### The Mizar article:

### The Jordan's Property for Certain Subsets of the Plane

**by****Yatsuka Nakamura, and****Jaroslaw Kotowicz**

- Received August 24, 1992
Copyright (c) 1992 Association of Mizar Users

- MML identifier: JORDAN1
- [ MML identifier index ]

environ vocabulary PRE_TOPC, FUNCT_1, RELAT_1, SUBSET_1, RELAT_2, ORDINAL2, BOOLE, RCOMP_1, BORSUK_1, FUNCOP_1, CONNSP_1, EUCLID, TOPREAL1, TOPS_2, ARYTM_1, MCART_1, ARYTM_3, SQUARE_1, PCOMPS_1, METRIC_1, JORDAN1, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, XREAL_0, REAL_1, RELAT_1, NAT_1, FUNCT_1, FUNCT_2, BINOP_1, PRE_TOPC, TOPS_2, CONNSP_1, SQUARE_1, RCOMP_1, STRUCT_0, METRIC_1, PCOMPS_1, BORSUK_1, TOPREAL1, EUCLID; constructors REAL_1, TOPS_2, CONNSP_1, SQUARE_1, RCOMP_1, TOPMETR, TOPREAL1, MEMBERED, NAT_1; clusters PRE_TOPC, BORSUK_1, STRUCT_0, EUCLID, TOPREAL1, XREAL_0, RELSET_1, MEMBERED, ZFMISC_1; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; definitions TARSKI, CONNSP_1, XBOOLE_0; theorems TARSKI, AXIOMS, REAL_1, FUNCOP_1, FUNCT_1, FUNCT_2, SUBSET_1, RCOMP_1, METRIC_1, PRE_TOPC, TOPS_1, TOPS_2, BORSUK_1, SEQ_2, CONNSP_1, SQUARE_1, REAL_2, TOPMETR, TOPREAL1, EUCLID, TOPREAL3, TREAL_1, RELAT_1, XBOOLE_0, XBOOLE_1, XREAL_0, XCMPLX_0, XCMPLX_1; begin :: :: Selected theorems on connected space :: reserve GX,GY for non empty TopSpace, x,y for Point of GX, r,s for Real; theorem Th1: for GX being TopStruct, A being Subset of GX holds the carrier of (GX|A) = A proof let GX be TopStruct, A be Subset of GX; A=[#](GX|A) by PRE_TOPC:def 10; hence thesis by PRE_TOPC:12; end; theorem Th2: for GX being non empty TopSpace st (for x,y being Point of GX ex GY st (GY is connected & ex f being map of GY,GX st f is continuous & x in rng(f)& y in rng(f))) holds GX is connected proof let GX; assume A1:(for x,y being Point of GX ex GY st (GY is connected & ex f being map of GY,GX st f is continuous & x in rng f & y in rng f)); for A,B being Subset of GX st [#](GX) = A \/ B & A <> {}(GX) & B <> {}(GX) & A is open & B is open holds A meets B proof let A,B be Subset of GX;assume that A2:[#](GX) = A \/ B and A3: A <> {}(GX) and A4: B <> {}(GX) and A5: A is open and A6: B is open; consider u being Element of A; u in [#](GX) by A2,A3,XBOOLE_0:def 2; then reconsider x=u as Point of GX; consider v being Element of B; v in [#](GX) by A2,A4,XBOOLE_0:def 2; then reconsider y=v as Point of GX; consider GY such that A7: GY is connected and A8: ex f being map of GY,GX st f is continuous & x in rng f & y in rng f by A1; consider f being map of GY,GX such that A9: f is continuous and A10: x in rng f and A11: y in rng f by A8; A12:f"A is open by A5,A9,TOPS_2:55; A13:f"B is open by A6,A9,TOPS_2:55; f"([#] GX)=[#] GY by TOPS_2:52; then A14:f"A \/ f"B = [#] GY by A2,RELAT_1:175; rng f /\ A <> {} by A3,A10,XBOOLE_0:def 3; then rng f meets A by XBOOLE_0:def 7; then A15:f"A <> {}(GY) by RELAT_1:173; rng f /\ B <> {} by A4,A11,XBOOLE_0:def 3; then rng f meets B by XBOOLE_0:def 7; then f"B <> {}(GY) by RELAT_1:173; then f"A meets f"B by A7,A12,A13,A14,A15,CONNSP_1:12; then f"A /\ f"B <> {} by XBOOLE_0:def 7; then f"(A /\ B) <> {} by FUNCT_1:137; then rng f meets (A /\ B) by RELAT_1:173; then consider u1 being set such that A16: u1 in rng f & u1 in A /\ B by XBOOLE_0:3; thus thesis by A16,XBOOLE_0:4; end; hence thesis by CONNSP_1:12; end; Lm1: 0 in [.0,1.] & 1 in [.0,1.] proof 0 in {r:0<=r & r<=1} & 1 in {s:0<=s & s<=1}; hence thesis by RCOMP_1:def 1; end; canceled; theorem Th4: ::Arcwise connectedness derives connectedness for GX being non empty TopSpace st (for x,y being Point of GX ex h being map of I[01],GX st h is continuous & x=h.0 & y=h.1) holds GX is connected proof let GX; assume A1: for x,y being Point of GX ex h being map of I[01],GX st h is continuous & x=h.0 & y=h.1; for x,y being Point of GX ex GY st (GY is connected & ex f being map of GY,GX st f is continuous & x in rng(f)& y in rng(f)) proof let x,y; now consider h being map of I[01],GX such that A2: h is continuous and A3: x=h.0 & y=h.1 by A1; [#] I[01] = the carrier of I[01] by PRE_TOPC:12; then 0 in dom h & 1 in dom h by Lm1,BORSUK_1:83,TOPS_2:51; then x in rng h & y in rng h by A3,FUNCT_1:def 5; hence thesis by A2,TREAL_1:22; end; hence thesis; end; hence thesis by Th2; end; theorem Th5: ::Arcwise connectedness derives connectedness for subset for A being Subset of GX st (for xa,ya being Point of GX st xa in A & ya in A & xa<>ya ex h being map of I[01],GX|A st h is continuous & xa=h.0 & ya=h.1) holds A is connected proof let A be Subset of GX;assume that A1: for xa,ya being Point of GX st xa in A & ya in A & xa<>ya ex h being map of I[01],GX|A st h is continuous & xa=h.0 & ya=h.1; per cases; suppose A is non empty; then reconsider A as non empty Subset of GX; A2: for xa,ya being Point of GX st xa in A & ya in A & xa = ya ex h being map of I[01],GX|A st h is continuous & xa=h.0 & ya=h.1 proof let xa,ya be Point of GX; assume A3: xa in A & ya in A & xa = ya; then xa in the carrier of (GX|A) by Th1; then reconsider xa' = xa as Element of GX|A; reconsider h = I[01] --> xa' as map of I[01], GX|A; take h; h = (the carrier of I[01]) --> xa' by BORSUK_1:def 3; hence thesis by A3,Lm1,BORSUK_1:36,83,FUNCOP_1:13; end; for xb,yb being Point of GX|A ex ha being map of I[01],GX|A st ha is continuous & xb=ha.0 & yb=ha.1 proof let xb,yb be Point of GX|A; xb in the carrier of (GX|A) & yb in the carrier of (GX|A); then xb in [#](GX|A) & yb in [#](GX|A) by PRE_TOPC:12; then A4: xb in A & yb in A by PRE_TOPC:def 10; per cases; suppose xb<>yb; hence thesis by A1,A4; suppose xb = yb; hence thesis by A2,A4; end; then GX|A is connected by Th4; hence thesis by CONNSP_1:def 3; suppose A is empty; then reconsider D = A as empty Subset of GX; let A1, B1 be Subset of GX|A such that A5: [#](GX|A) = A1 \/ B1 and A1,B1 are_separated; [#](GX|D) = D by PRE_TOPC:def 10; hence A1 = {}(GX|A) or B1 = {}(GX|A) by A5,XBOOLE_1:15; end; theorem Th6: for A0 being Subset of GX, A1 being Subset of GX st A0 is connected & A1 is connected & A0 meets A1 holds A0 \/ A1 is connected proof let A0 be Subset of GX, A1 be Subset of GX; assume that A1: A0 is connected & A1 is connected and A2: A0 meets A1; not A0,A1 are_separated by A2,CONNSP_1:2; hence thesis by A1,CONNSP_1:18; end; theorem Th7: for A0,A1,A2 being Subset of GX st A0 is connected & A1 is connected & A2 is connected & A0 meets A1 & A1 meets A2 holds A0 \/ A1 \/ A2 is connected proof let A0,A1,A2 be Subset of GX; assume that A1: A0 is connected & A1 is connected & A2 is connected and A2: A0 meets A1 & A1 meets A2; A3: A0 /\ A1 <> {} & A1 /\ A2 <> {} by A2,XBOOLE_0:def 7; A4: A0 \/ A1 is connected by A1,A2,Th6; (A0 \/ A1)/\ A2= A0 /\ A2 \/ A1 /\ A2 by XBOOLE_1:23; then (A0 \/ A1) /\ A2 <> {} by A3,XBOOLE_1:15; then (A0 \/ A1) meets A2 by XBOOLE_0:def 7; hence thesis by A1,A4,Th6; end; theorem Th8: for A0,A1,A2,A3 being Subset of GX st A0 is connected & A1 is connected & A2 is connected & A3 is connected & A0 meets A1 & A1 meets A2 & A2 meets A3 holds A0 \/ A1 \/ A2 \/ A3 is connected proof let A0,A1,A2,A3 be Subset of GX; assume that A1: A0 is connected & A1 is connected & A2 is connected & A3 is connected and A2: A0 meets A1 & A1 meets A2 & A2 meets A3; A3: A0 /\ A1 <> {} & A1 /\ A2 <> {} & A2 /\ A3 <> {} by A2,XBOOLE_0:def 7; A4: A0 \/ A1 \/ A2 is connected by A1,A2,Th7; (A0 \/ A1 \/ A2)/\ A3= (A0 \/ A1) /\ A3 \/ A2 /\ A3 by XBOOLE_1:23; then (A0 \/ A1 \/ A2) /\ A3 <> {} by A3,XBOOLE_1:15; then (A0 \/ A1 \/ A2) meets A3 by XBOOLE_0:def 7; hence thesis by A1,A4,Th6; end; begin :: :: Certain connected and open subsets in the Euclidean plane :: reserve Q,P1,P2 for Subset of TOP-REAL 2; reserve P for Subset of TOP-REAL 2; reserve w1,w2 for Point of TOP-REAL 2; definition let n be Nat, P be Subset of TOP-REAL n; attr P is convex means for w1,w2 being Point of TOP-REAL n st w1 in P & w2 in P holds LSeg(w1,w2) c= P; end; theorem Th9: for n being Nat, P being Subset of TOP-REAL n st P is convex holds P is connected proof let n be Nat, P be Subset of TOP-REAL n; assume that A1:for w3,w4 being Point of TOP-REAL n st w3 in P & w4 in P holds LSeg(w3,w4) c= P; for w1,w2 being Point of TOP-REAL n st w1 in P & w2 in P & w1<>w2 ex h being map of I[01],(TOP-REAL n)|P st h is continuous & w1=h.0 & w2=h.1 proof let w1,w2 be Point of TOP-REAL n; assume A2:w1 in P & w2 in P & w1<>w2; then A3: LSeg(w1,w2) c= P by A1; LSeg(w1,w2) is_an_arc_of w1,w2 by A2,TOPREAL1:15; then consider f being map of I[01], (TOP-REAL n)|LSeg(w1,w2) such that A4: f is_homeomorphism & f.0 = w1 & f.1 = w2 by TOPREAL1:def 2; A5:f is continuous by A4,TOPS_2:def 5; A6: rng f = [#]((TOP-REAL n)|LSeg(w1,w2)) by A4,TOPS_2:def 5; then A7: rng f c= P by A3,PRE_TOPC:def 10; then [#]((TOP-REAL n)|LSeg(w1,w2)) c= [#]((TOP-REAL n)|P) by A6,PRE_TOPC:def 10; then [#]((TOP-REAL n)|LSeg(w1,w2)) c=the carrier of ((TOP-REAL n)|P) by PRE_TOPC:12; then the carrier of ((TOP-REAL n)|LSeg(w1,w2)) c= the carrier of ((TOP-REAL n)|P) by PRE_TOPC:12; then A8:(TOP-REAL n)|LSeg(w1,w2) is SubSpace of (TOP-REAL n)|P by TOPMETR:4; dom f=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1; then reconsider g=f as Function of [.0,1.],P by A7,FUNCT_2:4; the carrier of (TOP-REAL n)|P = [#]((TOP-REAL n)|P) by PRE_TOPC:12 .