r=x as Point of (TOP-REAL 2); A10: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5; A11: g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5; f.r=(|.r.|)* ((r`1/|.r.|-cn)/(1-cn)) by A2,A9; hence thesis by A5,A11,A10; end; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th117: for cn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st -1 0.TOP-REAL 2) holds f is continuous proof let cn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=cn, b=1+cn; reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2; assume that A1: -1 0.TOP-REAL 2; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; 1+cn>0 by A1,XREAL_1:148; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 & g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and A6: g3 is continuous by A4,Th5; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; A8: for x being object st x in dom f holds f.x=g3.x proof let x be object; assume A9: x in dom f; then reconsider s=x as Point of (TOP-REAL 2)|K1; x in dom g3 by A7,A9; then x in K1 by A7,PRE_TOPC:8; then reconsider r=x as Point of (TOP-REAL 2); A10: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5; A11: g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5; f.r=(|.r.|)* ((r`1/|.r.|-cn)/(1+cn)) by A2,A9; hence thesis by A5,A11,A10; end; dom f=dom g3 by A7,FUNCT_2:def 1; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th118: for cn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st cn<1 & (for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*( -sqrt(1-((p`1/|.p.| -cn)/(1-cn))^2))) & (for q being Point of TOP-REAL 2 st q in the carrier of ( TOP-REAL 2)|K1 holds q`2<=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2) holds f is continuous proof let cn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=cn, b=1-cn; reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2; assume that A1: cn<1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2 )|K1 holds f.p=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) and A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2) |K1 holds q`2<=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; b>0 by A1,XREAL_1:149; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q =r1 & g1.q=r2 holds g3.q=r2*( -sqrt(|.1-((r1/r2-a)/b)^2.|)) and A6: g3 is continuous by A4,Th9; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A8: dom f=dom g3 by FUNCT_2:def 1; for x being object st x in dom f holds f.x=g3.x proof let x be object; A9: 1-cn>0 by A1,XREAL_1:149; assume A10: x in dom f; then x in K1 by A7,A8,PRE_TOPC:8; then reconsider r=x as Point of (TOP-REAL 2); A11: |.r.|<>0 by A3,A10,TOPRNS_1:24; |.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1; then A12: ((r`1) -(|.r.|))*((r`1)+|.r.|) =-(r`2)^2; (r`2)^2>=0 by XREAL_1:63; then r`1<= |.r.| by A12,XREAL_1:93; then r`1/|.r.| <= |.r.|/|.r.| by XREAL_1:72; then r`1/|.r.|<=1 by A11,XCMPLX_1:60; then A13: r`1/|.r.|-cn<=(1-cn) by XREAL_1:9; reconsider s=x as Point of (TOP-REAL 2)|K1 by A10; A14: now assume (1-cn)^2=0; then 1-cn+cn=0+cn by XCMPLX_1:6; hence contradiction by A1; end; cn-r`1/|.r.|<=0 by A3,A10,XREAL_1:47; then -(cn- r`1/|.r.|)>=-(1-cn) by A9,XREAL_1:24; then (1-cn)^2>=0 & (r`1/|.r.|-cn)^2<=(1-cn)^2 by A13,SQUARE_1:49,XREAL_1:63 ; then (r`1/|.r.|-cn)^2/(1-cn)^2<=(1-cn)^2/(1-cn)^2 by XREAL_1:72; then (r`1/|.r.|-cn)^2/(1-cn)^2<=1 by A14,XCMPLX_1:60; then ((r`1/|.r.|-cn)/(1-cn))^2<=1 by XCMPLX_1:76; then 1-((r`1/|.r.|-cn)/(1-cn))^2>=0 by XREAL_1:48; then |.1-((r`1/|.r.|-cn)/(1-cn))^2.| =1-((r`1/|.r.|-cn)/(1-cn))^2 by ABSVALUE:def 1; then A15: f.r=(|.r.|)*(-sqrt(|.1-((r`1/|.r.|-cn)/(1-cn))^2.|)) by A2,A10; A16: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5; g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5; hence thesis by A5,A15,A16; end; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th119: for cn being Real,K1 being non empty Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K1,R^1 st -1 0.TOP-REAL 2) holds f is continuous proof let cn be Real,K1 be non empty Subset of TOP-REAL 2, f be Function of ( TOP-REAL 2)|K1,R^1; reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5; set a=cn, b=1+cn; reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2; assume that A1: -1 0.TOP-REAL 2; A4: 1+cn>0 by A1,XREAL_1:148; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2 by A3; then for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6; then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q =r1 & g1.q=r2 holds g3.q=r2*( -sqrt(|.1-((r1/r2-a)/b)^2.|)) and A6: g3 is continuous by A4,Th9; A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A8: dom f=dom g3 by FUNCT_2:def 1; for x being object st x in dom f holds f.x=g3.x proof let x be object; assume A9: x in dom f; then x in K1 by A7,A8,PRE_TOPC:8; then reconsider r=x as Point of (TOP-REAL 2); reconsider s=x as Point of (TOP-REAL 2)|K1 by A9; A10: (1+cn)^2>0 by A4,SQUARE_1:12; A11: |.r.|<>0 by A3,A9,TOPRNS_1:24; |.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1; then A12: ((r`1) -(|.r.|))*((r`1)+|.r.|) =-(r`2)^2; (r`2)^2>=0 by XREAL_1:63; then -(|.r.|)<=r`1 by A12,XREAL_1:93; then r`1/|.r.| >= (-(|.r.|))/|.r.| by XREAL_1:72; then r`1/|.r.|>= -1 by A11,XCMPLX_1:197; then r`1/|.r.|-cn>=-1-cn by XREAL_1:9; then A13: r`1/|.r.|-cn>=-(1+cn); cn-r`1/|.r.|>=0 by A3,A9,XREAL_1:48; then -(cn-r`1/|.r.|)<=-0; then (r`1/|.r.|-cn)^2<=(1+cn)^2 by A4,A13,SQUARE_1:49; then (r`1/|.r.|-cn)^2/(1+cn)^2<=(1+cn)^2/(1+cn)^2 by A4,XREAL_1:72; then (r`1/|.r.|-cn)^2/(1+cn)^2<=1 by A10,XCMPLX_1:60; then ((r`1/|.r.|-cn)/(1+cn))^2<=1 by XCMPLX_1:76; then 1-((r`1/|.r.|-cn)/(1+cn))^2>=0 by XREAL_1:48; then |.1-((r`1/|.r.|-cn)/(1+cn))^2.| =1-((r`1/|.r.|-cn)/(1+cn))^2 by ABSVALUE:def 1; then A14: f.r=(|.r.|)*( -sqrt(|.1-((r`1/|.r.|-cn)/(1+cn))^2.|)) by A2,A9; A15: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5; g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5; hence thesis by A5,A14,A15; end; hence thesis by A6,A8,FUNCT_1:2; end; theorem Th120: for cn being Real, K0,B0 being Subset of TOP-REAL 2,f being Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1 0.TOP-REAL 2} & K0={p: p `1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2} holds f is continuous proof let cn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set sn=-sqrt(1-cn^2); set p0=|[cn,sn]|; A1: p0`2=sn by EUCLID:52; p0`1=cn by EUCLID:52; then A2: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1; assume A3: -1 0.TOP-REAL 2} & K0={p: p`1/|.p.|>=cn & p`2<=0 & p<>0. TOP-REAL 2}; then cn^2<1^2 by SQUARE_1:50; then A4: 1-cn^2>0 by XREAL_1:50; then A5: -sn>0 by SQUARE_1:25; A6: now assume p0=0.TOP-REAL 2; then --sn=-0 by EUCLID:52,JGRAPH_2:3; hence contradiction by A4,SQUARE_1:25; end; (-sn)^2=1-cn^2 by A4,SQUARE_1:def 2; then p0`1/|.p0.|=cn by A2,EUCLID:52,SQUARE_1:18; then A7: p0 in K0 by A3,A1,A6,A5; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; A8: rng (proj2*((cn-FanMorphS)|K1)) c= the carrier of R^1 by TOPMETR:17; A9: K0 c= B0 proof let x be object; assume x in K0; then ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1/|.p8.|>= cn & p8`2<=0 & p8<>0.TOP-REAL 2 by A3; hence thesis by A3; end; A10: dom ((cn-FanMorphS)|K1) c= dom (proj1*((cn-FanMorphS)|K1)) proof let x be object; assume A11: x in dom ((cn-FanMorphS)|K1); then x in dom (cn-FanMorphS) /\ K1 by RELAT_1:61; then x in dom (cn-FanMorphS) by XBOOLE_0:def 4; then A12: dom proj1 = (the carrier of TOP-REAL 2) & (cn-FanMorphS).x in rng (cn -FanMorphS) by FUNCT_1:3,FUNCT_2:def 1; ((cn-FanMorphS)|K1).x=(cn-FanMorphS).x by A11,FUNCT_1:47; hence thesis by A11,A12,FUNCT_1:11; end; A13: rng (proj1*((cn-FanMorphS)|K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj1*((cn-FanMorphS)|K1)) c= dom ((cn-FanMorphS)|K1) by RELAT_1:25; then dom (proj1*((cn-FanMorphS)|K1)) =dom ((cn-FanMorphS)|K1) by A10, XBOOLE_0:def 10 .=dom (cn-FanMorphS) /\ K1 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g2=proj1*((cn-FanMorphS)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A13,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g2.p=|.p.|* ((p`1/|.p.|-cn)/(1-cn)) proof let p be Point of TOP-REAL 2; A14: dom ((cn-FanMorphS)|K1)=dom (cn-FanMorphS) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A15: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A16: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|>= cn & p3`2<=0 & p3<>0.TOP-REAL 2 by A3,A15; then A17: (cn-FanMorphS).p =|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1- ((p`1/|.p.|-cn)/(1-cn))^2))]| by A3,Th115; ((cn-FanMorphS)|K1).p=(cn-FanMorphS).p by A16,A15,FUNCT_1:49; then g2.p=proj1.(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/ |.p.|-cn)/(1-cn))^2))]|) by A16,A14,A15,A17,FUNCT_1:13 .=(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/ (1-cn))^2))]|)`1 by PSCOMP_1:def 5 .=|.p.|* ((p`1/|.p.|-cn)/(1-cn)) by EUCLID:52; hence thesis; end; then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that A18: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f2.p=|.p.|* ((p`1/|.p.|-cn)/(1-cn)); A19: dom ((cn-FanMorphS)|K1) c= dom (proj2*((cn-FanMorphS)|K1)) proof let x be object; assume A20: x in dom ((cn-FanMorphS)|K1); then x in dom (cn-FanMorphS) /\ K1 by RELAT_1:61; then x in dom (cn-FanMorphS) by XBOOLE_0:def 4; then A21: dom proj2 = (the carrier of TOP-REAL 2) & (cn-FanMorphS).x in rng (cn -FanMorphS) by FUNCT_1:3,FUNCT_2:def 1; ((cn-FanMorphS)|K1).x=(cn-FanMorphS).x by A20,FUNCT_1:47; hence thesis by A20,A21,FUNCT_1:11; end; dom (proj2*((cn-FanMorphS)|K1)) c= dom ((cn-FanMorphS)|K1) by RELAT_1:25; then dom (proj2*((cn-FanMorphS)|K1)) =dom ((cn-FanMorphS)|K1) by A19, XBOOLE_0:def 10 .=dom (cn-FanMorphS) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g1=proj2*((cn-FanMorphS)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A8,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g1.p=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) proof let p be Point of TOP-REAL 2; A22: dom ((cn-FanMorphS)|K1)=dom (cn-FanMorphS) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A23: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A24: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|>= cn & p3`2<=0 & p3<>0.TOP-REAL 2 by A3,A23; then A25: (cn-FanMorphS).p=|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-( (p`1/|.p.