= P by PRE_TOPC:def 10; then reconsider gt=g as map of I[01],(TOP-REAL n)|P by BORSUK_1:83; gt is continuous by A5,A8,TOPMETR:7; hence thesis by A4; end; hence thesis by Th5; end; reserve pa,pb for Point of TOP-REAL 2, s1,t1,s2,t2,s,t,s3,t3,s4,t4,s5,t5,s6,t6, l,lambda,sa,sd,ta,td for Real; theorem Th10: s1<s3 & s1<s4 & 0 <= l & l <= 1 implies s1<(1-l)*s3+l*s4 proof assume A1:s1<s3 & s1<s4 & 0 <= l & l <= 1; now per cases; suppose l=0; hence thesis by A1; suppose l=1; hence thesis by A1; suppose A2: not(l=0 or l=1); then A3: l>0 & l<1 by A1,REAL_1:def 5; A4:l*s1<l*s4 by A1,A2,REAL_1:70; 1-l>0 by A3,SQUARE_1:11; then A5: (1-l)*s1<(1-l)*s3 by A1,REAL_1:70; (1-l)*s1+l*s1=(1-l+l)*s1 by XCMPLX_1:8 .=1*s1 by XCMPLX_1:27 .=s1; hence thesis by A4,A5,REAL_1:67; end; hence thesis; end; theorem Th11: s3<s1 & s4<s1 & 0 <= l & l <= 1 implies (1-l)*s3+l*s4<s1 proof assume A1:s1>s3 & s1>s4 & 0 <= l & l <= 1; now per cases; suppose l=0; hence thesis by A1; suppose l=1; hence thesis by A1; suppose A2: not(l=0 or l=1); then A3: l>0 & l<1 by A1,REAL_1:def 5; A4:l*s1>l*s4 by A1,A2,REAL_1:70; 1-l>0 by A3,SQUARE_1:11; then A5: (1-l)*s1>(1-l)*s3 by A1,REAL_1:70; (1-l)*s1+l*s1=(1-l+l)*s1 by XCMPLX_1:8 .=1*s1 by XCMPLX_1:27 .=s1; hence thesis by A4,A5,REAL_1:67; end; hence thesis; end; reserve s1a,t1a,s2a,t2a,s3a,t3a,sb,tb,sc,tc for Real; Lm2: the carrier of TOP-REAL 2 = REAL 2 by EUCLID:25; Lm3: for s1 holds { |[ sb,tb ]|:sb<s1} is Subset of REAL 2 proof let s1; { |[ sb,tb ]|:sb<s1} c= REAL 2 proof let y be set;assume y in { |[ sb,tb ]|:sb<s1}; then ex s7,t7 being Real st |[ s7,t7 ]|=y & s7<s1; hence thesis by Lm2; end; hence thesis; end; Lm4: for t1 holds { |[ sb,tb ]| :tb<t1} is Subset of REAL 2 proof let t1; { |[ sd,td ]|:td<t1} c= REAL 2 proof let y be set;assume y in { |[ sd,td ]|:td<t1}; then ex s7,t7 being Real st |[ s7,t7 ]|=y & t7<t1; hence thesis by Lm2; end; hence thesis; end; Lm5: for s2 holds { |[ sb,tb ]| :s2<sb} is Subset of REAL 2 proof let s2; { |[ sb,tb ]|:s2<sb} c= REAL 2 proof let y be set;assume y in { |[ sb,tb ]|:s2<sb}; then ex s7,t7 being Real st |[ s7,t7 ]|=y & s2<s7; hence thesis by Lm2; end; hence thesis; end; Lm6: for t2 holds { |[ sb,tb ]|:t2<tb} is Subset of REAL 2 proof let t2; { |[ sb,tb ]|:t2<tb} c= REAL 2 proof let y be set;assume y in { |[ sb,tb ]|:t2<tb}; then ex s7,t7 being Real st |[ s7,t7 ]|=y & t2<t7; hence thesis by Lm2; end; hence thesis; end; Lm7: for s1,s2,t1,t2 holds { |[ s,t ]| where s is Real,t is Real: s1<s & s<s2 & t1<t & t<t2} is Subset of REAL 2 proof let s1,s2,t1,t2; { |[ sb,tb ]| where sb is Real, tb is Real: s1<sb & sb<s2 & t1<tb & tb<t2} c= REAL 2 proof let y be set;assume y in { |[ sb,tb ]| where sb is Real,tb is Real: s1<sb & sb<s2 & t1<tb & tb<t2}; then ex s7,t7 being Real st |[ s7,t7 ]|=y & s1<s7 & s7<s2 & t1<t7 & t7<t2; hence thesis by Lm2; end; hence thesis; end; Lm8: for s1,s2,t1,t2 holds { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} is Subset of REAL 2 proof let s1,s2,t1,t2; { |[ sb,tb ]| where sb is Real,tb is Real: not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} c= REAL 2 proof let y be set;assume y in { |[ sb,tb ]| where sb is Real,tb is Real: not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)}; then ex s7,t7 being Real st |[ s7,t7 ]|=y & not (s1<=s7 & s7<=s2 & t1<=t7 & t7<=t2); hence thesis by Lm2; end; hence thesis; end; theorem Th12: {|[ s,t ]|: s1<s & s<s2 & t1<t & t<t2} = {|[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2} /\ {|[s5,t5]|:t1<t5} /\ {|[s6,t6]|:t6<t2} proof now let x be set;assume x in { |[ s,t ]|: s1<s & s<s2 & t1<t & t<t2}; then ex s,t st |[s,t]|=x & (s1<s & s<s2 & t1<t & t<t2); then x in { |[ s3,t3 ]|:s1<s3} & x in { |[ s4,t4 ]|:s4<s2} & x in { |[ s5,t5 ]|:t1<t5} & x in { |[ s6,t6 ]|:t6<t2}; then x in { |[ s3,t3 ]|:s1<s3} /\ { |[ s4,t4 ]|:s4<s2} & x in { |[ s5,t5 ]|:t1<t5} & x in { |[ s6,t6 ]|:t6<t2} by XBOOLE_0:def 3 ; then x in { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5} & x in { |[ s6,t6 ]|:t6<t2} by XBOOLE_0:def 3; hence x in { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5} /\ {|[s6,t6]|:t6<t2} by XBOOLE_0:def 3; end; then A1: { |[ s,t ]|: s1<s & s<s2 & t1<t & t<t2} c= { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5} /\ {|[s6,t6]|:t6<t2} by TARSKI:def 3; now let x be set;assume x in { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5} /\ {|[s6,t6]|:t6<t2}; then x in { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5} & x in { |[ s6,t6 ]|:t6<t2} by XBOOLE_0:def 3; then x in { |[ s3,t3 ]|:s1<s3} /\ { |[ s4,t4]|:s4<s2} & x in { |[ s5,t5 ]|:t1<t5} & x in { |[ s6,t6 ]|:t6<t2} by XBOOLE_0:def 3 ; then A2: x in { |[ s3,t3 ]|:s1<s3} & x in { |[ s4,t4 ]|:s4<s2} & x in { |[ s5,t5 ]|:t1<t5} & x in { |[ s6,t6 ]|:t6<t2} by XBOOLE_0:def 3 ; then A3:(ex sa,ta st |[sa,ta]|=x & s1<sa) & (ex sb,tb st |[sb,tb]|=x & sb<s2) & (ex sc,tc st |[sc,tc]|=x & t1<tc) & (ex sd,td st |[sd,td]|=x & td<t2); consider sa,ta such that A4:|[sa,ta]|=x & s1<sa by A2; reconsider p=x as Point of TOP-REAL 2 by A3; p`1=sa & p`2=ta by A4,EUCLID:56; then |[sa,ta]|=x & s1<sa & sa<s2 & t1<ta & ta<t2 by A3,A4,EUCLID:56; hence x in { |[ s,t ]|:s1<s & s<s2 & t1<t & t<t2}; end; then { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5} /\ {|[s6,t6]|:t6<t2}c={ |[ s,t ]|:s1<s & s<s2 & t1<t & t<t2} by TARSKI:def 3; hence thesis by A1,XBOOLE_0:def 10; end; theorem Th13: {|[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} = {|[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1} \/ {|[s5,t5]|:s2<s5} \/ {|[s6,t6]|:t2<t6} proof now let x be set;assume x in { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)}; then ex s,t st |[s,t]|=x & not (s1<=s & s<=s2 & t1<=t & t<=t2); then x in { |[ s3,t3 ]|:s3<s1}or x in { |[ s4,t4 ]|:t4<t1} or x in { |[ s5,t5 ]|:s2<s5} or x in { |[ s6,t6 ]|:t2<t6}; then x in { |[ s3,t3 ]|:s3<s1} \/ { |[ s4,t4 ]|:t4<t1} or x in { |[ s5,t5 ]|:s2<s5} or x in { |[ s6,t6 ]|:t2<t6} by XBOOLE_0:def 2; then x in { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5} or x in { |[ s6,t6 ]|:t2<t6} by XBOOLE_0:def 2; hence x in { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5} \/ {|[s6,t6]|:t2<t6} by XBOOLE_0:def 2; end; then A1: { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} c= { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5} \/ {|[s6,t6]|:t2<t6} by TARSKI:def 3; now let x be set;assume x in { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5} \/ {|[s6,t6]|:t2<t6}; then x in { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5} or x in { |[ s6,t6 ]|:t2<t6} by XBOOLE_0:def 2; then x in { |[ s3,t3 ]|:s3<s1} \/ { |[ s4,t4]|:t4<t1} or x in { |[ s5,t5 ]|:s2<s5} or x in { |[ s6,t6 ]|:t2<t6} by XBOOLE_0:def 2; then x in { |[ s3,t3 ]|:s3<s1} or x in { |[ s4,t4 ]|:t4<t1} or x in { |[ s5,t5 ]|:s2<s5} or x in { |[ s6,t6 ]|:t2<t6} by XBOOLE_0:def 2; then (ex sa,ta st |[sa,ta]|=x & sa<s1) or (ex sc,tc st |[sc,tc]|=x & tc<t1) or (ex sb,tb st |[sb,tb]|=x & s2<sb) or (ex sd,td st |[sd,td]|=x & t2<td); hence x in { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)}; end; then { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5} \/ {|[s6,t6]|:t2<t6}c={ |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} by TARSKI:def 3; hence thesis by A1,XBOOLE_0:def 10; end; theorem Th14: for s1,t1,s2,t2,P st P = {|[ s,t ]|:s1<s & s<s2 & t1<t & t<t2} holds P is convex proof let s1,t1,s2,t2,P; assume A1:P = { |[ s,t ]|:s1<s & s<s2 & t1<t & t<t2}; let w1,w2 such that A2: w1 in P & w2 in P; let x be set such that A3: x in LSeg(w1,w2); consider s3,t3 such that A4:|[ s3,t3 ]|=w1 & (s1<s3 & s3<s2 & t1<t3 & t3<t2) by A1,A2; A5: w1`1=s3 & w1`2=t3 by A4,EUCLID:56; consider s4,t4 such that A6:|[ s4,t4 ]|=w2 & (s1<s4 & s4<s2 & t1<t4 & t4<t2) by A1,A2; A7: w2`1=s4 & w2`2=t4 by A6,EUCLID:56; x in {(1-lambda)*w1 + lambda*w2: 0 <= lambda & lambda <= 1} by A3,TOPREAL1:def 4; then consider l such that A8: x = (1-l)*w1 + l*w2 & 0 <= l & l <= 1; set w = (1-l)*w1 + l*w2; A9: w = |[((1-l)*w1)`1+(l*w2)`1,((1-l)*w1)`2+ (l*w2)`2]| by EUCLID:59; (1-l)*w1=|[ (1-l)*w1`1 ,(1-l)*w1`2 ]| & l*w2=|[ l*w2`1 ,l*w2`2 ]| by EUCLID:61; then ((1-l)*w1)`1= (1-l)*w1`1 & ((1-l)*w1)`2= (1-l)*w1`2 & (l*w2)`1=l*w2`1 & (l*w2)`2=l*w2`2 by EUCLID:56; then w`1=(1-l)* w1`1+ l* w2`1 & w`2=(1-l)* w1`2+ l* w2`2 by A9,EUCLID:56; then s1< w`1 & w`1<s2 & t1< w`2 & w`2<t2 & w = |[w`1, w`2]| by A4,A5,A6,A7,A8,Th10,Th11,EUCLID:57; hence thesis by A1,A8; end; theorem Th15: for s1,t1,s2,t2,P st P = {|[ s,t ]|:s1<s & s<s2 & t1<t & t<t2} holds P is connected proof let s1,t1,s2,t2,P; assume P = { |[ s,t ]|:s1<s & s<s2 & t1<t & t<t2}; then P is convex by Th14; hence thesis by Th9; end; theorem Th16: for s1,P st P = { |[ s,t ]|:s1<s } holds P is convex proof let s1,P; assume A1:P = { |[ s,t ]|:s1<s }; let w1,w2 such that A2: w1 in P & w2 in P; let x be set such that A3: x in LSeg(w1,w2); consider s3,t3 such that A4:|[ s3,t3 ]|=w1 & s1<s3 by A1,A2; A5: w1`1=s3 & w1`2=t3 by A4,EUCLID:56; consider s4,t4 such that A6:|[ s4,t4 ]|=w2 & s1<s4 by A1,A2; A7: w2`1=s4 & w2`2=t4 by A6,EUCLID:56; x in {(1-lambda)*w1 + lambda*w2: 0 <= lambda & lambda <= 1} by A3,TOPREAL1:def 4; then consider l such that A8: x = (1-l)*w1 + l*w2& 0 <= l & l <= 1; set w = (1-l)*w1 + l*w2; A9: w= |[((1-l)*w1)`1+(l*w2)`1,((1-l)*w1)`2+ (l*w2)`2]| by EUCLID:59; (1-l)*w1=|[ (1-l)*w1`1 ,(1-l)*w1`2 ]| & l*w2=|[ l*w2`1 ,l*w2`2 ]| by EUCLID:61; then ((1-l)*w1)`1= (1-l)*w1`1 & ((1-l)*w1)`2= (1-l)*w1`2 & (l*w2)`1=l*w2`1 & (l*w2)`2=l*w2`2 by EUCLID:56; then w`1=(1-l)* w1`1+ l* w2`1 & w`2=(1-l)* w1`2+ l* w2`2 by A9,EUCLID:56; then s1< w`1 & w = |[w`1, w`2]| by A4,A5,A6,A7,A8,Th10,EUCLID:57; hence thesis by A1,A8; end; theorem Th17: for s1,P st P = { |[ s,t ]|:s1<s } holds P is connected proof let s1,P; assume A1:P = { |[ s,t ]|:s1<s }; set s3 =s1+1; s1<s3 by REAL_2:174; then |[ s3,0 ]| in { |[ s,t ]|:s1<s }; then P is convex non empty by A1,Th16; hence thesis by Th9; end; theorem Th18: for s2,P st P = {|[ s,t ]|:s<s2 } holds P is convex proof let s2,P; assume A1:P = { |[ s,t ]|:s<s2 }; let w1,w2 such that A2: w1 in P & w2 in P; let x be set such that A3: x in LSeg(w1,w2); consider s3,t3 such that A4:|[ s3,t3 ]|=w1 & s3<s2 by A1,A2; A5: w1`1=s3 & w1`2=t3 by A4,EUCLID:56; consider s4,t4 such that A6:|[ s4,t4 ]|=w2 & s4<s2 by A1,A2; A7: w2`1=s4 & w2`2=t4 by A6,EUCLID:56; x in {(1-lambda)*w1 + lambda*w2: 0 <= lambda & lambda <= 1} by A3,TOPREAL1:def 4; then consider l such that A8: x = (1-l)*w1 + l*w2& 0 <= l & l <= 1; set w = (1-l)*w1 + l*w2; A9: w= |[((1-l)*w1)`1+(l*w2)`1,((1-l)*w1)`2+ (l*w2)`2]| by EUCLID:59; (1-l)*w1=|[ (1-l)*w1`1 ,(1-l)*w1`2 ]| & l*w2=|[ l*w2`1 ,l*w2`2 ]| by EUCLID:61; then ((1-l)*w1)`1= (1-l)*w1`1 & ((1-l)*w1)`2= (1-l)*w1`2 & (l*w2)`1=l*w2`1 & (l*w2)`2=l*w2`2 by EUCLID:56; then w`1=(1-l)* w1`1+ l* w2`1 & w`2=(1-l)* w1`2+ l* w2`2 by A9,EUCLID:56; then s2> w`1 & w = |[w`1, w`2]| by A4,A5,A6,A7,A8,Th11,EUCLID:57; hence thesis by A1,A8; end; theorem Th19: for s2,P st P = {|[ s,t ]|:s<s2 } holds P is connected proof let s2,P; assume A1:P = { |[ s,t ]|:s<s2 }; set s3 =s2-1; s3<s2 by REAL_2:174; then |[ s3,0 ]| in { |[ s,t ]|:s<s2 }; then P is convex non empty by A1,Th18; hence thesis by Th9; end; theorem Th20: for t1,P st P = {|[ s,t ]|:t1<t } holds P is convex proof let t1,P; assume A1:P = { |[ s,t ]|:t1<t }; let w1,w2 such that A2: w1 in P & w2 in P; let x be set such that A3: x in LSeg(w1,w2); consider s3,t3 such that A4:|[ s3,t3 ]|=w1 & t1<t3 by A1,A2; A5: w1`1=s3 & w1`2=t3 by A4,EUCLID:56; consider s4,t4 such that A6:|[ s4,t4 ]|=w2 & t1<t4 by A1,A2; A7: w2`1=s4 & w2`2=t4 by A6,EUCLID:56; x in {(1-lambda)*w1 + lambda*w2: 0 <= lambda & lambda <= 1} by A3,TOPREAL1:def 4; then consider l such that A8: x = (1-l)*w1 + l*w2& 0 <= l & l <= 1; set w = (1-l)*w1 + l*w2; A9: w= |[((1-l)*w1)`1+(l*w2)`1,((1-l)*w1)`2+ (l*w2)`2]| by EUCLID:59; (1-l)*w1=|[ (1-l)*w1`1 ,(1-l)*w1`2 ]| & l*w2=|[ l*w2`1 ,l*w2`2 ]| by EUCLID:61; then ((1-l)*w1)`1= (1-l)*w1`1 & ((1-l)*w1)`2= (1-l)*w1`2 & (l*w2)`1=l*w2`1 & (l*w2)`2=l*w2`2 by EUCLID:56; then w`1=(1-l)* w1`1+ l* w2`1 & w`2=(1-l)* w1`2+ l* w2`2 by A9,EUCLID:56; then t1< w`2 & w = |[w`1, w`2]| by A4,A5,A6,A7,A8,Th10,EUCLID:57; hence thesis by A1,A8; end; theorem Th21: for t1,P st P = {|[ s,t ]|:t1<t } holds P is connected proof let t1,P; assume A1:P = { |[ s,t ]|:t1<t }; set t3 =t1+1; t1<t3 by REAL_2:174; then |[ 0,t3 ]| in { |[ s,t ]|:t1<t }; then P is convex non empty by A1,Th20; hence thesis by Th9; end; theorem Th22: for t2,P st P = { |[ s,t ]|: t<t2 } holds P is convex proof let t2,P; assume A1:P = { |[ s,t ]|:t<t2 }; let w1,w2 such that A2: w1 in P & w2 in P; let x be set such that A3: x in LSeg(w1,w2); consider s3,t3 such that A4:|[ s3,t3 ]|=w1 & t3<t2 by A1,A2; A5: w1`1=s3 & w1`2=t3 by A4,EUCLID:56; consider s4,t4 such that A6:|[ s4,t4 ]|=w2 & t4<t2 by A1,A2; A7: w2`1=s4 & w2`2=t4 by A6,EUCLID:56; x in {(1-lambda)*w1 + lambda*w2: 0 <= lambda & lambda <= 1} by A3,TOPREAL1:def 4; then consider l such that A8: x = (1-l)*w1 + l*w2& 0 <= l & l <= 1; set w = (1-l)*w1 + l*w2; A9: w= |[((1-l)*w1)`1+(l*w2)`1,((1-l)*w1)`2+ (l*w2)`2]| by EUCLID:59; (1-l)*w1=|[ (1-l)*w1`1 ,(1-l)*w1`2 ]| & l*w2=|[ l*w2`1 ,l*w2`2 ]| by EUCLID:61; then ((1-l)*w1)`1= (1-l)*w1`1 & ((1-l)*w1)`2= (1-l)*w1`2 & (l*w2)`1=l*w2`1 & (l*w2)`2=l*w2`2 by EUCLID:56; then w`1=(1-l)* w1`1+ l* w2`1 & w`2=(1-l)* w1`2+ l* w2`2 by A9,EUCLID:56; then t2> w`2 & w = |[w`1, w`2]| by A4,A5,A6,A7,A8,Th11,EUCLID:57; hence thesis by A1,A8; end; theorem Th23: for t2,P st P = { |[ s,t ]|: t<t2 } holds P is connected proof let t2,P; assume A1:P = { |[ s,t ]|:t<t2 }; set t3 =t2-1; t3<t2 by REAL_2:174; then |[ 0,t3 ]| in { |[ s,t ]|:t<t2 }; then P is convex non empty by A1,Th22; hence thesis by Th9; end; theorem Th24: for s1,t1,s2,t2,P st P = { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} holds P is connected proof let s1,t1,s2,t2,P;assume P={ |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)}; then A1: P={ |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5} \/ {|[s6,t6]|:t2<t6} by Th13; A2:{ |[ s,t ]|:s<s1} is Subset of TOP-REAL 2 by Lm2,Lm3; A3:{ |[ s,t ]|:t<t1} is Subset of TOP-REAL 2 by Lm2,Lm4; A4:{ |[ s,t ]|:s2<s} is Subset of TOP-REAL 2 by Lm2,Lm5; { |[ s,t ]|:t2<t} is Subset of TOP-REAL 2 by Lm2,Lm6; then reconsider A0={ |[ s,t ]|:s<s1}, A1={ |[ s,t ]|:t<t1}, A2={ |[ s,t ]|:s2<s}, A3={ |[ s,t ]|:t2<t} as Subset of TOP-REAL 2 by A2,A3,A4; A5:s1-1<s1 by REAL_2:174;A6:t1-1<t1 by REAL_2:174; A7: |[s1-1,t1-1]| in A0 by A5; |[s1-1,t1-1]| in A1 by A6; then A0 /\ A1 <> {} by A7,XBOOLE_0:def 3; then A8: A0 meets A1 by XBOOLE_0:def 7; A9: s2< s2+1 by REAL_2:174; A10:|[s2+1,t1-1]| in A1 by A6; |[s2+1,t1-1]| in A2 by A9; then A1 /\ A2 <> {} by A10,XBOOLE_0:def 3; then A11: A1 meets A2 by XBOOLE_0:def 7; A12: t2< t2+1 by REAL_2:174; A13:|[s2+1,t2+1]| in A2 by A9; |[s2+1,t2+1]| in A3 by A12; then A2 /\ A3 <> {} by A13,XBOOLE_0:def 3; then A14: A2 meets A3 by XBOOLE_0:def 7; A15:A0 is connected by Th19; A16:A1 is connected by Th23; A17:A2 is connected by Th17; A3 is connected by Th21; hence thesis by A1,A8,A11,A14,A15,A16,A17,Th8; end; Lm9: (s-r)^2=(r-s)^2 proof A1:(s-r)^2=s^2-2*s*r + r^2 by SQUARE_1:64 .= r^2+s^2-2*s*r by XCMPLX_1:29 .= r^2+s^2-2*(r*s) by XCMPLX_1:4; (r-s)^2=r^2-2*r*s + s^2 by SQUARE_1:64 .= r^2+s^2-2*r*s by XCMPLX_1:29 .= r^2+s^2-2*(r*s) by XCMPLX_1:4; hence thesis by A1; end; Lm10: TOP-REAL 2 =TopSpaceMetr(Euclid 2) by EUCLID:def 8; theorem Th25: for s1 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:s1<s } holds P is open proof let s1; let P be Subset of TOP-REAL 2 such that A1:P= { |[ s,t ]|:s1<s }; for pe being Point of Euclid 2 st pe in P ex r being real number st r>0 & Ball(pe,r) c= P proof let pe be Point of Euclid 2;assume pe in P; then consider s,t such that A2:|[ s,t ]|=pe & s1<s by A1; set r=(s-s1)/2; s-s1>0 by A2,SQUARE_1:11; then A3: r>0 by REAL_2:127; Ball(pe,r) c= P proof let x be set; assume x in Ball(pe,r); then x in {q where q is Element of Euclid 2:dist(pe,q)<r} by METRIC_1:18; then consider q being Element of Euclid 2 such that A4: q=x and A5: dist(pe,q)<r; reconsider ppe=pe, pq=q as Point of TOP-REAL 2 by TOPREAL3:13; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(pe,q)=dist(pe,q) by METRIC_1:def 1; then A6: sqrt ((ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2)<r by A5,TOPREAL3:12; A7: 0 <= (ppe`1 - pq`1)^2 & 0 <= (ppe`2 - pq`2)^2 by SQUARE_1:72; then A8: 0+0 <= (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 by REAL_1:55; then 0 <=sqrt ( (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2) by SQUARE_1:def 4; then (sqrt ((ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2))^2<r^2 by A6,SQUARE_1:78; then A9:(ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 < r^2 by A8,SQUARE_1:def 4; (ppe`1 - pq`1)^2 <= (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 by A7,REAL_2: 173 ; then (ppe`1 - pq`1)^2 <r^2 by A9,AXIOMS:22; then ppe`1 - pq`1 < r by A3,SQUARE_1:77; then ppe`1 <pq`1+r by REAL_1:84 ; then ppe`1 - r < pq`1 by REAL_1:84; then A10: s-(s-s1)/2 < pq`1 by A2,EUCLID:56; s-(s-s1)/2 = s-s1+s1-(s-s1)/2 by XCMPLX_1:27 .= (s-s1)-(s-s1)/2+s1 by XCMPLX_1:29 .= (s-s1)/2+(s-s1)/2-(s-s1)/2+s1 by XCMPLX_1:66 .= (s-s1)/2-(s-s1)/2+(s-s1)/2+s1 by XCMPLX_1:29 .=0+(s-s1)/2+s1 by XCMPLX_1:14 .