|-cn)/(1-cn))^2))]| by A3,Th115; ((cn-FanMorphS)|K1).p=(cn-FanMorphS).p by A24,A23,FUNCT_1:49; then g1.p=proj2.(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/ |.p.|-cn)/(1-cn))^2))]|) by A24,A22,A23,A25,FUNCT_1:13 .=(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/ (1-cn))^2))]|)`2 by PSCOMP_1:def 6 .= |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52; hence thesis; end; then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that A26: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f1.p=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)); for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A27: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|>= cn & p3`2<=0 & p3<>0.TOP-REAL 2 by A3,A27; hence thesis; end; then A28: f1 is continuous by A3,A26,Th118; A29: for x,y,s,r being Real st |[x,y]| in K1 & s=f2.(|[x,y]|) & r=f1. (|[x,y]|) holds f.(|[x,y]|)=|[s,r]| proof let x,y,s,r be Real; assume that A30: |[x,y]| in K1 and A31: s=f2.(|[x,y]|) & r=f1.(|[x,y]|); set p99=|[x,y]|; A32: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`1/|.p3.| >=cn & p3`2<=0 & p3<>0.TOP-REAL 2 by A3,A30; A33: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then A34: f1.p99=|.p99.|*( -sqrt(1-((p99`1/|.p99.|-cn)/(1-cn))^2)) by A26,A30; ((cn-FanMorphS)|K0).(|[x,y]|)=((cn-FanMorphS)).(|[x,y]|) by A30,FUNCT_1:49 .= |[ |.p99.|* ((p99`1/|.p99.|-cn)/(1-cn)), |.p99.|*( -sqrt(1-((p99`1/ |.p99.|-cn)/(1-cn))^2))]| by A3,A32,Th115 .=|[s,r]| by A18,A30,A31,A33,A34; hence thesis by A3; end; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<=0 & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A35: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|>= cn & p3`2<=0 & p3<>0.TOP-REAL 2 by A3,A35; hence thesis; end; then f2 is continuous by A3,A18,Th116; hence thesis by A7,A9,A28,A29,JGRAPH_2:35; end; theorem Th121: for cn being Real, K0,B0 being Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1 0.TOP-REAL 2} & K0={p: p `1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} holds f is continuous proof let cn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set sn=-sqrt(1-cn^2); set p0=|[cn,sn]|; A1: p0`2=sn by EUCLID:52; p0`1=cn by EUCLID:52; then A2: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1; assume A3: -1 0.TOP-REAL 2} & K0={p: p`1/|.p.|<=cn & p`2<=0 & p<>0. TOP-REAL 2}; then cn^2<1^2 by SQUARE_1:50; then A4: 1-cn^2>0 by XREAL_1:50; then A5: -sn>0 by SQUARE_1:25; A6: now assume p0=0.TOP-REAL 2; then --sn=-0 by EUCLID:52,JGRAPH_2:3; hence contradiction by A4,SQUARE_1:25; end; (-sn)^2=1-cn^2 by A4,SQUARE_1:def 2; then p0`1/|.p0.|=cn by A2,EUCLID:52,SQUARE_1:18; then A7: p0 in K0 by A3,A1,A6,A5; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; A8: rng (proj2*((cn-FanMorphS)|K1)) c= the carrier of R^1 by TOPMETR:17; A9: K0 c= B0 proof let x be object; assume x in K0; then ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1/|.p8.|<= cn & p8`2<=0 & p8<>0.TOP-REAL 2 by A3; hence thesis by A3; end; A10: dom ((cn-FanMorphS)|K1) c= dom (proj1*((cn-FanMorphS)|K1)) proof let x be object; assume A11: x in dom ((cn-FanMorphS)|K1); then x in dom (cn-FanMorphS) /\ K1 by RELAT_1:61; then x in dom (cn-FanMorphS) by XBOOLE_0:def 4; then A12: dom proj1 = (the carrier of TOP-REAL 2) & (cn-FanMorphS).x in rng (cn -FanMorphS) by FUNCT_1:3,FUNCT_2:def 1; ((cn-FanMorphS)|K1).x=(cn-FanMorphS).x by A11,FUNCT_1:47; hence thesis by A11,A12,FUNCT_1:11; end; A13: rng (proj1*((cn-FanMorphS)|K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj1*((cn-FanMorphS)|K1)) c= dom ((cn-FanMorphS)|K1) by RELAT_1:25; then dom (proj1*((cn-FanMorphS)|K1)) =dom ((cn-FanMorphS)|K1) by A10, XBOOLE_0:def 10 .=dom (cn-FanMorphS) /\ K1 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g2=proj1*((cn-FanMorphS)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A13,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g2.p=|.p.|* ((p`1/|.p.|-cn)/(1+cn)) proof let p be Point of TOP-REAL 2; A14: dom ((cn-FanMorphS)|K1)=dom (cn-FanMorphS) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A15: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A16: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|<= cn & p3`2<=0 & p3<>0.TOP-REAL 2 by A3,A15; then A17: (cn-FanMorphS).p =|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1- ((p`1/|.p.|-cn)/(1+cn))^2))]| by A3,Th115; ((cn-FanMorphS)|K1).p=(cn-FanMorphS).p by A16,A15,FUNCT_1:49; then g2.p=proj1.(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/ |.p.|-cn)/(1+cn))^2))]|) by A16,A14,A15,A17,FUNCT_1:13 .=(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/ (1+cn))^2))]|)`1 by PSCOMP_1:def 5 .=|.p.|* ((p`1/|.p.|-cn)/(1+cn)) by EUCLID:52; hence thesis; end; then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that A18: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f2.p=|.p.|* ((p`1/|.p.|-cn)/(1+cn)); A19: dom ((cn-FanMorphS)|K1) c= dom (proj2*((cn-FanMorphS)|K1)) proof let x be object; assume A20: x in dom ((cn-FanMorphS)|K1); then x in dom (cn-FanMorphS) /\ K1 by RELAT_1:61; then x in dom (cn-FanMorphS) by XBOOLE_0:def 4; then A21: dom proj2 = (the carrier of TOP-REAL 2) & (cn-FanMorphS).x in rng (cn -FanMorphS) by FUNCT_1:3,FUNCT_2:def 1; ((cn-FanMorphS)|K1).x=(cn-FanMorphS).x by A20,FUNCT_1:47; hence thesis by A20,A21,FUNCT_1:11; end; dom (proj2*((cn-FanMorphS)|K1)) c= dom ((cn-FanMorphS)|K1) by RELAT_1:25; then dom (proj2*((cn-FanMorphS)|K1)) =dom ((cn-FanMorphS)|K1) by A19, XBOOLE_0:def 10 .=dom (cn-FanMorphS) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8; then reconsider g1=proj2*((cn-FanMorphS)|K1) as Function of (TOP-REAL 2)|K1,R^1 by A8,FUNCT_2:2; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g1.p=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) proof let p be Point of TOP-REAL 2; A22: dom ((cn-FanMorphS)|K1)=dom (cn-FanMorphS) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1 .=K1 by XBOOLE_1:28; A23: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume A24: p in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|<= cn & p3`2<=0 & p3<>0.TOP-REAL 2 by A3,A23; then A25: (cn-FanMorphS).p=|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-( (p`1/|.p.|-cn)/(1+cn))^2))]| by A3,Th115; ((cn-FanMorphS)|K1).p=(cn-FanMorphS).p by A24,A23,FUNCT_1:49; then g1.p=proj2.(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/ |.p.|-cn)/(1+cn))^2))]|) by A24,A22,A23,A25,FUNCT_1:13 .=(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/ (1+cn))^2))]|)`2 by PSCOMP_1:def 6 .= |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52; hence thesis; end; then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that A26: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2) |K1 holds f1.p=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)); for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<=0 & q`1/|.q.|<=cn & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A27: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|<= cn & p3`2<=0 & p3<>0.TOP-REAL 2 by A3,A27; hence thesis; end; then A28: f1 is continuous by A3,A26,Th119; A29: for x,y,s,r being Real st |[x,y]| in K1 & s=f2.(|[x,y]|) & r=f1. (|[x,y]|) holds f.(|[x,y]|)=|[s,r]| proof let x,y,s,r be Real; assume that A30: |[x,y]| in K1 and A31: s=f2.(|[x,y]|) & r=f1.(|[x,y]|); set p99=|[x,y]|; A32: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`1/|.p3.| <=cn & p3`2<=0 & p3<>0.TOP-REAL 2 by A3,A30; A33: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then A34: f1.p99=|.p99.|*( -sqrt(1-((p99`1/|.p99.|-cn)/(1+cn))^2)) by A26,A30; ((cn-FanMorphS)|K0).(|[x,y]|)=((cn-FanMorphS)).(|[x,y]|) by A30,FUNCT_1:49 .= |[ |.p99.|* ((p99`1/|.p99.|-cn)/(1+cn)), |.p99.|*( -sqrt(1-((p99`1/ |.p99.|-cn)/(1+cn))^2))]| by A3,A32,Th115 .=|[s,r]| by A18,A30,A31,A33,A34; hence thesis by A3; end; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<=0 & q<>0.TOP-REAL 2 proof let q be Point of TOP-REAL 2; A35: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; assume q in the carrier of (TOP-REAL 2)|K1; then ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|<= cn & p3`2<=0 & p3<>0.TOP-REAL 2 by A3,A35; hence thesis; end; then f2 is continuous by A3,A18,Th117; hence thesis by A7,A9,A28,A29,JGRAPH_2:35; end; theorem Th122: for cn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p `1>=(cn)*(|.p.|) & p`2<=0} holds K03 is closed proof defpred Q[Point of TOP-REAL 2] means $1`2<=0; let sn be Real, K003 be Subset of TOP-REAL 2; assume A1: K003={p: p`1>=(sn)*(|.p.|) & p`2<=0}; reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; defpred P[Point of TOP-REAL 2] means ($1`1>=sn*|.$1.|); reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1 where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10; K1 is closed & KX is closed by Lm8,JORDAN6:8; hence thesis by A1,A2,TOPS_1:8; end; theorem Th123: for cn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p `1<=(cn)*(|.p.|) & p`2<=0} holds K03 is closed proof defpred Q[Point of TOP-REAL 2] means $1`2<=0; let sn be Real, K003 be Subset of TOP-REAL 2; assume A1: K003={p: p`1<=(sn)*(|.p.|) & p`2<=0}; reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; defpred P[Point of TOP-REAL 2] means ($1`1<=sn*|.$1.|); reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of TOP-REAL 2 from JGRAPH_2:sch 1; A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1 where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10; K1 is closed & KX is closed by Lm10,JORDAN6:8; hence thesis by A1,A2,TOPS_1:8; end; theorem Th124: for cn being Real, K0,B0 being Subset of TOP-REAL 2, f being Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1 0.TOP-REAL 2} holds f is continuous proof let cn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set sn=-sqrt(1-cn^2); set p0=|[cn,sn]|; A1: p0`2=sn by EUCLID:52; p0`1=cn by EUCLID:52; then A2: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1; assume A3: -1 0.TOP-REAL 2}; then cn^2<1^2 by SQUARE_1:50; then A4: 1-cn^2>0 by XREAL_1:50; then A5: -sn>0 by SQUARE_1:25; A6: now assume p0=0.TOP-REAL 2; then --sn=-0 by EUCLID:52,JGRAPH_2:3; hence contradiction by A4,SQUARE_1:25; end; then p0 in K0 by A3,A1,A5; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; (-sn)^2=1-cn^2 by A4,SQUARE_1:def 2; then A7: p0`1/|.p0.