=r+s1; then s1< s-(s-s1)/2 by A3,REAL_2:174; then |[pq`1,pq`2]|=x & s1<pq`1 by A4,A10,AXIOMS:22,EUCLID:57; hence thesis by A1; end; hence thesis by A3; end; hence thesis by Lm10,TOPMETR:22; end; theorem Th26: for s1 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:s1>s } holds P is open proof let s1; let P be Subset of TOP-REAL 2 such that A1:P= { |[ s,t ]|:s1>s }; for pe being Point of Euclid 2 st pe in P ex r being real number st r>0 & Ball(pe,r) c= P proof let pe be Point of Euclid 2;assume pe in P; then consider s,t such that A2:|[ s,t ]|=pe & s1>s by A1; set r=(s1-s)/2; s1-s>0 by A2,SQUARE_1:11; then A3: r>0 by REAL_2:127; Ball(pe,r) c= P proof let x be set; assume x in Ball(pe,r); then x in {q where q is Element of Euclid 2:dist(pe,q)<r} by METRIC_1:18; then consider q being Element of Euclid 2 such that A4: q=x and A5: dist(pe,q)<r; reconsider ppe=pe, pq=q as Point of TOP-REAL 2 by TOPREAL3:13; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(pe,q)=dist(pe,q) by METRIC_1:def 1; then A6: sqrt ((ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2)<r by A5,TOPREAL3:12; A7: 0 <= (ppe`1 - pq`1)^2 & 0 <= (ppe`2 - pq`2)^2 by SQUARE_1:72; then A8: 0+0 <= (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 by REAL_1:55; then 0 <=sqrt ( (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2) by SQUARE_1:def 4; then (sqrt ((ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2))^2<r^2 by A6,SQUARE_1:78; then A9:(ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 < r^2 by A8,SQUARE_1:def 4; (ppe`1 - pq`1)^2 <= (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 by A7,REAL_2: 173 ; then (ppe`1 - pq`1)^2 <r^2 by A9,AXIOMS:22; then (pq`1-ppe`1)^2 <r^2 by Lm9; then pq`1-ppe`1 < r by A3,SQUARE_1:77; then ppe`1 + r > pq`1 by REAL_1:84; then A10: s+(s1-s)/2 > pq`1 by A2,EUCLID:56; s+(s1-s)/2 = s-s1+s1+(s1-s)/2 by XCMPLX_1:27 .= s-s1 +(s1-s)/2+s1 by XCMPLX_1:1 .=-(s1-s)+(s1-s)/2+s1 by XCMPLX_1:143 .=-((s1-s)/2+(s1-s)/2)+(s1-s)/2+s1 by XCMPLX_1:66 .=-r-r+r+s1 by XCMPLX_1:161 .=-r+r-r+s1 by XCMPLX_1:29 .=0-r+s1 by XCMPLX_0:def 6 .=-r+s1 by XCMPLX_1:150 .=s1-r by XCMPLX_0:def 8; then s1> s+(s1-s)/2 by A3,REAL_2:174; then |[pq`1,pq`2]|=x & s1>pq`1 by A4,A10,AXIOMS:22,EUCLID:57; hence thesis by A1; end; hence thesis by A3; end; hence thesis by Lm10,TOPMETR:22; end; theorem Th27: for s1 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:s1<t } holds P is open proof let s1; let P be Subset of TOP-REAL 2 such that A1:P= { |[ s,t ]|:s1<t }; for pe being Point of Euclid 2 st pe in P ex r being real number st r>0 & Ball(pe,r) c= P proof let pe be Point of Euclid 2;assume pe in P; then consider s,t such that A2:|[ s,t ]|=pe & s1<t by A1; set r=(t-s1)/2; t-s1>0 by A2,SQUARE_1:11; then A3: r>0 by REAL_2:127; Ball(pe,r) c= P proof let x be set; assume x in Ball(pe,r); then x in {q where q is Element of Euclid 2:dist(pe,q)<r} by METRIC_1:18; then consider q being Element of Euclid 2 such that A4: q=x and A5: dist(pe,q)<r; reconsider ppe=pe, pq=q as Point of TOP-REAL 2 by TOPREAL3:13; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(pe,q)=dist(pe,q) by METRIC_1:def 1; then A6: sqrt ((ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2)<r by A5,TOPREAL3:12; A7: 0 <= (ppe`1 - pq`1)^2 & 0 <= (ppe`2 - pq`2)^2 by SQUARE_1:72; then A8: 0+0 <= (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 by REAL_1:55; then 0 <=sqrt ( (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2) by SQUARE_1:def 4; then (sqrt ((ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2))^2<r^2 by A6,SQUARE_1:78; then A9:(ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 < r^2 by A8,SQUARE_1:def 4; (ppe`2 - pq`2)^2 <= (ppe`2 - pq`2)^2+(ppe`1 - pq`1)^2 by A7,REAL_2:173; then (ppe`2 - pq`2)^2 <r^2 by A9,AXIOMS:22; then ppe`2 - pq`2 < r by A3,SQUARE_1:77; then ppe`2 <pq`2+r by REAL_1:84 ; then ppe`2 - r < pq`2 by REAL_1:84; then A10: t-(t-s1)/2 < pq`2 by A2,EUCLID:56; t-(t-s1)/2 = t-s1+s1-(t-s1)/2 by XCMPLX_1:27 .= (t-s1)-(t-s1)/2+s1 by XCMPLX_1:29 .= (t-s1)/2+(t-s1)/2-(t-s1)/2+s1 by XCMPLX_1:66 .= (t-s1)/2-(t-s1)/2+(t-s1)/2+s1 by XCMPLX_1:29 .=0+(t-s1)/2+s1 by XCMPLX_1:14 .=r+s1; then s1< t-(t-s1)/2 by A3,REAL_2:174; then |[pq`1,pq`2]|=x & s1<pq`2 by A4,A10,AXIOMS:22,EUCLID:57; hence thesis by A1; end; hence thesis by A3; end; hence thesis by Lm10,TOPMETR:22; end; theorem Th28: for s1 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:s1>t } holds P is open proof let s1; let P be Subset of TOP-REAL 2 such that A1:P= { |[ s,t ]|:s1>t }; for pe being Point of Euclid 2 st pe in P ex r being real number st r>0 & Ball(pe,r) c= P proof let pe be Point of Euclid 2;assume pe in P; then consider s,t such that A2:|[ s,t ]|=pe & s1>t by A1; set r=(s1-t)/2; s1-t>0 by A2,SQUARE_1:11; then A3: r>0 by REAL_2:127; Ball(pe,r) c= P proof let x be set; assume x in Ball(pe,r); then x in {q where q is Element of Euclid 2:dist(pe,q)<r} by METRIC_1:18; then consider q being Element of Euclid 2 such that A4: q=x and A5: dist(pe,q)<r; reconsider ppe=pe, pq=q as Point of TOP-REAL 2 by TOPREAL3:13; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(pe,q)=dist(pe,q) by METRIC_1:def 1; then A6: sqrt ((ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2)<r by A5,TOPREAL3:12; A7: 0 <= (ppe`1 - pq`1)^2 & 0 <= (ppe`2 - pq`2)^2 by SQUARE_1:72; then A8: 0+0 <= (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 by REAL_1:55; then 0 <=sqrt ( (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2) by SQUARE_1:def 4; then (sqrt ((ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2))^2<r^2 by A6,SQUARE_1:78; then A9:(ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 < r^2 by A8,SQUARE_1:def 4; (ppe`2 - pq`2)^2 <= (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 by A7,REAL_2: 173 ; then (ppe`2 - pq`2)^2 <r^2 by A9,AXIOMS:22; then (pq`2-ppe`2)^2 <r^2 by Lm9; then pq`2-ppe`2 < r by A3,SQUARE_1:77; then ppe`2 + r > pq`2 by REAL_1:84; then A10: t+(s1-t)/2 > pq`2 by A2,EUCLID:56; t+(s1-t)/2 = t-s1+s1+(s1-t)/2 by XCMPLX_1:27 .= t-s1 +(s1-t)/2+s1 by XCMPLX_1:1 .=-(s1-t)+(s1-t)/2+s1 by XCMPLX_1:143 .=-((s1-t)/2+(s1-t)/2)+(s1-t)/2+s1 by XCMPLX_1:66 .=-r-r+r+s1 by XCMPLX_1:161 .=-r+r-r+s1 by XCMPLX_1:29 .=0-r+s1 by XCMPLX_0:def 6 .=-r+s1 by XCMPLX_1:150 .=s1-r by XCMPLX_0:def 8; then s1> t+(s1-t)/2 by A3,REAL_2:174; then |[pq`1,pq`2]|=x & s1>pq`2 by A4,A10,AXIOMS:22,EUCLID:57; hence thesis by A1; end; hence thesis by A3; end; hence thesis by Lm10,TOPMETR:22; end; theorem Th29: for s1,t1,s2,t2 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:s1<s & s<s2 & t1<t & t<t2} holds P is open proof let s1,t1,s2,t2; let P be Subset of TOP-REAL 2; assume A1:P = { |[ s,t ]|:s1<s & s<s2 & t1<t & t<t2}; {|[s,t]|:s1<s} is Subset of TOP-REAL 2 by Lm2,Lm5; then reconsider P1={|[s,t]|:s1<s} as Subset of TOP-REAL 2; {|[s,t]|:s<s2} is Subset of TOP-REAL 2 by Lm2,Lm3; then reconsider P2={|[s,t]|:s<s2} as Subset of TOP-REAL 2; {|[s,t]|:t1<t} is Subset of TOP-REAL 2 by Lm2,Lm6; then reconsider P3={|[s,t]|:t1<t} as Subset of TOP-REAL 2; {|[s,t]|:t<t2} is Subset of TOP-REAL 2 by Lm2,Lm4; then reconsider P4={|[s,t]|:t<t2} as Subset of TOP-REAL 2; A2: P=P1 /\ P2 /\ P3 /\ P4 by A1,Th12; P1 is open & P2 is open by Th25,Th26; then A3: P1 /\ P2 is open by TOPS_1:38; A4: P3 is open & P4 is open by Th27,Th28; then P1 /\ P2 /\ P3 is open by A3,TOPS_1:38; hence thesis by A2,A4,TOPS_1:38; end; theorem Th30: for s1,t1,s2,t2 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} holds P is open proof let s1,t1,s2,t2; let P be Subset of TOP-REAL 2; assume P={ |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)}; then A1: P={ |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5} \/ {|[s6,t6]|:t2<t6} by Th13; { |[ s,t ]|:s<s1} is Subset of TOP-REAL 2 by Lm2,Lm3; then reconsider A0={ |[ s,t ]|:s<s1} as Subset of TOP-REAL 2; { |[ s,t ]|:t<t1} is Subset of TOP-REAL 2 by Lm2,Lm4; then reconsider A1={ |[ s,t ]|:t<t1} as Subset of TOP-REAL 2; { |[ s,t ]|:s2<s} is Subset of TOP-REAL 2 by Lm2,Lm5; then reconsider A2={ |[ s,t ]|:s2<s} as Subset of TOP-REAL 2; { |[ s,t ]|:t2<t} is Subset of TOP-REAL 2 by Lm2,Lm6; then reconsider A3={ |[ s,t ]|:t2<t} as Subset of TOP-REAL 2; A0 is open & A1 is open by Th26,Th28; then A2: A0 \/ A1 is open by TOPS_1:37; A2 is open by Th25; then A3: A0 \/ A1 \/ A2 is open by A2,TOPS_1:37; A3 is open by Th27; hence thesis by A1,A3,TOPS_1:37; end; theorem Th31: for s1,t1,s2,t2,P,Q st P = { |[ sa,ta ]|:s1<sa & sa<s2 & t1<ta & ta<t2} & Q = { |[ sb,tb ]|:not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} holds P misses Q proof let s1,t1,s2,t2,P,Q; assume that A1: P = { |[ sa,ta ]|:s1<sa & sa<s2 & t1<ta & ta<t2} and A2: Q = { |[ sb,tb ]|:not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)}; assume not thesis; then consider x be set such that A3: x in P & x in Q by XBOOLE_0:3; consider sa,ta such that A4:|[sa,ta]|=x and A5: (s1<sa & sa<s2 & t1<ta & ta<t2) by A1,A3; consider sb,tb such that A6:|[sb,tb]|=x and A7:not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2) by A2,A3; set p= |[sa,ta]|; p`1=sa & p`1=sb & p`2=ta & p`2=tb by A4,A6,EUCLID:56; hence contradiction by A5,A7; end; theorem Th32: for s1,s2,t1,t2 be Real holds {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2} = {|[sa,ta]|:s1<sa & sa<s2 & t1<ta & ta<t2} proof let s1,s2,t1,t2; now let x be set; A1: now assume x in {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2}; then consider pp being Point of TOP-REAL 2 such that A2:pp=x & s1<pp`1 & pp`1<s2 & t1<pp`2 & pp`2<t2; |[pp`1,pp`2]|=x & s1<pp`1 & pp`1<s2 & t1<pp`2 & pp`2<t2 by A2,EUCLID:57; hence x in {|[s1a,t1a]|:s1<s1a & s1a<s2 & t1<t1a & t1a<t2}; end; now assume x in {|[sa,ta]|:s1<sa & sa<s2 & t1<ta & ta<t2}; then consider sa,ta being Real such that A3: |[sa,ta]|=x & (s1<sa & sa<s2 & t1<ta & ta<t2); set pa=|[sa,ta]|; pa=x & (s1<pa`1 & pa`1<s2 & t1<pa`2 & pa`2<t2) by A3,EUCLID:56; hence x in {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2}; end;hence x in {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2} iff x in {|[sa,ta]|:s1<sa & sa<s2 & t1<ta & ta<t2} by A1;end; hence thesis by TARSKI:2; end; theorem Th33: for s1,s2,t1,t2 holds {qc where qc is Point of TOP-REAL 2: not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)} = {|[sb,tb]| : not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} proof let s1,s2,t1,t2; now let x be set; A1: now assume x in {qc where qc is Point of TOP-REAL 2: not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)}; then consider q being Point of TOP-REAL 2 such that A2:q=x & not (s1<=q`1 & q`1<=s2 & t1<=q`2 & q`2<=t2); |[q`1,q`2]|=x & not (s1<=q`1 & q`1<=s2 & t1<=q`2 & q`2<=t2) by A2,EUCLID:57; hence x in {|[s2a,t2a]| : not (s1<=s2a & s2a<=s2 & t1<=t2a & t2a<=t2)}; end; now assume x in {|[sb,tb]| : not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)}; then consider sb,tb being Real such that A3: |[sb,tb]|=x & not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2); set qa=|[sb,tb]|; qa=x & not (s1<=qa`1 & qa`1<=s2 & t1<=qa`2 & qa`2<=t2) by A3,EUCLID:56; hence x in {qc where qc is Point of TOP-REAL 2: not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)}; end; hence x in {qc where qc is Point of TOP-REAL 2: not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)} iff x in {|[sb,tb]| : not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} by A1; end; hence thesis by TARSKI:2; end; theorem Th34: for s1,s2,t1,t2 holds { p0 where p0 is Point of TOP-REAL 2:s1<p0`1 & p0`1<s2 & t1<p0`2 & p0`2<t2} is Subset of TOP-REAL 2 proof let s1,s2,t1,t2; { |[ sc,tc ]|: s1<sc & sc<s2 & t1<tc & tc<t2} is Subset of TOP-REAL 2 by Lm2,Lm7; hence thesis by Th32; end; theorem Th35: for s1,s2,t1,t2 holds { pq where pq is Point of TOP-REAL 2: not (s1<=pq`1 & pq`1<=s2 & t1<=pq`2 & pq`2<=t2)} is Subset of TOP-REAL 2 proof let s1,s2,t1,t2; { |[ sb,tb ]|:not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} is Subset of TOP-REAL 2 by Lm2,Lm8; hence thesis by Th33; end; theorem Th36: for s1,t1,s2,t2,P st P = { p0 where p0 is Point of TOP-REAL 2:s1<p0`1 & p0`1<s2 & t1<p0`2 & p0`2<t2} holds P is connected proof let s1,t1,s2,t2,P;assume P = { p0 where p0 is Point of TOP-REAL 2:s1<p0`1 & p0`1<s2 & t1<p0`2 & p0`2<t2}; then P={|[sa,ta]|:s1<sa & sa<s2 & t1<ta & ta<t2} by Th32; hence thesis by Th15; end; theorem Th37: for s1,t1,s2,t2,P st P = { pq where pq is Point of TOP-REAL 2: not (s1<=pq`1 & pq`1<=s2 & t1<=pq`2 & pq`2<=t2)} holds P is connected proof let s1,t1,s2,t2,P;assume P= { pq where pq is Point of TOP-REAL 2: not (s1<=pq`1 & pq`1<=s2 & t1<=pq`2 & pq`2<=t2)}; then P = {|[sb,tb]| : not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} by Th33; hence thesis by Th24; end; theorem Th38: for s1,t1,s2,t2 for P being Subset of TOP-REAL 2 st P = { p0 where p0 is Point of TOP-REAL 2:s1<p0`1 & p0`1<s2 & t1<p0`2 & p0`2<t2} holds P is open proof let s1,t1,s2,t2; let P be Subset of TOP-REAL 2; assume P = { p0 where p0 is Point of TOP-REAL 2:s1<p0`1 & p0`1<s2 & t1<p0`2 & p0`2<t2}; then P={|[sa,ta]|:s1<sa & sa<s2 & t1<ta & ta<t2} by Th32; hence thesis by Th29; end; theorem Th39: for s1,t1,s2,t2 for P being Subset of TOP-REAL 2 st P = { pq where pq is Point of TOP-REAL 2: not (s1<=pq`1 & pq`1<=s2 & t1<=pq`2 & pq`2<=t2)} holds P is open proof let s1,t1,s2,t2; let P be Subset of TOP-REAL 2;assume P= { pq where pq is Point of TOP-REAL 2: not (s1<=pq`1 & pq`1<=s2 & t1<=pq`2 & pq`2<=t2)}; then P = {|[sb,tb]| : not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} by Th33; hence thesis by Th30; end; theorem Th40: for s1,t1,s2,t2,P,Q st P = {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2} & Q = {qc where qc is Point of TOP-REAL 2: not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)} holds P misses Q proof let s1,t1,s2,t2,P,Q; assume that A1: P= {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2} and A2:Q= {qc where qc is Point of TOP-REAL 2: not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)}; A3:P={|[sa,ta]|:s1<sa & sa<s2 & t1<ta & ta<t2} by A1,Th32; Q= {|[sb,tb]| : not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} by A2,Th33; hence thesis by A3,Th31; end; theorem Th41: for s1,t1,s2,t2 for P,P1,P2 being Subset of TOP-REAL 2 st s1<s2 & t1<t2 & P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)} holds P`= P1 \/ P2 & P`<> {} & P1 misses P2 & for P1A,P2B being Subset of (TOP-REAL 2)|P` st P1A = P1 & P2B = P2 holds P1A is_a_component_of (TOP-REAL 2)|P` & P2B is_a_component_of (TOP-REAL 2)|P` proof let s1,t1,s2,t2; let P,P1,P2 be Subset of TOP-REAL 2; assume that A1:s1<s2 & t1<t2 and A2:P ={p where p is Point of TOP-REAL 2:p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} and A3: P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} and A4: P2 = {pc where pc is Point of TOP-REAL 2: not( s1<=pc`1 & pc`1<=s2 & t1<=pc`2 & pc`2<=t2)}; now let x be set;assume A5: x in P`; then x in P`; then x in [#](TOP-REAL 2) \ P by PRE_TOPC:17; then A6:x in [#](TOP-REAL 2) & not x in P by XBOOLE_0:def 4; reconsider pd=x as Point of TOP-REAL 2 by A5; not (pd`1 = s1 & pd`2 <= t2 & pd`2 >= t1 or pd`1 <= s2 & pd`1 >= s1 & pd`2 = t2 or pd`1 <= s2 & pd`1 >= s1 & pd`2 = t1 or pd`1 = s2 & pd`2 <= t2 & pd`2 >= t1)by A2,A6; then s1<pd`1 & pd`1<s2 & t1 <pd`2 & pd`2 < t2 or not( s1<=pd`1 & pd`1<=s2 & t1<=pd`2 & pd`2<=t2) by REAL_1:def 5; then x in P1 or x in P2 by A3,A4;hence x in P1 \/ P2 by XBOOLE_0:def 2 ; end; then A7:P` c= P1 \/ P2 by TARSKI:def 3; now let x be set such that A8: x in P1 \/ P2; now per cases by A8,XBOOLE_0:def 2; suppose A9: x in P1; then A10: ex pa st pa=x & s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2 by A3; now assume x in P; then ex pb st pb=x &(pb`1 = s1 & pb`2 <= t2 & pb`2 >= t1 or pb`1 <= s2 & pb`1 >= s1 & pb`2 = t2 or pb`1 <= s2 & pb`1 >= s1 & pb`2 = t1 or pb`1 = s2 & pb`2 <= t2 & pb`2 >= t1) by A2; hence contradiction by A10; end; hence x in P` by A9,SUBSET_1:50; suppose x in P2; then consider pc being Point of TOP-REAL 2 such that A11:pc=x and A12: not( s1<=pc`1 & pc`1<=s2 & t1<=pc`2 & pc`2<=t2) by A4; now assume pc in P; then ex p being Point of TOP-REAL 2 st p=pc & ( p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1) by A2; hence contradiction by A1,A12; end; hence x in P`by A11,SUBSET_1:50; end; hence x in P`; end; then P1 \/ P2 c= P` by TARSKI:def 3; hence A13: P`=P1 \/ P2 by A7,XBOOLE_0:def 10; set s3 =(s1+s2)/2, t3=(t1+t2)/2; s1+s1<s1+s2 & t1+t1<t1+t2 by A1,REAL_1:53; then (s1+s1)/2<s3 & (t1+t1)/2<t3 by REAL_1:73; then A14: s1<s3 & t1<t3 by XCMPLX_1:65; s1+s2<s2+s2 & t1+t2<t2+t2 by A1,REAL_1:53; then s3<(s2+s2)/2 & t3<(t2+t2)/2 by REAL_1:73; then A15: s3<s2 & t3<t2 by XCMPLX_1:65; set pp=|[s3,t3]|; pp`1=s3 & pp`2=t3 by EUCLID:56; then A16: |[ s3,t3 ]| in { pp2 where pp2 is Point of TOP-REAL 2: s1<pp2`1 & pp2`1<s2 & t1<pp2`2 & pp2`2<t2} by A14,A15; then consider x be set such that A17:x in P1 by A3; thus P`<>{} by A13,A17,XBOOLE_0:def 2; set P' = P`; P1 misses P2 by A3,A4,Th40; hence A18: P1 /\ P2 = {} by XBOOLE_0:def 7; now let P1A,P2B be Subset of (TOP-REAL 2)|P'; assume A19: P1A=P1 & P2B=P2; P1 is connected by A3,Th36; then A20: P1A is connected by A19,CONNSP_1:24; A21:P2 is connected by A4,Th37; A22: P2={ |[ sa,ta ]|:not (s1<=sa & sa<=s2 & t1<=ta & ta<=t2)} by A4,Th33; reconsider A0={ |[ s3a,t3a ]|:s3a<s1} as Subset of TOP-REAL 2 by Lm2,Lm3; reconsider A1={ |[ s4,t4 ]|:t4<t1} as Subset of TOP-REAL 2 by Lm2,Lm4; reconsider A2={ |[ s5,t5 ]|:s2<s5} as Subset of TOP-REAL 2 by Lm2,Lm5; reconsider A3={ |[ s6,t6 ]|:t2<t6} as Subset of TOP-REAL 2 by Lm2,Lm6; A23: P2=A0 \/ A1 \/ A2 \/ A3 by A22,Th13; t2+1>t2 by REAL_2:174; then |[s2+1,t2+1]| in A3; then A24: P2B <>{} by A19,A23,XBOOLE_1:15; A25: P2B is connected by A19,A21,CONNSP_1:24; A26: for CP being Subset of (TOP-REAL 2)|(P') st CP is connected holds P1A c= CP implies P1A = CP proof let CP be Subset of (TOP-REAL 2)|(P'); assume CP is connected; then A27:((TOP-REAL 2)|P')|CP is connected by CONNSP_1:def 3; now assume A28:P1A c= CP; P1A /\ CP c= the carrier of ((TOP-REAL 2)|P')|CP proof P1A /\ CP c= CP by XBOOLE_1:17; hence thesis by Th1; end; then reconsider AP=P1A /\ CP as Subset of ((TOP-REAL 2)|P')|CP; A29: P1 /\ P` =P1A by A13,A19,XBOOLE_1:21; P1 is open by A3,Th38; then A30: P1 in the topology of TOP-REAL 2 by PRE_TOPC:def 5; A31: P`= [#]((TOP-REAL 2)|P') by PRE_TOPC:def 10; P1 /\ [#]((TOP-REAL 2)|P')=P1A by A29,PRE_TOPC:def 10; then A32: P1A in the topology of (TOP-REAL 2)|P' by A30,PRE_TOPC:def 9; A33: CP=[#](((TOP-REAL 2)|P')|CP) by PRE_TOPC:def 10; A34:AP<>{}(((TOP-REAL 2)|P')|CP) by A3,A16,A19,A28,XBOOLE_1:28; AP in the topology of ((TOP-REAL 2)|P')|CP by A32,A33,PRE_TOPC:def 9; then A35:AP is open by PRE_TOPC:def 5; P2B /\ CP c= the carrier of ((TOP-REAL 2)|P')|CP proof P2B /\ CP c= CP by XBOOLE_1:17; hence thesis by Th1; end; then reconsider BP=P2B /\ CP as Subset of ((TOP-REAL 2)|P')|CP; A36: P2 /\ P` =P2B by A13,A19,XBOOLE_1:21; P2 is open by A4,Th39; then A37: P2 in the topology of TOP-REAL 2 by PRE_TOPC:def 5; P2 /\ [#]((TOP-REAL 2)|P')=P2B by A36,PRE_TOPC:def 10; then A38: P2B in the topology of (TOP-REAL 2)|P' by A37,PRE_TOPC:def 9; CP=[#](((TOP-REAL 2)|P')|CP) by PRE_TOPC:def 10; then BP in the topology of ((TOP-REAL 2)|P')|CP by A38,PRE_TOPC:def 9; then A39:BP is open by PRE_TOPC:def 5; CP c= P` by A31,PRE_TOPC:14; then A40: CP=(P1A \/ P2B) /\ CP by A13,A19,XBOOLE_1:28 .