|=cn by A2,EUCLID:52,SQUARE_1:18; then A8: p0 in {p: p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2} by A1,A6,A5; not p0 in {0.TOP-REAL 2} by A6,TARSKI:def 1; then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A3,XBOOLE_0:def 5; K1 c= D proof let x be object; assume A9: x in K1; then ex p6 being Point of TOP-REAL 2 st p6=x & p6`2<=0 & p6 <>0.TOP-REAL 2 by A3; then not x in {0.TOP-REAL 2} by TARSKI:def 1; hence thesis by A3,A9,XBOOLE_0:def 5; end; then D=K1 \/ D by XBOOLE_1:12; then A10: (TOP-REAL 2)|K1 is SubSpace of (TOP-REAL 2)|D by TOPMETR:4; A11: {p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} c= K1 proof let x be object; assume x in {p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2}; then ex p st p=x & p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2; hence thesis by A3; end; A12: {p: p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2} c= K1 proof let x be object; assume x in {p: p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2}; then ex p st p=x & p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2; hence thesis by A3; end; then reconsider K00={p: p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2} as non empty Subset of ((TOP-REAL 2)|K1) by A8,PRE_TOPC:8; the carrier of (TOP-REAL 2)|D = D by PRE_TOPC:8; then A13: rng (f|K00) c=D; p0 in {p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} by A1,A6,A5,A7; then reconsider K11={p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} as non empty Subset of ((TOP-REAL 2)|K1) by A11,PRE_TOPC:8; the carrier of (TOP-REAL 2)|D = D by PRE_TOPC:8; then A14: rng (f|K11) c=D; the carrier of (TOP-REAL 2)|B0=the carrier of (TOP-REAL 2)|D; then A15: dom f=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1 .=K1 by PRE_TOPC:8; then dom (f|K00)=K00 by A12,RELAT_1:62 .= the carrier of ((TOP-REAL 2)|K1)|K00 by PRE_TOPC:8; then reconsider f1=f|K00 as Function of ((TOP-REAL 2)|K1)|K00,(TOP-REAL 2)|D by A13,FUNCT_2:2 ; dom (f|K11)=K11 by A11,A15,RELAT_1:62 .= the carrier of ((TOP-REAL 2)|K1)|K11 by PRE_TOPC:8; then reconsider f2=f|K11 as Function of ((TOP-REAL 2)|K1)|K11,(TOP-REAL 2)|D by A14,FUNCT_2:2 ; A16: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; defpred P[Point of TOP-REAL 2] means $1`1/|.$1.|>=cn & $1`2<=0 & $1<>0. TOP-REAL 2; A17: dom f2=the carrier of ((TOP-REAL 2)|K1)|K11 by FUNCT_2:def 1 .=K11 by PRE_TOPC:8; {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; then reconsider K001={p: p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2} as non empty Subset of TOP-REAL 2 by A8; A18: the carrier of (TOP-REAL 2)|K1 =K1 by PRE_TOPC:8; defpred P[Point of TOP-REAL 2] means $1`1>=cn*(|.$1.|) & $1`2<=0; {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; then reconsider K003={p: p`1>=(cn)*(|.p.|) & p`2<=0} as Subset of TOP-REAL 2; defpred P[Point of TOP-REAL 2] means $1`1/|.$1.|<=cn & $1`2<=0 & $1<>0. TOP-REAL 2; A19: {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; A20: rng ((cn-FanMorphS)|K001) c= K1 proof let y be object; assume y in rng ((cn-FanMorphS)|K001); then consider x being object such that A21: x in dom ((cn-FanMorphS)|K001) and A22: y=((cn-FanMorphS)|K001).x by FUNCT_1:def 3; x in dom (cn-FanMorphS) by A21,RELAT_1:57; then reconsider q=x as Point of TOP-REAL 2; A23: y=(cn-FanMorphS).q by A21,A22,FUNCT_1:47; dom ((cn-FanMorphS)|K001)=(dom (cn-FanMorphS))/\ K001 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K001 by FUNCT_2:def 1 .=K001 by XBOOLE_1:28; then A24: ex p2 being Point of TOP-REAL 2 st p2=q & p2`1/|.p2.|>= cn & p2`2<=0 & p2<>0.TOP-REAL 2 by A21; then A25: (q`1/|.q.|-cn)>= 0 by XREAL_1:48; |.q.|<>0 by A24,TOPRNS_1:24; then A26: (|.q.|)^2>0^2 by SQUARE_1:12; set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn) /(1-cn))^2))]|; A27: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52; A28: 1-cn>0 by A3,XREAL_1:149; 0<=(q`2)^2 by XREAL_1:63; then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`1)^2 <= (|.q.|)^2 by JGRAPH_3:1; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A26,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`1/|.q.| by SQUARE_1:51; then 1-cn>=q`1/|.q.|-cn by XREAL_1:9; then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A28,XREAL_1:72; then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A28,XCMPLX_1:197; then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A28,A25,SQUARE_1:49; then A29: 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then A30: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187; sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>=0 by A29,SQUARE_1:def 2; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)>=0; then A31: sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76; A32: q4`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; then A33: (q4`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A30,SQUARE_1:def 2; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A27,A33; then A34: q4<>0.TOP-REAL 2 by A26,TOPRNS_1:23; cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-(( q`1/|.q.|-cn)/(1-cn))^2))]| by A3,A24,Th115; hence thesis by A3,A23,A32,A31,A34; end; A35: dom (cn-FanMorphS)=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then dom ((cn-FanMorphS)|K001)=K001 by RELAT_1:62 .= the carrier of (TOP-REAL 2)|K001 by PRE_TOPC:8; then reconsider f3=(cn-FanMorphS)|K001 as Function of (TOP-REAL 2)|K001,(TOP-REAL 2)|K1 by A18,A20,FUNCT_2:2; A36: K003 is closed by Th122; defpred P[Point of TOP-REAL 2] means $1`1<=(cn)*(|.$1.|) & $1`2<=0; {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7; then reconsider K004={p: p`1<=(cn)*(|.p.|) & p`2<=0} as Subset of (TOP-REAL 2); A37: K004 /\ K1 c= K11 proof let x be object; assume A38: x in K004 /\ K1; then x in K004 by XBOOLE_0:def 4; then consider q1 being Point of TOP-REAL 2 such that A39: q1=x and A40: q1`1<=(cn)*(|.q1.|) and q1`2<=0; x in K1 by A38,XBOOLE_0:def 4; then A41: ex q2 being Point of TOP-REAL 2 st q2=x & q2`2<=0 & q2 <>0.TOP-REAL 2 by A3; q1`1/|.q1.|<=(cn)*(|.q1.|)/|.q1.| by A40,XREAL_1:72; then q1`1/|.q1.|<=(cn) by A39,A41,TOPRNS_1:24,XCMPLX_1:89; hence thesis by A39,A41; end; A42: K004 is closed by Th123; the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then ((TOP-REAL 2)|K1)|K00=(TOP-REAL 2)|K001 & f1= f3 by A3,FUNCT_1:51 ,GOBOARD9:2; then A43: f1 is continuous by A3,A10,Th120,PRE_TOPC:26; A44: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5; p0 in {p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} by A1,A6,A5,A7; then reconsider K111={p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} as non empty Subset of TOP-REAL 2 by A19; A45: rng ((cn-FanMorphS)|K111) c= K1 proof let y be object; assume y in rng ((cn-FanMorphS)|K111); then consider x being object such that A46: x in dom ((cn-FanMorphS)|K111) and A47: y=((cn-FanMorphS)|K111).x by FUNCT_1:def 3; x in dom (cn-FanMorphS) by A46,RELAT_1:57; then reconsider q=x as Point of TOP-REAL 2; A48: y=(cn-FanMorphS).q by A46,A47,FUNCT_1:47; dom ((cn-FanMorphS)|K111)=(dom (cn-FanMorphS))/\ K111 by RELAT_1:61 .=(the carrier of TOP-REAL 2)/\ K111 by FUNCT_2:def 1 .=K111 by XBOOLE_1:28; then A49: ex p2 being Point of TOP-REAL 2 st p2=q & p2`1/|.p2.|<= cn & p2`2<=0 & p2<>0.TOP-REAL 2 by A46; then A50: (q`1/|.q.|-cn)<=0 by XREAL_1:47; |.q.|<>0 by A49,TOPRNS_1:24; then A51: (|.q.|)^2>0^2 by SQUARE_1:12; set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn) /(1+cn))^2))]|; A52: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52; A53: 1+cn>0 by A3,XREAL_1:148; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1 ,XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A51,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`1/|.q.| by SQUARE_1:51; then -1-cn<=q`1/|.q.|-cn by XREAL_1:9; then (-(1+cn))/(1+cn)<=(( q`1/|.q.|-cn))/(1+cn) by A53,XREAL_1:72; then -1<=(( q`1/|.q.|-cn))/(1+cn) by A53,XCMPLX_1:197; then A54: ( (q`1/|.q.|-cn) /(1+cn))^2<=1^2 by A53,A50,SQUARE_1:49; then A55: 1-((q`1/|.q.|-cn)/(1+cn))^2>=0 by XREAL_1:48; 1-(-((q`1/|.q.|-cn)/(1+cn)))^2>=0 by A54,XREAL_1:48; then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187; then sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(q`1/|.q.|-cn)^2/(1+cn)^2)>=0; then A56: sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76; A57: q4`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52; then A58: (q4`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A55,SQUARE_1:def 2; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A52,A58; then A59: q4<>0.TOP-REAL 2 by A51,TOPRNS_1:23; cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-(( q`1/|.q.|-cn)/(1+cn))^2))]| by A3,A49,Th115; hence thesis by A3,A48,A57,A56,A59; end; dom ((cn-FanMorphS)|K111)=K111 by A35,RELAT_1:62 .= the carrier of (TOP-REAL 2)|K111 by PRE_TOPC:8; then reconsider f4=(cn-FanMorphS)|K111 as Function of (TOP-REAL 2)|K111,(TOP-REAL 2)|K1 by A16,A45,FUNCT_2:2; the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8; then ((TOP-REAL 2)|K1)|K11=(TOP-REAL 2)|K111 & f2= f4 by A3,FUNCT_1:51 ,GOBOARD9:2; then A60: f2 is continuous by A3,A10,Th121,PRE_TOPC:26; set T1= ((TOP-REAL 2)|K1)|K00,T2=((TOP-REAL 2)|K1)|K11; A61: [#](((TOP-REAL 2)|K1)|K11)=K11 by PRE_TOPC:def 5; K11 c= K004 /\ K1 proof let x be object; assume x in K11; then consider p such that A62: p=x and A63: p`1/|.p.|<=cn and A64: p`2<=0 and A65: p<>0.TOP-REAL 2; p`1/|.p.|*|.p.|<=(cn)*(|.p.|) by A63,XREAL_1:64; then p`1<=(cn)*(|.p.|) by A65,TOPRNS_1:24,XCMPLX_1:87; then A66: x in K004 by A62,A64; x in K1 by A3,A62,A64,A65; hence thesis by A66,XBOOLE_0:def 4; end; then K11=K004 /\ [#]((TOP-REAL 2)|K1) by A44,A37,XBOOLE_0:def 10; then A67: K11 is closed by A42,PRE_TOPC:13; A68: K003 /\ K1 c= K00 proof let x be object; assume A69: x in K003 /\ K1; then x in K003 by XBOOLE_0:def 4; then consider q1 being Point of TOP-REAL 2 such that A70: q1=x and A71: q1`1>=(cn)*(|.q1.|) and q1`2<=0; x in K1 by A69,XBOOLE_0:def 4; then A72: ex q2 being Point of TOP-REAL 2 st q2=x & q2`2<=0 & q2 <>0.TOP-REAL 2 by A3; q1`1/|.q1.|>=(cn)*(|.q1.|)/|.q1.| by A71,XREAL_1:72; then q1`1/|.q1.|>=(cn) by A70,A72,TOPRNS_1:24,XCMPLX_1:89; hence thesis by A70,A72; end; A73: the carrier of ((TOP-REAL 2)|K1)=K0 by PRE_TOPC:8; A74: D<>{}; A75: [#](((TOP-REAL 2)|K1)|K00)=K00 by PRE_TOPC:def 5; A76: for p being object st p in ([#]T1)/\([#]T2) holds f1.p = f2.p proof let p be object; assume A77: p in ([#]T1)/\([#]T2); then p in K00 by A75,XBOOLE_0:def 4; hence f1.p=f.p by FUNCT_1:49 .=f2.p by A61,A77,FUNCT_1:49; end; K00 c= K003 /\ K1 proof let x be object; assume x in K00; then consider p such that A78: p=x and A79: p`1/|.p.|>=cn and A80: p`2<=0 and A81: p<>0.TOP-REAL 2; p`1/|.p.|*|.p.|>=(cn)*(|.p.|) by A79,XREAL_1:64; then p`1>=(cn)*(|.p.|) by A81,TOPRNS_1:24,XCMPLX_1:87; then A82: x in K003 by A78,A80; x in K1 by A3,A78,A80,A81; hence thesis by A82,XBOOLE_0:def 4; end; then K00=K003 /\ [#]((TOP-REAL 2)|K1) by A44,A68,XBOOLE_0:def 10; then A83: K00 is closed by A36,PRE_TOPC:13; A84: K1 c= K00 \/ K11 proof let x be object; assume x in K1; then consider p such that A85: p=x & p`2<=0 & p<>0.TOP-REAL 2 by A3; per cases; suppose p`1/|.p.|>=cn; then x in K00 by A85; hence thesis by XBOOLE_0:def 3; end; suppose p`1/|.p.| =0 & p<>0.TOP-REAL 2} holds f is continuous proof let cn be Real,K0,B0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)| K0,(TOP-REAL 2)|B0; set sn=sqrt(1-cn^2); set p0=|[cn,sn]|; A1: p0`2=sn by EUCLID:52; assume A2: -1 =0 & p<>0.TOP-REAL 2}; then cn^2<1^2 by SQUARE_1:50; then A3: 1-cn^2>0 by XREAL_1:50; then sn>0 by SQUARE_1:25; then p0 in K0 by A2,A1,JGRAPH_2:3; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; p0`2>0 by A1,A3,SQUARE_1:25; then not p0 in {0.TOP-REAL 2} by JGRAPH_2:3,TARSKI:def 1; then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A2,XBOOLE_0:def 5; A4: K1 c= D proof let x be object; assume x in K1; then consider p2 being Point of TOP-REAL 2 such that A5: p2=x and p2`2>=0 and A6: p2<>0.TOP-REAL 2 by A2; not p2 in {0.TOP-REAL 2} by A6,TARSKI:def 1; hence thesis by A2,A5,XBOOLE_0:def 5; end; for p being Point of (TOP-REAL 2)|K1,V being Subset of (TOP-REAL 2)|D st f.p in V & V is open holds ex W being Subset of (TOP-REAL 2)|K1 st p in W & W is open & f.:W c= V proof let p be Point of (TOP-REAL 2)|K1,V be Subset of (TOP-REAL 2)|D; assume that A7: f.p in V and A8: V is open; consider V2 being Subset of TOP-REAL 2 such that A9: V2 is open and A10: V2 /\ [#]((TOP-REAL 2)|D)=V by A8,TOPS_2:24; reconsider W2=V2 /\ [#]((TOP-REAL 2)|K1) as Subset of (TOP-REAL 2)| K1; A11: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5; then A12: f.p=(cn-FanMorphS).p by A2,FUNCT_1:49; A13: f.:W2 c= V proof let y be object; assume y in f.:W2; then consider x being object such that A14: x in dom f and A15: x in W2 and A16: y=f.x by FUNCT_1:def 6; f is Function of (TOP-REAL 2)|K1,(TOP-REAL 2)|D; then dom f= K1 by A11,FUNCT_2:def 1; then consider p4 being Point of TOP-REAL 2 such that A17: x=p4 and A18: p4`2>=0 and p4<>0.TOP-REAL 2 by A2,A14; A19: p4 in V2 by A15,A17,XBOOLE_0:def 4; p4 in [#]((TOP-REAL 2)|K1) by A14,A17; then p4 in D by A4,A11; then A20: p4 in [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5; f.p4=(cn-FanMorphS).p4 by A2,A11,A14,A17,FUNCT_1:49 .=p4 by A18,Th113; hence thesis by A10,A16,A17,A19,A20,XBOOLE_0:def 4; end; p in the carrier of (TOP-REAL 2)|K1; then consider q being Point of TOP-REAL 2 such that A21: q=p and A22: q`2>=0 and q <>0.TOP-REAL 2 by A2,A11; (cn-FanMorphS).q=q by A22,Th113; then p in V2 by A7,A10,A12,A21,XBOOLE_0:def 4; then A23: p in W2 by XBOOLE_0:def 4; W2 is open by A9,TOPS_2:24; hence thesis by A23,A13; end; hence thesis by JGRAPH_2:10; end; theorem Th126: for cn being Real, B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2)|B0,f being Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2 )|B0) st -1 0.TOP-REAL 2} holds f is continuous proof let cn be Real, B0 be Subset of TOP-REAL 2, K0 be Subset of (TOP-REAL 2)|B0,f be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0); the carrier of (TOP-REAL 2)|B0= B0 by PRE_TOPC:8; then reconsider K1=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1; assume A1: -1 0.TOP-REAL 2 }; K0 c= B0 proof let x be object; assume x in K0; then A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2<=0 & p8 <>0.TOP-REAL 2 by A1; then not x in {0.TOP-REAL 2} by TARSKI:def 1; hence thesis by A1,A2,XBOOLE_0:def 5; end; then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7; hence thesis by A1,Th124; end; theorem Th127: for cn being Real, B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2)|B0,f being Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2 )|B0) st -1 =0 & p<>0.TOP-REAL 2} holds f is continuous proof let cn be Real, B0 be Subset of TOP-REAL 2, K0 be Subset of (TOP-REAL 2)|B0,f be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0); the carrier of (TOP-REAL 2)|B0=B0 by PRE_TOPC:8; then reconsider K1=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1; assume A1: -1 =0 & p<>0.TOP-REAL 2}; K0 c= B0 proof let x be object; assume x in K0; then A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2>=0 & p8 <>0.TOP-REAL 2 by A1; then not x in {0.TOP-REAL 2} by TARSKI:def 1; hence thesis by A1,A2,XBOOLE_0:def 5; end; then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7; hence thesis by A1,Th125; end; theorem Th128: for cn being Real,p being Point of TOP-REAL 2 holds |.(cn -FanMorphS).p.|=|.p.| proof let cn be Real,p be Point of TOP-REAL 2; set f=cn-FanMorphS; set z=f.p; set q=p; reconsider qz=z as Point of TOP-REAL 2; per cases; suppose A1: q`1/|.q.|>=cn & q`2<0; then A2: (cn-FanMorphS).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-( (q`1/|.q.|-cn)/(1-cn))^2))]| by Th113; then A3: qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; A4: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by A2,EUCLID:52; A5: (q`1/|.q.|-cn)>=0 by A1,XREAL_1:48; A6: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; |.q.|<>0 by A1,JGRAPH_2:3,TOPRNS_1:24; then A7: (|.q.|)^2>0 by SQUARE_1:12; 0<=(q`2)^2 by XREAL_1:63; then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A6,XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A7,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`1/|.q.| by SQUARE_1:51; then A8: 1-cn>=q`1/|.q.|-cn by XREAL_1:9; per cases; suppose A9: 1-cn=0; A10: ((q`1/|.q.|-cn)/(1-cn))=(q`1/|.q.|-cn)*(1-cn)" by XCMPLX_0:def 9 .= (q`1/|.q.|-cn)*0 by A9 .=0; then 1-((q`1/|.q.|-cn)/(1-cn))^2=1; then (cn-FanMorphS).q= |[ |.q.|*0,|.q.|*(-1)]| by A1,A10,Th113, SQUARE_1:18 .=|[0,-(|.q.|)]|; then ((cn-FanMorphS).q)`2=(- |.q.|) & ((cn-FanMorphS).q)`1=0 by EUCLID:52 ; then |.(cn-FanMorphS).p.|=sqrt((-(|.q.|))^2+0^2) by JGRAPH_3:1 .=sqrt(((|.q.|))^2) .=|.q.| by SQUARE_1:22; hence thesis; end; suppose A11: 1-cn<>0; per cases by A11; suppose A12: 1-cn>0; -(1-cn)<= -( q`1/|.q.|-cn) by A8,XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A12,XREAL_1:72; then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A12,XCMPLX_1:197; then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A5,A12,SQUARE_1:49; then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then A13: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187; A14: (qz`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2 by A3 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A13,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A4,A14; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; hence thesis by SQUARE_1:22; end; suppose A15: 1-cn<0; 0+(q`1)^2<(q`1)^2+(q`2)^2 by A1,SQUARE_1:12,XREAL_1:8; then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,A6,XREAL_1:74; then (q`1)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60; then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76; then A16: 1 > q`1/|.p.| by SQUARE_1:52; q`1/|.q.|-cn>=0 by A1,XREAL_1:48; hence thesis by A15,A16,XREAL_1:9; end; end; end; suppose A17: q`1/|.q.| 0 by JGRAPH_2:3,TOPRNS_1:24; then A18: (|.q.|)^2>0 by SQUARE_1:12; A19: (q`1/|.q.|-cn)<0 by A17,XREAL_1:49; A20: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; 0<=(q`2)^2 by XREAL_1:63; then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A20,XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A18,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`1/|.q.| by SQUARE_1:51; then A21: -1-cn<=q`1/|.q.|-cn by XREAL_1:9; A22: (cn-FanMorphS).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1- ((q`1/|.q.|-cn)/(1+cn))^2))]| by A17,Th114; then A23: qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52; A24: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by A22,EUCLID:52; per cases; suppose A25: 1+cn=0; ((q`1/|.q.|-cn)/(1+cn))=(q`1/|.q.|-cn)*(1+cn)" by XCMPLX_0:def 9 .= (q`1/|.q.|-cn)*0 by A25 .=0; then ((cn-FanMorphS).q)`2=-(|.q.|) & ((cn-FanMorphS).q)`1=0 by A22, EUCLID:52,SQUARE_1:18; then |.(cn-FanMorphS).p.|=sqrt((-(|.q.|))^2+0^2) by JGRAPH_3:1 .=sqrt(((|.q.|))^2) .=|.q.| by SQUARE_1:22; hence thesis; end; suppose A26: 1+cn<>0; per cases by A26; suppose A27: 1+cn>0; then (-(1+cn))/(1+cn)<=(( q`1/|.q.|-cn))/(1+cn) by A21,XREAL_1:72; then -1<=(( q`1/|.q.|-cn))/(1+cn) by A27,XCMPLX_1:197; then ( (q`1/|.q.|-cn) /(1+cn))^2<=1^2 by A19,A27,SQUARE_1:49; then A28: 1-(((q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48; A29: (qz`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2 by A23 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A28,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A24,A29; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; hence thesis by SQUARE_1:22; end; suppose A30: 1+cn<0; 0+(q`1)^2<(q`1)^2+(q`2)^2 by A17,SQUARE_1:12,XREAL_1:8; then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A18,A20,XREAL_1:74; then (q`1)^2/(|.q.|)^2 < 1 by A18,XCMPLX_1:60; then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76; then -1 < q`1/|.p.| by SQUARE_1:52; then A31: q`1/|.q.