=AP \/ BP by XBOOLE_1:23; now assume A41: BP<>{}; A42:AP /\ BP = P1A /\ (P2B /\ CP) /\ CP by XBOOLE_1:16 .= P1A /\ P2B /\ CP /\ CP by XBOOLE_1:16 .= P1A /\ P2B /\ (CP /\ CP) by XBOOLE_1:16 .= P1A /\ P2B /\ CP; P1 misses P2 by A3,A4,Th40; then P1 /\ P2 = {} by XBOOLE_0:def 7; then AP misses BP by A19,A42,XBOOLE_0:def 7; hence contradiction by A27,A33,A34,A35,A39,A40,A41,CONNSP_1:12; end; hence thesis by A40,XBOOLE_1:28; end; hence thesis; end; hence P1A is_a_component_of (TOP-REAL 2)|P' by A20,CONNSP_1:def 5; for CP being Subset of (TOP-REAL 2)|(P') st CP is connected holds P2B c= CP implies P2B = CP proof let CP be Subset of (TOP-REAL 2)|(P'); assume CP is connected; then A43:((TOP-REAL 2)|P')|CP is connected by CONNSP_1:def 3; assume A44:P2B c= CP; P2B /\ CP c= the carrier of ((TOP-REAL 2)|P')|CP proof P2B /\ CP c= CP by XBOOLE_1:17; hence thesis by Th1; end; then reconsider BP=P2B /\ CP as Subset of ((TOP-REAL 2)|P')|CP; A45: P2 /\ P` =P2B by A13,A19,XBOOLE_1:21; P2 is open by A4,Th39; then A46: P2 in the topology of TOP-REAL 2 by PRE_TOPC:def 5; A47: P`= [#]((TOP-REAL 2)|P') by PRE_TOPC:def 10; P2 /\ [#]((TOP-REAL 2)|P')=P2B by A45,PRE_TOPC:def 10; then A48: P2B in the topology of (TOP-REAL 2)|P' by A46,PRE_TOPC:def 9; A49: CP=[#](((TOP-REAL 2)|P')|CP) by PRE_TOPC:def 10; A50:BP<>{}(((TOP-REAL 2)|P')|CP) by A24,A44,XBOOLE_1:28; BP in the topology of ((TOP-REAL 2)|P')|CP by A48,A49,PRE_TOPC:def 9; then A51:BP is open by PRE_TOPC:def 5; P1A /\ CP c= the carrier of ((TOP-REAL 2)|P')|CP proof P1A /\ CP c= CP by XBOOLE_1:17; hence thesis by Th1; end; then reconsider AP=P1A /\ CP as Subset of ((TOP-REAL 2)|P')|CP; A52: P1 /\ P` =P1A by A13,A19,XBOOLE_1:21; P1 is open by A3,Th38; then A53: P1 in the topology of TOP-REAL 2 by PRE_TOPC:def 5; P1 /\ [#]((TOP-REAL 2)|P')=P1A by A52,PRE_TOPC:def 10; then A54: P1A in the topology of (TOP-REAL 2)|P' by A53,PRE_TOPC:def 9; CP=[#](((TOP-REAL 2)|P')|CP) by PRE_TOPC:def 10; then AP in the topology of ((TOP-REAL 2)|P')|CP by A54,PRE_TOPC:def 9; then A55:AP is open by PRE_TOPC:def 5; CP c= P` by A47,PRE_TOPC:14; then A56: CP=(P1A \/ P2B) /\ CP by A13,A19,XBOOLE_1:28 .=AP \/ BP by XBOOLE_1:23; now assume A57: AP<>{}; AP /\ BP = P1A /\ (P2B /\ CP) /\ CP by XBOOLE_1:16 .= P1A /\ P2B /\ CP /\ CP by XBOOLE_1:16 .= P1A /\ P2B /\ (CP /\ CP) by XBOOLE_1:16 .= P1A /\ P2B /\ CP; then AP misses BP by A18,A19,XBOOLE_0:def 7; hence contradiction by A43,A49,A50,A51,A55,A56,A57,CONNSP_1:12; end; hence thesis by A44,A56,XBOOLE_1:28; end; hence P1A is_a_component_of (TOP-REAL 2)|P' & P2B is_a_component_of (TOP-REAL 2)|P' by A20,A25,A26,CONNSP_1:def 5; end; hence thesis; end; theorem Th42: for s1,t1,s2,t2 for P,P1,P2 being Subset of TOP-REAL 2 st s1<s2 & t1<t2 & P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)} holds P = (Cl P1) \ P1 & P = (Cl P2) \P2 proof let s1,t1,s2,t2; let P,P1,P2 be Subset of TOP-REAL 2; assume that A1: s1<s2 & t1<t2 and A2:P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)}; reconsider PP = P` as Subset of TOP-REAL 2; A3: PP=P1 \/ P2 & PP <> {} & P1 misses P2 by A1,A2,Th41; then A4:P2 c= P1` & P1 c= P2` by PRE_TOPC:21; P= (P1 \/ P2)` by A3; then P=[#](TOP-REAL 2) \ (P1 \/ P2) by PRE_TOPC:17; then A5:P=([#](TOP-REAL 2)\P1) /\ ([#](TOP-REAL 2)\P2) by XBOOLE_1:53; then P c=[#](TOP-REAL 2)\P2 by XBOOLE_1:17; then P c= P2` by PRE_TOPC:17; then A6:P c= P2`; A7:[#](TOP-REAL 2)=P \/ (P2 \/ P1) by A3,PRE_TOPC:18 .=P \/ P2 \/ P1 by XBOOLE_1:4; A8: P1`= P \/ P2 proof now let x be set;assume x in P1`; then x in P1`; then x in [#](TOP-REAL 2)\P1 by PRE_TOPC:17; then x in [#](TOP-REAL 2) & not x in P1 by XBOOLE_0:def 4; hence x in P \/ P2 by A7,XBOOLE_0:def 2; end; then A9:P1` c=P \/ P2 by TARSKI:def 3; now let x be set;assume x in P \/ P2; then x in P or x in P2 by XBOOLE_0:def 2; then x in [#](TOP-REAL 2)\P1 or x in P2 by A5,XBOOLE_0:def 3; then x in P1` by A4,PRE_TOPC:17; hence x in P1`; end; then P \/ P2 c=P1` by TARSKI:def 3; hence thesis by A9,XBOOLE_0:def 10; end; A10:P1 is open by A2,Th38; [#](TOP-REAL 2) \ P1` = P1`` by PRE_TOPC:17 .=P1; then A11: P \/ P2 is closed by A8,A10,PRE_TOPC:def 6; A12:P2 c= P \/ P2 by XBOOLE_1:7; Cl P2 c= P \/ P2 proof let x be set;assume x in Cl P2; hence thesis by A11,A12,PRE_TOPC:45; end; then A13:(Cl P2) \ P2 c= P \/ P2 \P2 by XBOOLE_1:33; (P \/ P2) \ P2 c= P proof let x be set;assume x in (P \/ P2) \ P2; then x in P \/ P2 & not x in P2 by XBOOLE_0:def 4; hence thesis by XBOOLE_0:def 2; end; then A14:(Cl P2)\P2 c= P by A13,XBOOLE_1:1; A15: P c= (Cl P2)\P2 proof P c= Cl P2 proof let x be set;assume x in P; then consider p being Point of TOP-REAL 2 such that A16:p=x and A17: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1 by A2; reconsider q=p as Point of Euclid 2 by TOPREAL3:13; now per cases by A17; suppose A18: p`1 = s1 & p`2 <= t2 & p`2 >= t1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P2 meets Q proof let Q be Subset of TOP-REAL 2;assume that A19:Q is open and A20:p in Q; consider r be real number such that A21: r>0 and A22:Ball(q,r) c= Q by A19,A20,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; set pa=|[p`1-r/2,p`2]|; A23:pa`1=p`1-r/2 & pa`2=p`2 by EUCLID:56; A24: r/2>0 by A21,SEQ_2:3; then not( s1<=pa`1 & pa`1<=s2 & t1<=pa`2 & pa`2<=t2) by A18,A23, REAL_2:174; then A25:pa in P2 by A2; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A26: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A27:(p`2 - pa`2)^2 =0 by A23,SQUARE_1:60,XCMPLX_1:14; A28: p`1-pa`1 = r/2 by A23,XCMPLX_1:18; then p`1 - pa`1 < r by A21,SEQ_2:4; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A24,A27,A28,SQUARE_1:78; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A26,SQUARE_1:def 4; then A29:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A21,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A29,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; then P2 /\ Q <> {} by A22,A25,XBOOLE_0:def 3; hence thesis by XBOOLE_0:def 7; end; hence x in Cl P2 by A16,PRE_TOPC:def 13; suppose A30: p`1 <= s2 & p`1 >= s1 & p`2 = t2; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P2 meets Q proof let Q be Subset of TOP-REAL 2;assume that A31:Q is open and A32:p in Q; consider r being real number such that A33: r>0 and A34:Ball(q,r) c= Q by A31,A32,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; set pa=|[p`1,p`2+r/2]|; A35:pa`1=p`1 & pa`2=p`2+r/2 by EUCLID:56; A36: r/2>0 by A33,SEQ_2:3; then not( s1<=pa`1 & pa`1<=s2 & t1<=pa`2 & pa`2<=t2) by A30,A35, REAL_2:174; then A37:pa in P2 by A2; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A38: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A39:(p`1 - pa`1)^2 =0 by A35,SQUARE_1:60,XCMPLX_1:14; A40: pa`2-p`2 = r/2 by A35,XCMPLX_1:26; then A41:pa`2 - p`2 < r by A33,SEQ_2:4; (pa`2-p`2)^2=(p`2-pa`2)^2 by Lm9; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A36,A39,A40,A41,SQUARE_1:78; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A38,SQUARE_1:def 4; then A42:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A33,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A42,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; then P2 /\ Q <> {} by A34,A37,XBOOLE_0:def 3; hence thesis by XBOOLE_0:def 7; end; hence x in Cl P2 by A16,PRE_TOPC:def 13; suppose A43: p`1 <= s2 & p`1 >= s1 & p`2 = t1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P2 meets Q proof let Q be Subset of TOP-REAL 2;assume that A44:Q is open and A45:p in Q; consider r being real number such that A46: r>0 and A47:Ball(q,r) c= Q by A44,A45,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; set pa=|[p`1,p`2-r/2]|; A48:pa`1=p`1 & pa`2=p`2-r/2 by EUCLID:56; A49: r/2>0 by A46,SEQ_2:3; then not( s1<=pa`1 & pa`1<=s2 & t1<=pa`2 & pa`2<=t2) by A43,A48,REAL_2:174; then A50:pa in P2 by A2; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A51: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A52:(p`1 - pa`1)^2 =0 by A48,SQUARE_1:60,XCMPLX_1:14; A53: p`2-pa`2 = r/2 by A48,XCMPLX_1:18; then p`2 - pa`2 < r by A46,SEQ_2:4; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A49,A52,A53,SQUARE_1:78; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A51,SQUARE_1:def 4; then A54:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A46,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A54,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A47,A50,XBOOLE_0:3; end; hence x in Cl P2 by A16,PRE_TOPC:def 13; suppose A55: p`1 = s2 & p`2 <= t2 & p`2 >= t1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P2 meets Q proof let Q be Subset of TOP-REAL 2;assume that A56:Q is open and A57:p in Q; consider r being real number such that A58: r>0 and A59:Ball(q,r) c= Q by A56,A57,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; set pa=|[p`1+r/2,p`2]|; A60:pa`1=p`1+r/2 & pa`2=p`2 by EUCLID:56; A61: r/2>0 by A58,SEQ_2:3; then not( s1<=pa`1 & pa`1<=s2 & t1<=pa`2 & pa`2<=t2) by A55,A60,REAL_2:174; then A62:pa in P2 by A2; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A63: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A64:(p`2 - pa`2)^2 =0 by A60,SQUARE_1:60,XCMPLX_1:14; A65: pa`1-p`1 = r/2 by A60,XCMPLX_1:26; then A66:pa`1 - p`1 < r by A58,SEQ_2:4; (pa`1-p`1)^2=(p`1-pa`1)^2 by Lm9; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A61,A64,A65,A66,SQUARE_1:78; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A63,SQUARE_1:def 4; then A67:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A58,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A67,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A59,A62,XBOOLE_0:3; end; hence x in Cl P2 by A16,PRE_TOPC:def 13; end; hence x in Cl P2; end; then P c= Cl P2 /\ P2` by A6,XBOOLE_1:19; hence thesis by SUBSET_1:32; end; P c=[#](TOP-REAL 2)\P1 by A5,XBOOLE_1:17; then P c= P1` by PRE_TOPC:17; then A68:P c= P1`; [#](TOP-REAL 2)=P \/ (P1 \/ P2) by A3,PRE_TOPC:18; then A69:[#](TOP-REAL 2)=P \/ P1 \/ P2 by XBOOLE_1:4; A70: P2`= P \/ P1 proof now let x be set;assume x in P2`; then x in P2`; then x in [#](TOP-REAL 2)\P2 by PRE_TOPC:17; then x in [#](TOP-REAL 2) & not x in P2 by XBOOLE_0:def 4; hence x in P \/ P1 by A69,XBOOLE_0:def 2; end; then A71:P2` c=P \/ P1 by TARSKI:def 3; now let x be set;assume x in P \/ P1; then x in P or x in P1 by XBOOLE_0:def 2; then x in [#](TOP-REAL 2)\P2 or x in P1 by A5,XBOOLE_0:def 3; then x in P2` by A4,PRE_TOPC:17; hence x in P2`; end; then P \/ P1 c=P2` by TARSKI:def 3; hence thesis by A71,XBOOLE_0:def 10; end; A72:P2 is open by A2,Th39; [#](TOP-REAL 2) \ P2` = P2`` by PRE_TOPC:17 .=P2; then A73: P \/ P1 is closed by A70,A72,PRE_TOPC:def 6; A74:P1 c= P \/ P1 by XBOOLE_1:7; Cl P1 c= P \/ P1 proof let x be set;assume x in Cl P1; hence thesis by A73,A74,PRE_TOPC:45; end; then A75:(Cl P1) \ P1 c= P \/ P1 \P1 by XBOOLE_1:33; (P \/ P1) \ P1 c= P proof let x be set;assume x in (P \/ P1) \ P1; then x in P \/ P1 & not x in P1 by XBOOLE_0:def 4; hence thesis by XBOOLE_0:def 2; end; then A76:(Cl P1)\P1 c= P by A75,XBOOLE_1:1; P c= (Cl P1)\P1 proof P c= Cl P1 proof let x be set;assume x in P; then consider p being Point of TOP-REAL 2 such that A77:p=x and A78: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1 by A2; reconsider q=p as Point of Euclid 2 by TOPREAL3:13; now per cases by A78; suppose A79: p`1 = s1 & p`2 <= t2 & p`2 >= t1; now per cases by A79,REAL_1:def 5; suppose A80: p`1 = s1 & p`2 < t2 & p`2 > t1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A81:Q is open and A82:p in Q; consider r be real number such that A83: r>0 and A84:Ball(q,r) c= Q by A81,A82,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A85:r/2>0 by A83,SEQ_2:3; A86:r/2<r by A83,SEQ_2:4; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A87:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A88:r2<= r/2 & r2<=(s2-s1)/2 & r2<=(t2-t1)/2 by A87,AXIOMS:22; A89:r2<r by A86,A87,AXIOMS:22; A90: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A91:0<r2 by A85,SQUARE_1:38; set pa=|[p`1+r2,p`2]|; A92:pa`1=p`1+r2 & pa`2=p`2 by EUCLID:56; (s2-s1)/2<s2-s1 by A90,SEQ_2:4; then r2<s2-p`1 by A80,A88,AXIOMS:22; then s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2 by A80,A91,A92,REAL_1: 86,REAL_2:174; then A93:pa in P1 by A2; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A94: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A95:(p`2 - pa`2)^2 =0 by A92,SQUARE_1:60,XCMPLX_1:14; A96: pa`1-p`1 = r2 by A92,XCMPLX_1:26; (pa`1-p`1)^2=(p`1-pa`1)^2 by Lm9; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A89,A91,A95,A96,SQUARE_1:78; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A94,SQUARE_1:def 4; then A97:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A83,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A97,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A84,A93,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; suppose A98: p`1 = s1 & p`2 = t1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A99:Q is open and A100:p in Q; consider r be real number such that A101: r>0 and A102:Ball(q,r) c= Q by A99,A100,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A103:r/2>0 by A101,SEQ_2:3; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A104:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; A105: r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A106:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A104,AXIOMS:22; A107: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A108:0<r2 by A103,SQUARE_1:38; set pa=|[p`1+r2,p`2+r2]|; A109:pa`1=p`1+r2 & pa`2=p`2+r2 by EUCLID:56; then A110:pa`1> s1 by A98,A108,REAL_2:174; A111:pa`2> t1 by A98,A108,A109,REAL_2:174; (s2-s1)/2<s2-s1 by A107,SEQ_2:4; then r2<s2-p`1 by A98,A106,AXIOMS:22; then A112: pa`1<s2 by A109,REAL_1:86; (t2-t1)/2<t2-t1 by A107,SEQ_2:4; then r2<t2-p`2 by A98,A106,AXIOMS:22; then pa`2<t2 by A109,REAL_1:86; then A113:pa in P1 by A2,A110,A111,A112; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A114: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A115: pa`1-p`1 = r2 by A109,XCMPLX_1:26; (pa`1-p`1)^2=(p`1-pa`1)^2 by Lm9; then A116: (p`1 - pa`1)^2 <= (r/2)^2 by A105,A108,A115,SQUARE_1:77; A117: pa`2-p`2 = r2 by A109,XCMPLX_1:26; A118:(pa`2-p`2)^2=(p`2-pa`2)^2 by Lm9; (pa`2-p`2)^2<=(r/2)^2 by A105,A108,A117,SQUARE_1:77; then A119:(p`1 - pa`1)^2 + (p`2 - pa`2)^2 <= (r/2)^2 +(r/2)^2 by A116,A118,REAL_1:55; A120:r^2=(r/2+r/2)^2 by XCMPLX_1:66 .=(r/2)^2+2*(r/2)*(r/2)+(r/2)^2 by SQUARE_1:63 .=(r/2)^2+(r/2)^2+2*(r/2)*(r/2) by XCMPLX_1:1; 2*(r/2)>0 by A103,REAL_2:122; then 2*(r/2)*(r/2)>0 by A103,REAL_2:122; then r^2>(r/2)^2+(r/2)^2 by A120,REAL_2:174; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A119,AXIOMS:22; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A114,SQUARE_1:def 4; then A121:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A101,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A121,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A102,A113,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; suppose A122: p`1 = s1 & p`2 = t2; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A123:Q is open and A124:p in Q; consider r be real number such that A125: r>0 and A126:Ball(q,r) c= Q by A123,A124,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A127:r/2>0 by A125,SEQ_2:3; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A128:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; A129: r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A130:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A128,AXIOMS:22; A131: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A132:0<r2 by A127,SQUARE_1:38; set pa=|[p`1+r2,p`2-r2]|; A133:pa`1=p`1+r2 & pa`2=p`2-r2 by EUCLID:56; then A134:pa`1> s1 by A122,A132,REAL_2:174; A135:pa`2< t2 by A122,A132,A133,REAL_2:174; (s2-s1)/2<s2-s1 by A131,SEQ_2:4; then r2<s2-p`1 by A122,A130,AXIOMS:22; then A136: pa`1<s2 by A133,REAL_1:86; (t2-t1)/2<t2-t1 by A131,SEQ_2:4; then r2<p`2-t1 by A122,A130,AXIOMS:22; then pa`2>t1 by A133,REAL_2:165; then A137:pa in P1 by A2,A134,A135,A136; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A138: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A139: pa`1-p`1 = r2 by A133,XCMPLX_1:26; A140:(pa`1-p`1)^2=(p`1-pa`1)^2 by Lm9; A141:r2^2<= (r/2)^2 by A129,A132,SQUARE_1:77; p`2-pa`2 =r2 by A133,XCMPLX_1:18; then A142:(p`1 - pa`1)^2 + (p`2 - pa`2)^2 <= (r/2)^2 +(r/2)^2 by A139,A140,A141,REAL_1:55; A143:r^2=(r/2+r/2)^2 by XCMPLX_1:66 .=(r/2)^2+2*(r/2)*(r/2)+(r/2)^2 by SQUARE_1:63 .