|-cn>-1-cn by XREAL_1:9; -(1+cn)>-0 by A30,XREAL_1:24; hence thesis by A17,A31,XREAL_1:49; end; end; end; suppose q`2>=0; hence thesis by Th113; end; end; theorem Th129: for cn being Real,x,K0 being set st -1 0.TOP-REAL 2} holds (cn-FanMorphS).x in K0 proof let cn be Real,x,K0 be set; assume A1: -1 0.TOP-REAL 2}; then consider p such that A2: p=x and A3: p`2<=0 and A4: p<>0.TOP-REAL 2; A5: now assume |.p.|<=0; then |.p.|=0; hence contradiction by A4,TOPRNS_1:24; end; then A6: (|.p.|)^2>0 by SQUARE_1:12; per cases; suppose A7: p`1/|.p.|<=cn; reconsider p9= (cn-FanMorphS).p as Point of TOP-REAL 2; (cn-FanMorphS).p= |[ |.p.|*((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-( (p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A3,A4,A7,Th115; then A8: p9`2=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52; A9: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; A10: 1+cn>0 by A1,XREAL_1:148; per cases; suppose p`2=0; hence thesis by A1,A2,Th113; end; suppose p`2<>0; then 0+(p`1)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8; then (p`1)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A9,XREAL_1:74; then (p`1)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60; then ((p`1)/|.p.|)^2 < 1 by XCMPLX_1:76; then -1 < p`1/|.p.| by SQUARE_1:52; then -1-cn< p`1/|.p.|-cn by XREAL_1:9; then (-1)*(1+cn)/(1+cn)< (p`1/|.p.|-cn)/(1+cn) by A10,XREAL_1:74; then A11: -1< (p`1/|.p.|-cn)/(1+cn) by A10,XCMPLX_1:89; p`1/|.p.|-cn<=0 by A7,XREAL_1:47; then 1^2> ((p`1/|.p.|-cn)/(1+cn))^2 by A10,A11,SQUARE_1:50; then 1-((p`1/|.p.|-cn)/(1+cn))^2>0 by XREAL_1:50; then --sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)>0 by SQUARE_1:25; then -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)<0; then |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))<0 by A5,XREAL_1:132; hence thesis by A1,A2,A8,JGRAPH_2:3; end; end; suppose A12: p`1/|.p.|>cn; reconsider p9= (cn-FanMorphS).p as Point of TOP-REAL 2; (cn-FanMorphS).p= |[ |.p.|*((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-( (p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A3,A4,A12,Th115; then A13: p9`2=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52; A14: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; A15: 1-cn>0 by A1,XREAL_1:149; per cases; suppose p`2=0; hence thesis by A1,A2,Th113; end; suppose p`2<>0; then 0+(p`1)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8; then (p`1)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A14,XREAL_1:74; then (p`1)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60; then ((p`1)/|.p.|)^2 < 1 by XCMPLX_1:76; then p`1/|.p.|<1 by SQUARE_1:52; then (p`1/|.p.|-cn)<1-cn by XREAL_1:9; then (p`1/|.p.|-cn)/(1-cn)<(1-cn)/(1-cn) by A15,XREAL_1:74; then A16: (p`1/|.p.|-cn)/(1-cn)<1 by A15,XCMPLX_1:60; -(1-cn)< -0 & p`1/|.p.|-cn>=cn-cn by A12,A15,XREAL_1:9,24; then (-1)*(1-cn)/(1-cn)< (p`1/|.p.|-cn)/(1-cn) by A15,XREAL_1:74; then -1< (p`1/|.p.|-cn)/(1-cn) by A15,XCMPLX_1:89; then 1^2> ((p`1/|.p.|-cn)/(1-cn))^2 by A16,SQUARE_1:50; then 1-((p`1/|.p.|-cn)/(1-cn))^2>0 by XREAL_1:50; then --sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)>0 by SQUARE_1:25; then -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)<0; then p9`2<0 by A5,A13,XREAL_1:132; hence thesis by A1,A2,JGRAPH_2:3; end; end; end; theorem Th130: for cn being Real,x,K0 being set st -1 =0 & p<>0.TOP-REAL 2} holds (cn-FanMorphS).x in K0 proof let cn be Real,x,K0 be set; assume A1: -1 =0 & p<>0.TOP-REAL 2}; then ex p st p=x & p`2>=0 & p<>0.TOP-REAL 2; hence thesis by A1,Th113; end; theorem Th131: for cn being Real, D being non empty Subset of TOP-REAL 2 st -1 0.TOP-REAL 2 by EUCLID:52,JGRAPH_2:3; then A5: |[0,-1]| in {p where p is Point of TOP-REAL 2: p`2<=0 & p<>0.TOP-REAL 2}; Y1`2=1 by EUCLID:52; then A6: Y1 in {p where p is Point of TOP-REAL 2: p`2>=0 & p<>0.TOP-REAL 2} by JGRAPH_2:3; A7: D =B0` by A2 .=(NonZero TOP-REAL 2) by SUBSET_1:def 4; {p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from InclSub(A7); then reconsider K0={p:p`2<=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL 2)|D by A5; A8: K0=the carrier of ((TOP-REAL 2)|D)|K0 by PRE_TOPC:8; defpred P[Point of TOP-REAL 2] means $1`2>=0; {p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from InclSub(A7); then reconsider K1={p: p`2>=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL 2)|D by A6; A9: K0 is closed & K1 is closed by A7,Th62,Th63; A10: the carrier of ((TOP-REAL 2)|D) =D by PRE_TOPC:8; A11: rng ((cn-FanMorphS)|K0) c= the carrier of ((TOP-REAL 2)|D)|K0 proof let y be object; assume y in rng ((cn-FanMorphS)|K0); then consider x being object such that A12: x in dom ((cn-FanMorphS)|K0) and A13: y=((cn-FanMorphS)|K0).x by FUNCT_1:def 3; x in (dom (cn-FanMorphS)) /\ K0 by A12,RELAT_1:61; then A14: x in K0 by XBOOLE_0:def 4; K0 c= the carrier of TOP-REAL 2 by A10,XBOOLE_1:1; then reconsider p=x as Point of TOP-REAL 2 by A14; (cn-FanMorphS).p=y by A13,A14,FUNCT_1:49; then y in K0 by A1,A14,Th129; hence thesis by PRE_TOPC:8; end; A15: K0 c= (the carrier of TOP-REAL 2) proof let z be object; assume z in K0; then ex p8 being Point of TOP-REAL 2 st p8=z & p8`2<=0 & p8 <>0.TOP-REAL 2; hence thesis; end; dom ((cn-FanMorphS)|K0)= dom ((cn-FanMorphS)) /\ K0 by RELAT_1:61 .=((the carrier of TOP-REAL 2)) /\ K0 by FUNCT_2:def 1 .=K0 by A15,XBOOLE_1:28; then reconsider f=(cn-FanMorphS)|K0 as Function of ((TOP-REAL 2)|D)|K0, (( TOP-REAL 2)|D) by A8,A11,FUNCT_2:2,XBOOLE_1:1; A16: K1=the carrier of ((TOP-REAL 2)|D)|K1 by PRE_TOPC:8; A17: rng ((cn-FanMorphS)|K1) c= the carrier of ((TOP-REAL 2)|D)|K1 proof let y be object; assume y in rng ((cn-FanMorphS)|K1); then consider x being object such that A18: x in dom ((cn-FanMorphS)|K1) and A19: y=((cn-FanMorphS)|K1).x by FUNCT_1:def 3; x in (dom (cn-FanMorphS)) /\ K1 by A18,RELAT_1:61; then A20: x in K1 by XBOOLE_0:def 4; K1 c= the carrier of TOP-REAL 2 by A10,XBOOLE_1:1; then reconsider p=x as Point of TOP-REAL 2 by A20; (cn-FanMorphS).p=y by A19,A20,FUNCT_1:49; then y in K1 by A1,A20,Th130; hence thesis by PRE_TOPC:8; end; A21: K1 c= (the carrier of TOP-REAL 2) proof let z be object; assume z in K1; then ex p8 being Point of TOP-REAL 2 st p8=z & p8`2>=0 & p8 <>0.TOP-REAL 2; hence thesis; end; dom ((cn-FanMorphS)|K1)= dom ((cn-FanMorphS)) /\ K1 by RELAT_1:61 .=((the carrier of TOP-REAL 2)) /\ K1 by FUNCT_2:def 1 .=K1 by A21,XBOOLE_1:28; then reconsider g=(cn-FanMorphS)|K1 as Function of ((TOP-REAL 2)|D)|K1, (( TOP-REAL 2)|D) by A16,A17,FUNCT_2:2,XBOOLE_1:1; A22: K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5; A23: D c= K0 \/ K1 proof let x be object; assume A24: x in D; then reconsider px=x as Point of TOP-REAL 2; not x in {0.TOP-REAL 2} by A7,A24,XBOOLE_0:def 5; then px`2>=0 & px<>0.TOP-REAL 2 or px`2<=0 & px<>0.TOP-REAL 2 by TARSKI:def 1; then x in K1 or x in K0; hence thesis by XBOOLE_0:def 3; end; A25: dom f=K0 by A8,FUNCT_2:def 1; A26: K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5; A27: for x be object st x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1))) holds f.x = g.x proof let x be object; assume A28: x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1)) ); then x in K0 by A26,XBOOLE_0:def 4; then f.x=(cn-FanMorphS).x by FUNCT_1:49; hence thesis by A22,A28,FUNCT_1:49; end; D= [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5; then A29: ([#](((TOP-REAL 2)|D)|K0)) \/ ([#](((TOP-REAL 2)|D)|K1)) = [#](( TOP-REAL 2)|D) by A26,A22,A23,XBOOLE_0:def 10; A30: f is continuous & g is continuous by A1,A7,Th126,Th127; then consider h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that A31: h= f+*g and h is continuous by A26,A22,A29,A9,A27,JGRAPH_2:1; A32: dom g=K1 by A16,FUNCT_2:def 1; K0=[#](((TOP-REAL 2)|D)|K0) & K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5; then A33: f tolerates g by A27,A25,A32,PARTFUN1:def 4; A34: the carrier of ((TOP-REAL 2)|D) =(NonZero TOP-REAL 2) by A7,PRE_TOPC:8; A35: for x being object st x in dom h holds h.x=((cn-FanMorphS)|D).x proof let x be object; assume A36: x in dom h; then reconsider p=x as Point of TOP-REAL 2 by A34,XBOOLE_0:def 5; not x in {0.TOP-REAL 2} by A7,A3,A36,XBOOLE_0:def 5; then A37: x <>0.TOP-REAL 2 by TARSKI:def 1; per cases; suppose A38: x in K0; A39: (cn-FanMorphS)|D.p=(cn-FanMorphS).p by A3,A36,FUNCT_1:49 .=f.p by A38,FUNCT_1:49; h.p=(g+*f).p by A31,A33,FUNCT_4:34 .=f.p by A25,A38,FUNCT_4:13; hence thesis by A39; end; suppose not x in K0; then not p`2<=0 by A37; then A40: x in K1 by A37; (cn-FanMorphS)|D.p=(cn-FanMorphS).p by A3,A36,FUNCT_1:49 .=g.p by A40,FUNCT_1:49; hence thesis by A31,A32,A40,FUNCT_4:13; end; end; dom h=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1; then f+*g=(cn-FanMorphS)|D by A31,A4,A35,FUNCT_1:2; hence thesis by A26,A22,A29,A30,A9,A27,JGRAPH_2:1; end; theorem Th132: for cn being Real st -1 f.(0.TOP-REAL 2) proof let p be Point of (TOP-REAL 2)|D; A5: [#]((TOP-REAL 2)|D)=D by PRE_TOPC:def 5; then reconsider q=p as Point of TOP-REAL 2 by XBOOLE_0:def 5; not p in {0.TOP-REAL 2} by A5,XBOOLE_0:def 5; then A6: p<>0.TOP-REAL 2 by TARSKI:def 1; per cases; suppose A7: q`1/|.q.|>=cn & q`2<=0; set q9= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|- cn)/(1-cn))^2))]|; A8: q9`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52; A9: q9`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; now assume A10: q9=0.TOP-REAL 2; A11: |.q.|<>0^2 by A6,TOPRNS_1:24; then -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)= -sqrt(1-0) by A8,A10, JGRAPH_2:3,XCMPLX_1:6 .=-1 by SQUARE_1:18; hence contradiction by A9,A10,A11,JGRAPH_2:3,XCMPLX_1:6; end; hence thesis by A1,A2,A3,A6,A7,Th115; end; suppose A12: q`1/|.q.| 0^2 by A6,TOPRNS_1:24; then -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)= -sqrt(1-0) by A13,A15, JGRAPH_2:3,XCMPLX_1:6 .=-1 by SQUARE_1:18; hence contradiction by A14,A15,A16,JGRAPH_2:3,XCMPLX_1:6; end; hence thesis by A1,A2,A3,A6,A12,Th115; end; suppose q`2>0; then f.p=p by Th113; hence thesis by A6,Th113,JGRAPH_2:3; end; end; A17: for V being Subset of (TOP-REAL 2) st f.(0.TOP-REAL 2) in V & V is open ex W being Subset of (TOP-REAL 2) st 0.TOP-REAL 2 in W & W is open & f.:W c= V proof reconsider u0=0.TOP-REAL 2 as Point of Euclid 2 by EUCLID:67; let V be Subset of TOP-REAL 2; reconsider VV = V as Subset of TopSpaceMetr Euclid 2 by Lm11; assume that A18: f.(0.TOP-REAL 2) in V and A19: V is open; VV is open by A19,Lm11,PRE_TOPC:30; then consider r being Real such that A20: r>0 and A21: Ball(u0,r) c= V by A3,A18,TOPMETR:15; reconsider r as Real; the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8; then reconsider W1=Ball(u0,r) as Subset of TOP-REAL 2; A22: W1 is open by GOBOARD6:3; A23: f.:W1 c= W1 proof let z be object; assume z in f.:W1; then consider y being object such that A24: y in dom f and A25: y in W1 and A26: z=f.y by FUNCT_1:def 6; z in rng f by A24,A26,FUNCT_1:def 3; then reconsider qz=z as Point of TOP-REAL 2; reconsider pz=qz as Point of Euclid 2 by EUCLID:67; reconsider q=y as Point of TOP-REAL 2 by A24; reconsider qy=q as Point of Euclid 2 by EUCLID:67; dist(u0,qy) =0; hence thesis by A25,A26,Th113; end; suppose A28: q<>0.TOP-REAL 2 & q`1/|.q.|>=cn & q`2<=0; then A29: (q`1/|.q.|-cn)>= 0 by XREAL_1:48; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then A30: (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; A31: 1-cn>0 by A2,XREAL_1:149; |.q.|<>0 by A28,TOPRNS_1:24; then (|.q.|)^2>0 by SQUARE_1:12; then (q`1)^2/(|.q.|)^2 <= 1 by A30,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`1/|.q.| by SQUARE_1:51; then 1-cn>=q`1/|.q.|-cn by XREAL_1:9; then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A31,XREAL_1:72; then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A31,XCMPLX_1:197; then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A31,A29,SQUARE_1:49; then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then A32: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187; A33: (cn-FanMorphS).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)) , |.q.|*( - sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))]| by A1,A2,A28,Th115; then A34: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by A26,EUCLID:52; qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by A26,A33,EUCLID:52; then A35: (qz`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A32,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A34,A35; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; then A36: |.qz.|=|.q.| by SQUARE_1:22; |.- q.| 0.TOP-REAL 2 & q`1/|.q.| 0 by A1,XREAL_1:148; |.q.|<>0 by A37,TOPRNS_1:24; then (|.q.|)^2>0 by SQUARE_1:12; then (q`1)^2/(|.q.|)^2 <= 1 by A38,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`1/|.q.| by SQUARE_1:51; then --1>=-q`1/|.q.| by XREAL_1:24; then 1+cn>=-q`1/|.q.|+cn by XREAL_1:7; then A40: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A39,XREAL_1:185; (cn-q`1/|.q.|)>=0 by A37,XREAL_1:48; then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A39; then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A40,SQUARE_1:49; then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48; then A41: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187; A42: (cn-FanMorphS).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( - sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))]| by A1,A2,A37,Th115; then A43: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by A26,EUCLID:52; qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by A26,A42,EUCLID:52; then A44: (qz`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A41,SQUARE_1:def 2; (|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A43,A44; then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22; then A45: |.qz.|=|.q.| by SQUARE_1:22; |.- q.| 0 by A2,XREAL_1:149; per cases by JGRAPH_2:3; suppose A7: q`2>=0; then A8: (cn-FanMorphS).q=q by Th113; per cases by JGRAPH_2:3; suppose p`2>=0; hence thesis by A5,A8,Th113; end; suppose A9: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2<=0; then A10: |.p.|<>0 by TOPRNS_1:24; then A11: (|.p.|)^2>0 by SQUARE_1:12; A12: (p`1/|.p.|-cn)>=0 by A9,XREAL_1:48; A13: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1; 0<=(p`2)^2 by XREAL_1:63; then 0+(p`1)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7; then (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A13,XREAL_1:72; then (p`1)^2/(|.p.|)^2 <= 1 by A11,XCMPLX_1:60; then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76; then 1>=p`1/|.p.| by SQUARE_1:51; then 1-cn>=p`1/|.p.|-cn by XREAL_1:9; then -(1-cn)<= -( p`1/|.p.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XREAL_1:72; then A14: -1<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XCMPLX_1:197; A15: cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt( 1-((p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A2,A9,Th115; then A16: q`2=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by A5,A8,EUCLID:52; (p`1/|.p.|-cn)>= 0 by A9,XREAL_1:48; then ((-(p`1/|.p.|-cn))/(1-cn))^2<=1^2 by A6,A14,SQUARE_1:49; then A17: 1-((-(p`1/|.p.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then sqrt(1-((-(p`1/|.p.|-cn))/(1-cn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(p`1/|.p.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(p`1/|.p.|-cn)^2/(1-cn)^2)>=0; then sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76; then q`2=0 by A5,A7,A8,A15,EUCLID:52; then A18: -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)=-0 by A16,A10,XCMPLX_1:6; 1-(-((p`1/|.p.|-cn))/(1-cn))^2>=0 by A17,XCMPLX_1:187; then 1-((p`1/|.p.|-cn)/(1-cn))^2=0 by A18,SQUARE_1:24; then 1= (p`1/|.p.|-cn)/(1-cn) by A6,A12,SQUARE_1:18,22; then 1 *(1-cn)=(p`1/|.p.|-cn) by A6,XCMPLX_1:87; then 1 *|.p.|=p`1 by A9,TOPRNS_1:24,XCMPLX_1:87; then p`2=0 by A13,XCMPLX_1:6; hence thesis by A5,A8,Th113; end; suppose A19: p<>0.TOP-REAL 2 & p`1/|.p.| 0 by A19,TOPRNS_1:24; then A24: (|.p.|)^2>0 by SQUARE_1:12; A25: 1+cn>0 by A1,XREAL_1:148; A26: (p`1/|.p.|-cn)<=0 by A19,XREAL_1:47; then A27: -1<=(-( p`1/|.p.|-cn))/(1+cn) by A25; 0<=(p`2)^2 by XREAL_1:63; then 0+(p`1)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7; then (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A22,XREAL_1:72; then (p`1)^2/(|.p.|)^2 <= 1 by A24,XCMPLX_1:60; then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76; then (-((p`1)/|.p.|))^2 <= 1; then 1>= -p`1/|.p.| by SQUARE_1:51; then (1+cn)>= -p`1/|.p.|+cn by XREAL_1:7; then (-(p`1/|.p.|-cn))/(1+cn)<=1 by A25,XREAL_1:185; then ((-(p`1/|.p.|-cn))/(1+cn))^2<=1^2 by A27,SQUARE_1:49; then A28: 1-((-(p`1/|.p.|-cn))/(1+cn))^2>=0 by XREAL_1:48; then sqrt(1-((-(p`1/|.p.|-cn))/(1+cn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(p`1/|.p.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-((p`1/|.p.|-cn))^2/(1+cn)^2)>=0; then sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76; then q`2=0 by A5,A7,A8,A20,EUCLID:52; then A29: -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)=-0 by A21,A23,XCMPLX_1:6; 1-(-((p`1/|.p.|-cn))/(1+cn))^2>=0 by A28,XCMPLX_1:187; then 1-((p`1/|.p.|-cn)/(1+cn))^2=0 by A29,SQUARE_1:24; then 1=(-((p`1/|.p.|-cn)/(1+cn)))^2; then 1= -((p`1/|.p.|-cn)/(1+cn)) by A25,A26,SQUARE_1:18,22; then 1= ((-(p`1/|.p.|-cn))/(1+cn)) by XCMPLX_1:187; then 1 *(1+cn)=-(p`1/|.p.|-cn) by A25,XCMPLX_1:87; then 1+cn-cn=-p`1/|.p.|; then 1=(-p`1)/|.p.| by XCMPLX_1:187; then 1 *|.p.|=-p`1 by A19,TOPRNS_1:24,XCMPLX_1:87; then (p`1)^2-(p`1)^2 =(p`2)^2 by A22,XCMPLX_1:26; then p`2=0 by XCMPLX_1:6; hence thesis by A5,A8,Th113; end; end; suppose A30: q`1/|.q.|>=cn & q`2<=0 & q<>0.TOP-REAL 2; then |.q.|<>0 by TOPRNS_1:24; then A31: (|.q.|)^2>0 by SQUARE_1:12; set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|- cn)/(1-cn))^2))]|; A32: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52; A33: cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1- ((q`1/|.q.|-cn)/(1-cn))^2))]| by A1,A2,A30,Th115; per cases by JGRAPH_2:3; suppose A34: p`2>=0; then A35: (cn-FanMorphS).p=p by Th113; then A36: p`2=|.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by A5,A33,EUCLID:52; A37: (q`1/|.q.|-cn)>=0 by A30,XREAL_1:48; A38: 1-cn>0 by A2,XREAL_1:149; A39: |.q.|<>0 by A30,TOPRNS_1:24; then A40: (|.q.|)^2>0 by SQUARE_1:12; A41: (q`1/|.q.|-cn)>= 0 by A30,XREAL_1:48; A42: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; 0<=(q`2)^2 by XREAL_1:63; then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A42,XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A40,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`1/|.q.| by SQUARE_1:51; then 1-cn>=q`1/|.q.|-cn by XREAL_1:9; then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A38,XREAL_1:72; then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A38,XCMPLX_1:197; then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A38,A41,SQUARE_1:49; then A43: 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>=0 by SQUARE_1:def 2; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)>=0; then sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76; then p`2=0 by A5,A33,A34,A35,EUCLID:52; then A44: -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)=-0 by A36,A39,XCMPLX_1:6; 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by A43,XCMPLX_1:187; then 1-((q`1/|.q.|-cn)/(1-cn))^2=0 by A44,SQUARE_1:24; then 1= (q`1/|.q.|-cn)/(1-cn) by A38,A37,SQUARE_1:18,22; then 1 *(1-cn)=(q`1/|.q.|-cn) by A38,XCMPLX_1:87; then 1 *|.q.|=q`1 by A30,TOPRNS_1:24,XCMPLX_1:87; then q`2=0 by A42,XCMPLX_1:6; hence thesis by A5,A35,Th113; end; suppose A45: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2<=0; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A31,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then 1>=q`1/|.q.| by SQUARE_1:51; then 1-cn>=q`1/|.q.|-cn by XREAL_1:9; then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A6,XREAL_1:72; then A46: -1<=(-( q`1/|.q.|-cn))/(1-cn) by A6,XCMPLX_1:197; (q`1/|.q.|-cn)>= 0 by A30,XREAL_1:48; then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A6,A46,SQUARE_1:49; then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then A47: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187; q4`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; then A48: (q4`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A47,SQUARE_1:def 2; A49: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A49,A48; then A50: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22; then A51: |.q4.|=|.q.| by SQUARE_1:22; 0<=(p`2)^2 by XREAL_1:63; then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`1)^2<=(p`1)^2+(p`2)^2 by JGRAPH_3:1,XREAL_1:7; then A52: (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72; |.p.|<>0 by A45,TOPRNS_1:24; then (|.p.|)^2>0 by SQUARE_1:12; then (p`1)^2/(|.p.|)^2 <= 1 by A52,XCMPLX_1:60; then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76; then 1>=p`1/|.p.| by SQUARE_1:51; then 1-cn>=p`1/|.p.|-cn by XREAL_1:9; then -(1-cn)<= -( p`1/|.p.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XREAL_1:72; then A53: -1<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XCMPLX_1:197; (p`1/|.p.|-cn)>= 0 by A45,XREAL_1:48; then ((-(p`1/|.p.|-cn))/(1-cn))^2<=1^2 by A6,A53,SQUARE_1:49; then 1-((-(p`1/|.p.|-cn))/(1-cn))^2>=0 by XREAL_1:48; then A54: 1-(-((p`1/|.p.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187; set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/|.p.| -cn)/(1-cn))^2))]|; A55: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1-cn)) by EUCLID:52; p4`2= |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52; then A56: (p4`2)^2= (|.p.|)^2*(sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))^2 .= (|.p.|)^2*(1-((p`1/|.p.|-cn)/(1-cn))^2) by A54,SQUARE_1:def 2; (|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1 .=(|.p.|)^2 by A55,A56; then A57: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22; then A58: |.p4.|=|.p.| by SQUARE_1:22; A59: cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt (1-((p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A2,A45,Th115; then ((p`1/|.p.|-cn)/(1-cn)) =|.q.|* ((q`1/|.q.|-cn)/(1-cn))/|.p.| by A5,A33,A32,A45,A55,TOPRNS_1:24,XCMPLX_1:89; then (p`1/|.p.|-cn)/(1-cn)=(q`1/|.q.|-cn)/(1-cn) by A5,A33,A45,A59,A50 ,A57,TOPRNS_1:24,XCMPLX_1:89; then (p`1/|.p.|-cn)/(1-cn)*(1-cn)=q`1/|.q.|-cn by A6,XCMPLX_1:87; then p`1/|.p.|-cn=q`1/|.q.|-cn by A6,XCMPLX_1:87; then p`1/|.p.|*|.p.|=q`1 by A5,A33,A45,A59,A51,A58,TOPRNS_1:24 ,XCMPLX_1:87; then A60: p`1=q`1 by A45,TOPRNS_1:24,XCMPLX_1:87; |.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1; then (-p`2)^2=(q`2)^2 by A5,A33,A59,A51,A58,A60; then -p`2=sqrt((-q`2)^2) by A45,SQUARE_1:22; then A61: --p`2=--q`2 by A30,SQUARE_1:22; p=|[p`1,p`2]| by EUCLID:53; hence thesis by A60,A61,EUCLID:53; end; suppose A62: p<>0.TOP-REAL 2 & p`1/|.p.| =0 by A30,EUCLID:52 ,XREAL_1:48; A65: 1-cn>0 by A2,XREAL_1:149; cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt (1-((p`1/|.p .|-cn)/(1+cn))^2))]| & |.p.|<>0 by A1,A2,A62,Th115,TOPRNS_1:24; hence thesis by A5,A33,A32,A63,A64,A65,XREAL_1:132; end; end; suppose A66: q`1/|.q.| 0.TOP-REAL 2; then A67: |.q.|<>0 by TOPRNS_1:24; then A68: (|.q.|)^2>0 by SQUARE_1:12; set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|- cn)/(1+cn))^2))]|; A69: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52; A70: cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1 -((q`1/|.q.|-cn)/(1+cn))^2))]| by A1,A2,A66,Th115; per cases by JGRAPH_2:3; suppose A71: p`2>=0; then A72: (cn-FanMorphS).p=p by Th113; then A73: p`2=|.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by A5,A70,EUCLID:52; A74: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1; A75: 1+cn>0 by A1,XREAL_1:148; 0<=(q`2)^2 by XREAL_1:63; then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A74,XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A68,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then (-((q`1)/|.q.|))^2 <= 1; then 1>= -q`1/|.q.| by SQUARE_1:51; then (1+cn)>= -q`1/|.q.|+cn by XREAL_1:7; then A76: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A75,XREAL_1:185; A77: (q`1/|.q.|-cn)<=0 by A66,XREAL_1:47; then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A75; then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A76,SQUARE_1:49; then A78: 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48; then A79: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187; sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>=0 by A78,SQUARE_1:def 2; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76; then sqrt(1-((q`1/|.q.|-cn))^2/(1+cn)^2)>=0; then sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76; then p`2=0 by A5,A70,A71,A72,EUCLID:52; then -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)=-0 by A67,A73,XCMPLX_1:6; then 1-((q`1/|.q.|-cn)/(1+cn))^2=0 by A79,SQUARE_1:24; then 1=(-((q`1/|.q.|-cn)/(1+cn)))^2; then 1= -((q`1/|.q.|-cn)/(1+cn)) by A75,A77,SQUARE_1:18,22; then 1= ((-(q`1/|.q.|-cn))/(1+cn)) by XCMPLX_1:187; then 1 *(1+cn)=-(q`1/|.q.|-cn) by A75,XCMPLX_1:87; then 1+cn-cn=-q`1/|.q.|; then 1=(-q`1)/|.q.| by XCMPLX_1:187; then 1 *|.q.|=-q`1 by A66,TOPRNS_1:24,XCMPLX_1:87; then (q`1)^2-(q`1)^2 =(q`2)^2 by A74,XCMPLX_1:26; then q`2=0 by XCMPLX_1:6; hence thesis by A5,A72,Th113; end; suppose A80: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2<=0; set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/|.p.| -cn)/(1-cn))^2))]|; A81: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1-cn)) & |.q.|<>0 by A66,EUCLID:52 ,TOPRNS_1:24; q`1/|.q.|-cn<0 by A66,XREAL_1:49; then A82: ((q`1/|.q.|-cn)/(1+cn))<0 by A1,XREAL_1:141,148; A83: 1-cn>0 by A2,XREAL_1:149; cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt (1-((p`1/|.p .|-cn)/(1-cn))^2))]| & p`1/|.p.|-cn>=0 by A1,A2,A80,Th115, XREAL_1:48; hence thesis by A5,A70,A69,A82,A81,A83,XREAL_1:132; end; suppose A84: p<>0.TOP-REAL 2 & p`1/|.p.| 0 by A1,XREAL_1:148; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A68,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then -1<=q`1/|.q.| by SQUARE_1:51; then -1-cn<=q`1/|.q.|-cn by XREAL_1:9; then -(-1-cn)>= -(q`1/|.q.|-cn) by XREAL_1:24; then A87: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A86,XREAL_1:185; (q`1/|.q.|-cn)<=0 by A66,XREAL_1:47; then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A86; then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A87,SQUARE_1:49; then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48; then A88: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187; q4`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52; then A89: (q4`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2 .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A88,SQUARE_1:def 2; A90: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52; set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/|.p.| -cn)/(1+cn))^2))]|; A91: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1+cn)) by EUCLID:52; |.p.|<>0 by A84,TOPRNS_1:24; then (|.p.|)^2>0 by SQUARE_1:12; then (p`1)^2/(|.p.|)^2 <= 1 by A85,XCMPLX_1:60; then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76; then -1<=p`1/|.p.| by SQUARE_1:51; then -1-cn<=p`1/|.p.|-cn by XREAL_1:9; then -(-1-cn)>= -(p`1/|.p.|-cn) by XREAL_1:24; then A92: (-(p`1/|.p.|-cn))/(1+cn)<=1 by A86,XREAL_1:185; (p`1/|.p.|-cn)<=0 by A84,XREAL_1:47; then -1<=(-( p`1/|.p.|-cn))/(1+cn) by A86; then ((-(p`1/|.p.|-cn))/(1+cn))^2<=1^2 by A92,SQUARE_1:49; then 1-((-(p`1/|.p.|-cn))/(1+cn))^2>=0 by XREAL_1:48; then A93: 1-(-((p`1/|.p.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187; p4`2= |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52; then A94: (p4`2)^2= (|.p.|)^2*(sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))^2 .= (|.p.|)^2*(1-((p`1/|.p.|-cn)/(1+cn))^2) by A93,SQUARE_1:def 2; (|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1 .=(|.p.|)^2 by A91,A94; then A95: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22; then A96: |.p4.|=|.p.| by SQUARE_1:22; (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1 .=(|.q.|)^2 by A90,A89; then A97: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22; then A98: |.q4.|=|.q.| by SQUARE_1:22; A99: cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt (1-((p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A2,A84,Th115; then ((p`1/|.p.|-cn)/(1+cn)) =|.q.|* ((q`1/|.q.|-cn)/(1+cn))/|.p.| by A5,A70,A69,A84,A91,TOPRNS_1:24,XCMPLX_1:89; then (p`1/|.p.|-cn)/(1+cn)=(q`1/|.q.|-cn)/(1+cn) by A5,A70,A84,A99,A97 ,A95,TOPRNS_1:24,XCMPLX_1:89; then (p`1/|.p.|-cn)/(1+cn)*(1+cn)=q`1/|.q.|-cn by A86,XCMPLX_1:87; then p`1/|.p.|-cn=q`1/|.q.|-cn by A86,XCMPLX_1:87; then p`1/|.p.|*|.p.|=q`1 by A5,A70,A84,A99,A98,A96,TOPRNS_1:24 ,XCMPLX_1:87; then A100: p`1=q`1 by A84,TOPRNS_1:24,XCMPLX_1:87; |.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1; then (-p`2)^2=(q`2)^2 by A5,A70,A99,A98,A96,A100; then -p`2=sqrt((-q`2)^2) by A84,SQUARE_1:22; then A101: --p`2=--q`2 by A66,SQUARE_1:22; p=|[p`1,p`2]| by EUCLID:53; hence thesis by A100,A101,EUCLID:53; end; end; end; hence thesis by FUNCT_1:def 4; end; theorem Th134: for cn being Real st -1 =0; then y=(cn-FanMorphS).q by Th113; hence ex x being set st x in dom (cn-FanMorphS) & y=(cn-FanMorphS).x by A3,A4; end; case A5: q`1/|.q.|>=0 & q`2<=0 & q<>0.TOP-REAL 2; --(1+cn)>0 by A1,XREAL_1:148; then A6: -(-1-cn)>0; A7: 1-cn>=0 by A2,XREAL_1:149; then q`1/|.q.|*(1-cn)>=0 by A5; then -1-cn<= q`1/|.q.|*(1-cn) by A6; then A8: -1-cn+cn<= q`1/|.q.|*(1-cn)+cn by XREAL_1:7; set px=|[ |.q.|*(q`1/|.q.|*(1-cn)+cn), -((|.q.|)*sqrt(1-(q`1/|.q.|*( 1-cn)+cn)^2))]|; A9: px`1 = |.q.|*(q`1/|.q.|*(1-cn)+cn) by EUCLID:52; |.q.|<>0 by A5,TOPRNS_1:24; then A10: |.q.|^2>0 by SQUARE_1:12; A11: dom (cn-FanMorphS)=the carrier of TOP-REAL 2 by FUNCT_2:def 1; A12: 1-cn>0 by A2,XREAL_1:149; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A10,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then q`1/|.q.|<=1 by SQUARE_1:51; then q`1/|.q.|*(1-cn) <=1 *(1-cn) by A12,XREAL_1:64; then q`1/|.q.|*(1-cn)+cn-cn <=1-cn; then (q`1/|.q.|*(1-cn)+cn) <=1 by XREAL_1:9; then 1^2>=(q`1/|.q.|*(1-cn)+cn)^2 by A8,SQUARE_1:49; then A13: 1-(q`1/|.q.|*(1-cn)+cn)^2>=0 by XREAL_1:48; then A14: sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2)>=0 by SQUARE_1:def 2; A15: px`2 = -(|.q.|)*sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2) by EUCLID:52; then |.px.|^2=((- |.q.|)*sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2))^2 +(|.q.|* (q`1/|.q.|*(1-cn)+cn))^2 by A9,JGRAPH_3:1 .=(|.q.|)^2*(sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2))^2 +(|.q.|)^2*((q`1 /|.q.|*(1-cn)+cn))^2; then A16: |.px.|^2=(|.q.|)^2*(1-(q`1/|.q.|*(1-cn)+cn)^2) +(|.q.|)^2*((q`1 /|.q.|*(1-cn)+cn))^2 by A13,SQUARE_1:def 2 .= (|.q.|)^2; then A17: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22 .=|.q.| by SQUARE_1:22; then A18: px<>0.TOP-REAL 2 by A5,TOPRNS_1:23,24; (q`1/|.q.|*(1-cn)+cn)>=0+cn by A5,A7,XREAL_1:7; then px`1/|.px.| >=cn by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89; then A19: (cn-FanMorphS).px =|[ |.px.|* ((px`1/|.px.|-cn)/(1-cn) ), |.px .|*( -sqrt(1-((px`1/|.px.|-cn)/(1-cn))^2))]| by A1,A2,A15,A14,A18,Th115; |.px.|*( sqrt((-(q`2/|.q.|))^2))=|.q.|*(-(q`2/|.q.|)) by A5,A17, SQUARE_1:22 .= (-q`2)/|.q.|*|.q.| by XCMPLX_1:187 .=-q`2 by A5,TOPRNS_1:24,XCMPLX_1:87; then A20: |.px.|*(- sqrt((-(q`2/|.q.|))^2))=q`2; A21: |.px.|* ((px`1/|.px.|-cn)/(1-cn)) =|.q.|* (( ((q`1/|.q.|*(1-cn) +cn))-cn)/(1-cn)) by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89 .=|.q.|* ( q`1/|.q.|) by A12,XCMPLX_1:89 .= q`1 by A5,TOPRNS_1:24,XCMPLX_1:87; then |.px.|*( -sqrt(1-((px`1/|.px.|-cn)/(1-cn))^2)) = |.px.|*( -sqrt (1-(q`1/|.px.|)^2)) by A5,A17,TOPRNS_1:24,XCMPLX_1:89 .= |.px.|*( -sqrt(1-(q`1)^2/(|.px.|)^2)) by XCMPLX_1:76 .= |.px.|*( -sqrt( (|.px.|)^2/(|.px.|)^2-(q`1)^2/(|.px.|)^2)) by A10,A16,XCMPLX_1:60 .= |.px.|*( -sqrt( ((|.px.|)^2-(q`1)^2)/(|.px.|)^2)) by XCMPLX_1:120 .= |.px.|*( -sqrt( ((q`1)^2+(q`2)^2-(q`1)^2)/(|.px.|)^2)) by A16, JGRAPH_3:1 .= |.px.|*( -sqrt((q`2/|.q.|)^2)) by A17,XCMPLX_1:76; hence ex x being set st x in dom (cn-FanMorphS) & y=(cn-FanMorphS).x by A19,A21,A20,A11,EUCLID:53; end; case A22: q`1/|.q.|<0 & q`2<=0 & q<>0.TOP-REAL 2; A23: 1+cn>=0 by A1,XREAL_1:148; 1-cn>0 by A2,XREAL_1:149; then A24: 1-cn+cn>= q`1/|.q.|*(1+cn)+cn by A22,A23,XREAL_1:7; A25: 1+cn>0 by A1,XREAL_1:148; |.q.|<>0 by A22,TOPRNS_1:24; then A26: |.q.|^2>0 by SQUARE_1:12; 0<=(q`2)^2 by XREAL_1:63; then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1,XREAL_1:7; then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72; then (q`1)^2/(|.q.|)^2 <= 1 by A26,XCMPLX_1:60; then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76; then q`1/|.q.|>=-1 by SQUARE_1:51; then q`1/|.q.|*(1+cn) >=(-1)*(1+cn) by A25,XREAL_1:64; then q`1/|.q.|*(1+cn)+cn-cn >=-1-cn; then (q`1/|.q.|*(1+cn)+cn) >=-1 by XREAL_1:9; then 1^2>=(q`1/|.q.|*(1+cn)+cn)^2 by A24,SQUARE_1:49; then A27: 1-(q`1/|.q.|*(1+cn)+cn)^2>=0 by XREAL_1:48; then A28: sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2)>=0 by SQUARE_1:def 2; A29: dom (cn-FanMorphS)=the carrier of TOP-REAL 2 by FUNCT_2:def 1; set px=|[ |.q.|*(q`1/|.q.|*(1+cn)+cn), -(|.q.|)*(sqrt(1-(q`1/|.q.|*( 1+cn)+cn)^2))]|; A30: px`1 = |.q.|*(q`1/|.q.|*(1+cn)+cn) by EUCLID:52; A31: px`2 = -(|.q.|)*sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2) by EUCLID:52; then |.px.|^2=(-(|.q.|)*sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2))^2 +(|.q.|*( q`1/|.q.|*(1+cn)+cn))^2 by A30,JGRAPH_3:1 .=(|.q.|)^2*(sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2))^2 +(|.q.|)^2*((q`1 /|.q.|*(1+cn)+cn))^2; then A32: |.px.|^2=(|.q.|)^2*(1-(q`1/|.q.|*(1+cn)+cn)^2) +(|.q.|)^2*((q`1 /|.q.|*(1+cn)+cn))^2 by A27,SQUARE_1:def 2 .= (|.q.|)^2; then A33: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22 .=|.q.| by SQUARE_1:22; then A34: px<>0.TOP-REAL 2 by A22,TOPRNS_1:23,24; (q`1/|.q.|*(1+cn)+cn)<=0+cn by A22,A23,XREAL_1:7; then px`1/|.px.| <=cn by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89; then A35: (cn-FanMorphS).px =|[ |.px.|* ((px`1/|.px.|-cn)/(1+cn) ), |.px .|*( -sqrt(1-((px`1/|.px.|-cn)/(1+cn))^2))]| by A1,A2,A31,A28,A34,Th115; A36: |.px.|*( -sqrt((q`2/|.q.|)^2)) = |.px.|*( -sqrt((-q`2/|.q.|)^2) ) .=|.px.|*(--(q`2/|.q.|)) by A22,SQUARE_1:22 .=q`2 by A22,A33,TOPRNS_1:24,XCMPLX_1:87; A37: |.px.|* ((px`1/|.px.|-cn)/(1+cn)) =|.q.|* (( ((q`1/|.q.|*(1+cn) +cn))-cn)/(1+cn)) by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89 .=|.q.|* ( q`1/|.q.|) by A25,XCMPLX_1:89 .= q`1 by A22,TOPRNS_1:24,XCMPLX_1:87; then |.px.|*( -sqrt(1-((px`1/|.px.|-cn)/(1+cn))^2)) = |.px.|*( -sqrt (1-(q`1/|.px.|)^2)) by A22,A33,TOPRNS_1:24,XCMPLX_1:89 .= |.px.|*( -sqrt(1-(q`1)^2/(|.px.|)^2)) by XCMPLX_1:76 .= |.px.|*( -sqrt( (|.px.|)^2/(|.px.|)^2-(q`1)^2/(|.px.|)^2)) by A26,A32,XCMPLX_1:60 .= |.px.|*( -sqrt( ((|.px.|)^2-(q`1)^2)/(|.px.|)^2)) by XCMPLX_1:120 .= |.px.|*( -sqrt( ((q`1)^2+(q`2)^2-(q`1)^2)/(|.px.|)^2)) by A32, JGRAPH_3:1 .= |.px.|*( -sqrt((q`2/|.q.|)^2)) by A33,XCMPLX_1:76; hence ex x being set st x in dom (cn-FanMorphS) & y=(cn-FanMorphS).x by A35,A37,A36,A29,EUCLID:53; end; end; hence thesis by A3,FUNCT_1:def 3; end; hence thesis by A3,XBOOLE_0:def 10; end; hence thesis; end; theorem Th135: for cn being Real,p2 being Point of TOP-REAL 2 st -1 =cn holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphS).q holds p`2<0 & p`1>=0 proof let cn be Real,q be Point of TOP-REAL 2; assume that A1: cn<1 and A2: q`2<0 and A3: q`1/|.q.|>=cn; A4: 1-cn>0 by A1,XREAL_1:149; let p be Point of TOP-REAL 2; set qz=p; assume p=(cn-FanMorphS).q; then A5: p=|[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1- cn))^2))]| by A2,A3,Th113; then A6: qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; A7: (q`1/|.q.|-cn)>= 0 by A3,XREAL_1:48; A8: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24; then A9: (|.q.|)^2>0 by SQUARE_1:12; (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1 ,SQUARE_1:12,XREAL_1:8; then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A9,XREAL_1:74; then (q`1)^2/(|.q.|)^2 < 1 by A9,XCMPLX_1:60; then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76; then 1>q`1/|.q.| by SQUARE_1:52; then 1-cn>q`1/|.q.|-cn by XREAL_1:9; then -(1-cn)< -( q`1/|.q.|-cn) by XREAL_1:24; then (-(1-cn))/(1-cn)<(-( q`1/|.q.|-cn))/(1-cn) by A4,XREAL_1:74; then -1<(-( q`1/|.q.|-cn))/(1-cn) by A4,XCMPLX_1:197; then ((-(q`1/|.q.|-cn))/(1-cn))^2<1^2 by A4,A7,SQUARE_1:50; hence thesis by A5,A8,A4,A6,A7,Lm13,EUCLID:52,XREAL_1:132; end; theorem Th138: for cn being Real,q being Point of TOP-REAL 2 st -1 0 by A1,XREAL_1:148; A5: (q`1/|.q.|-cn)< 0 by A3,XREAL_1:49; then -( q`1/|.q.|-cn)>0 by XREAL_1:58; then (-(1+cn))/(1+cn)<(-( q`1/|.q.|-cn))/(1+cn) by A4,XREAL_1:74; then A6: -1<(-( q`1/|.q.|-cn))/(1+cn) by A4,XCMPLX_1:197; A7: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24; then A8: (|.q.|)^2>0 by SQUARE_1:12; (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1 ,SQUARE_1:12,XREAL_1:8; then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A8,XREAL_1:74; then (q`1)^2/(|.q.|)^2 < 1 by A8,XCMPLX_1:60; then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76; then -1 -(q`1/|.q.|-cn) by XREAL_1:24; then (-(q`1/|.q.|-cn))/(1+cn)<1 by A4,XREAL_1:191; then ((-(q`1/|.q.|-cn))/(1+cn))^2<1^2 by A6,SQUARE_1:50; then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>0 by XREAL_1:50; then sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>0 by SQUARE_1:25; then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)> 0 by XCMPLX_1:76; then sqrt(1-(q`1/|.q.|-cn)^2/(1+cn)^2)> 0; then --sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)> 0 by XCMPLX_1:76; then A9: -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)< 0; let p be Point of TOP-REAL 2; set qz=p; assume p=(cn-FanMorphS).q; then p=|[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+ cn))^2))]| by A2,A3,Th114; then A10: qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) & qz`1= |.q.|* ((q`1/ |.q.| -cn)/(1+cn)) by EUCLID:52; ((q`1/|.q.|-cn)/(1+cn))<0 by A1,A5,XREAL_1:141,148; hence thesis by A7,A10,A9,XREAL_1:132; end; theorem Th139: for cn being Real,q1,q2 being Point of TOP-REAL 2 st cn<1 & q1 `2<0 & q1`1/|.q1.|>=cn & q2`2<0 & q2`1/|.q2.|>=cn & q1`1/|.q1.|=cn and A4: q2`2<0 and A5: q2`1/|.q2.|>=cn and A6: q1`1/|.q1.| 0 by A1,A6,XREAL_1:9,149; let p1,p2 be Point of TOP-REAL 2; assume that A8: p1=(cn-FanMorphS).q1 and A9: p2=(cn-FanMorphS).q2; A10: |.p2.|=|.q2.| by A9,Th128; p2=|[ |.q2.|* ((q2`1/|.q2.|-cn)/(1-cn)), |.q2.|*( -sqrt(1-((q2`1/|.q2.| -cn)/(1-cn))^2))]| by A4,A5,A9,Th113; then A11: p2`1= |.q2.|* ((q2`1/|.q2.|-cn)/(1-cn)) by EUCLID:52; |.q2.|>0 by A4,Lm1,JGRAPH_2:3; then A12: p2`1/|.p2.|= ((q2`1/|.q2.|-cn)/(1-cn)) by A11,A10,XCMPLX_1:89; p1=|[ |.q1.|* ((q1`1/|.q1.|-cn)/(1-cn)), |.q1.|*( -sqrt(1-((q1`1/|.q1.|- cn)/(1-cn))^2))]| by A2,A3,A8,Th113; then A13: p1`1= |.q1.|* ((q1`1/|.q1.|-cn)/(1-cn)) by EUCLID:52; A14: |.p1.|=|.q1.| by A8,Th128; |.q1.|>0 by A2,Lm1,JGRAPH_2:3; then p1`1/|.p1.|= ((q1`1/|.q1.|-cn)/(1-cn)) by A13,A14,XCMPLX_1:89; hence thesis by A12,A7,XREAL_1:74; end; theorem Th140: for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1 0 by A1,A6,XREAL_1:9,148; let p1,p2 be Point of TOP-REAL 2; assume that A8: p1=(cn-FanMorphS).q1 and A9: p2=(cn-FanMorphS).q2; A10: |.p2.|=|.q2.| by A9,Th128; p2=|[ |.q2.|* ((q2`1/|.q2.|-cn)/(1+cn)), |.q2.|*( -sqrt(1-((q2`1/|.q2.| -cn)/(1+cn))^2))]| by A4,A5,A9,Th114; then A11: p2`1= |.q2.|* ((q2`1/|.q2.|-cn)/(1+cn)) by EUCLID:52; |.q2.|>0 by A4,Lm1,JGRAPH_2:3; then A12: p2`1/|.p2.|= (q2`1/|.q2.|-cn)/(1+cn) by A11,A10,XCMPLX_1:89; p1=|[ |.q1.|* ((q1`1/|.q1.|-cn)/(1+cn)), |.q1.|*( -sqrt(1-((q1`1/|.q1.|- cn)/(1+cn))^2))]| by A2,A3,A8,Th114; then A13: p1`1= |.q1.|* ((q1`1/|.q1.|-cn)/(1+cn)) by EUCLID:52; A14: |.p1.|=|.q1.| by A8,Th128; |.q1.|>0 by A2,Lm1,JGRAPH_2:3; then p1`1/|.p1.|= (q1`1/|.q1.|-cn)/(1+cn) by A13,A14,XCMPLX_1:89; hence thesis by A12,A7,XREAL_1:74; end; theorem for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1 =cn & q2`1/|.q2.|>=cn; hence thesis by A2,A3,A4,A5,A6,A7,Th139; end; suppose q1`1/|.q1.|>=cn & q2`1/|.q2.| =cn; then p2`1>=0 by A2,A4,A7,Th137; then A9: p2`1/|.p2.|>=0; p1`1<0 by A1,A3,A6,A8,Th138; hence thesis by A9,Lm1,JGRAPH_2:3,XREAL_1:141; end; suppose q1`1/|.q1.| 0 by A1,JGRAPH_2:3,TOPRNS_1:24; assume p=(cn-FanMorphS).q; then A4: p=|[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1- cn))^2))]| by A1,A2,Th113; then p`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52; hence thesis by A2,A4,A3,Lm13,EUCLID:52,XREAL_1:132; end; theorem 0.TOP-REAL 2 = (a-FanMorphS).(0.TOP-REAL 2) by Th113,JGRAPH_2:3;