=(r/2)^2+(r/2)^2+2*(r/2)*(r/2) by XCMPLX_1:1; 2*(r/2)>0 by A127,REAL_2:122; then 2*(r/2)*(r/2)>0 by A127,REAL_2:122; then r^2>(r/2)^2+(r/2)^2 by A143,REAL_2:174; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A142,AXIOMS:22; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A138,SQUARE_1:def 4; then A144:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A125,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A144,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A126,A137,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; end; hence x in Cl P1 by A77; suppose A145: p`1 <= s2 & p`1 >= s1 & p`2 = t2; now per cases by A145,REAL_1:def 5; suppose A146: p`2 = t2 & p`1 < s2 & p`1 > s1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A147:Q is open and A148:p in Q; consider r be real number such that A149: r>0 and A150:Ball(q,r) c= Q by A147,A148,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A151:r/2>0 by A149,SEQ_2:3; A152:r/2<r by A149,SEQ_2:4; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A153:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A154:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A153,AXIOMS:22; A155:r2<r by A152,A153,AXIOMS:22; A156: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A157:0<r2 by A151,SQUARE_1:38; set pa=|[p`1,p`2-r2]|; A158:pa`1=p`1 & pa`2=p`2-r2 by EUCLID:56; (t2-t1)/2<t2-t1 by A156,SEQ_2:4; then r2<p`2-t1 by A146,A154,AXIOMS:22; then s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2 by A146,A157,A158, REAL_2:165,174; then A159:pa in P1 by A2; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A160: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A161:(p`1 - pa`1)^2 =0 by A158,SQUARE_1:60,XCMPLX_1:14; p`2-pa`2 =r2 by A158,XCMPLX_1:18; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A155,A157,A161,SQUARE_1:78; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A160,SQUARE_1:def 4; then A162:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A149,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A162,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A150,A159,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; suppose A163: p`2 = t2 & p`1 = s1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A164:Q is open and A165:p in Q; consider r be real number such that A166: r>0 and A167:Ball(q,r) c= Q by A164,A165,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A168:r/2>0 by A166,SEQ_2:3; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A169:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; A170: r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A171:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A169,AXIOMS:22; A172: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A173:0<r2 by A168,SQUARE_1:38; set pa=|[p`1+r2,p`2-r2]|; A174:pa`1=p`1+r2 & pa`2=p`2-r2 by EUCLID:56; then A175:pa`1> s1 by A163,A173,REAL_2:174; A176:pa`2< t2 by A163,A173,A174,REAL_2:174; (t2-t1)/2<t2-t1 by A172,SEQ_2:4; then r2<p`2-t1 by A163,A171,AXIOMS:22; then A177: pa`2>t1 by A174,REAL_2:165; (s2-s1)/2<s2-s1 by A172,SEQ_2:4; then r2<s2-p`1 by A163,A171,AXIOMS:22; then pa`1<s2 by A174,REAL_1:86; then A178:pa in P1 by A2,A175,A176,A177; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A179: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; p`2-pa`2 = r2 by A174,XCMPLX_1:18; then A180: (p`2 - pa`2)^2 <= (r/2)^2 by A170,A173,SQUARE_1:77; A181: pa`1-p`1 = r2 by A174,XCMPLX_1:26; A182:(pa`1-p`1)^2=(p`1-pa`1)^2 by Lm9; (pa`1-p`1)^2<=(r/2)^2 by A170,A173,A181,SQUARE_1:77; then A183:(p`1 - pa`1)^2 + (p`2 - pa`2)^2 <= (r/2)^2 +(r/2)^2 by A180,A182,REAL_1:55; A184:r^2=(r/2+r/2)^2 by XCMPLX_1:66 .=(r/2)^2+2*(r/2)*(r/2)+(r/2)^2 by SQUARE_1:63 .=(r/2)^2+(r/2)^2+2*(r/2)*(r/2) by XCMPLX_1:1; 2*(r/2)>0 by A168,REAL_2:122; then 2*(r/2)*(r/2)>0 by A168,REAL_2:122; then r^2>(r/2)^2+(r/2)^2 by A184,REAL_2:174; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A183,AXIOMS:22; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A179,SQUARE_1:def 4; then A185:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A166,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A185,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A167,A178,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; suppose A186: p`2 = t2 & p`1 = s2; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A187:Q is open and A188:p in Q; consider r be real number such that A189: r>0 and A190:Ball(q,r) c= Q by A187,A188,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A191:r/2>0 by A189,SEQ_2:3; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A192:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; A193: r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A194:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A192,AXIOMS:22; A195: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A196:0<r2 by A191,SQUARE_1:38; set pa=|[p`1-r2,p`2-r2]|; A197:pa`1=p`1-r2 & pa`2=p`2-r2 by EUCLID:56; then A198:pa`1< s2 by A186,A196,REAL_2:174; A199:pa`2< t2 by A186,A196,A197,REAL_2:174; (s2-s1)/2<s2-s1 by A195,SEQ_2:4; then r2<p`1-s1 by A186,A194,AXIOMS:22; then A200: pa`1>s1 by A197,REAL_2:165; (t2-t1)/2<t2-t1 by A195,SEQ_2:4; then r2<p`2-t1 by A186,A194,AXIOMS:22; then pa`2>t1 by A197,REAL_2:165; then A201:pa in P1 by A2,A198,A199,A200; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A202: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A203: p`1-pa`1 = r2 by A197,XCMPLX_1:18; A204:r2^2<= (r/2)^2 by A193,A196,SQUARE_1:77; p`2-pa`2 =r2 by A197,XCMPLX_1:18; then A205:(p`1 - pa`1)^2 + (p`2 - pa`2)^2 <= (r/2)^2 +(r/2)^2 by A203,A204,REAL_1:55; A206:r^2=(r/2+r/2)^2 by XCMPLX_1:66 .=(r/2)^2+2*(r/2)*(r/2)+(r/2)^2 by SQUARE_1:63 .=(r/2)^2+(r/2)^2+2*(r/2)*(r/2) by XCMPLX_1:1; 2*(r/2)>0 by A191,REAL_2:122; then 2*(r/2)*(r/2)>0 by A191,REAL_2:122; then r^2>(r/2)^2+(r/2)^2 by A206,REAL_2:174; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A205,AXIOMS:22; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A202,SQUARE_1:def 4; then A207:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A189,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A207,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A190,A201,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; end; hence x in Cl P1 by A77; suppose A208: p`1 <= s2 & p`1 >= s1 & p`2 = t1; now per cases by A208,REAL_1:def 5; suppose A209: p`2 = t1 & p`1 < s2 & p`1 > s1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A210:Q is open and A211:p in Q; consider r be real number such that A212: r>0 and A213:Ball(q,r) c= Q by A210,A211,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A214:r/2>0 by A212,SEQ_2:3; A215:r/2<r by A212,SEQ_2:4; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A216:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A217:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A216,AXIOMS:22; A218:r2<r by A215,A216,AXIOMS:22; A219: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A220:0<r2 by A214,SQUARE_1:38; set pa=|[p`1,p`2+r2]|; A221:pa`2=p`2+r2 & pa`1=p`1 by EUCLID:56; (t2-t1)/2<t2-t1 by A219,SEQ_2:4; then r2<t2-p`2 by A209,A217,AXIOMS:22; then s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2 by A209,A220,A221, REAL_1:86,REAL_2:174; then A222:pa in P1 by A2; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A223: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A224:(p`1 - pa`1)^2 =0 by A221,SQUARE_1:60,XCMPLX_1:14; A225: pa`2-p`2 = r2 by A221,XCMPLX_1:26; (pa`2-p`2)^2=(p`2-pa`2)^2 by Lm9; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A218,A220,A224,A225, SQUARE_1:78; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A223,SQUARE_1:def 4; then A226:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A212,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A226,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A213,A222,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; suppose A227: p`2 = t1 & p`1 = s1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A228:Q is open and A229:p in Q; consider r be real number such that A230: r>0 and A231:Ball(q,r) c= Q by A228,A229,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A232:r/2>0 by A230,SEQ_2:3; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A233:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; A234: r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A235:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A233,AXIOMS:22; A236: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A237:0<r2 by A232,SQUARE_1:38; set pa=|[p`1+r2,p`2+r2]|; A238:pa`1=p`1+r2 & pa`2=p`2+r2 by EUCLID:56; then A239:pa`1> s1 by A227,A237,REAL_2:174; A240:pa`2> t1 by A227,A237,A238,REAL_2:174; (s2-s1)/2<s2-s1 by A236,SEQ_2:4; then r2<s2-p`1 by A227,A235,AXIOMS:22; then A241: pa`1<s2 by A238,REAL_1:86; (t2-t1)/2<t2-t1 by A236,SEQ_2:4; then r2<t2-p`2 by A227,A235,AXIOMS:22; then pa`2<t2 by A238,REAL_1:86; then A242:pa in P1 by A2,A239,A240,A241; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A243: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A244: pa`1-p`1 = r2 by A238,XCMPLX_1:26; (pa`1-p`1)^2=(p`1-pa`1)^2 by Lm9; then A245: (p`1 - pa`1)^2 <= (r/2)^2 by A234,A237,A244,SQUARE_1:77; A246: pa`2-p`2 = r2 by A238,XCMPLX_1:26; A247:(pa`2-p`2)^2=(p`2-pa`2)^2 by Lm9; (pa`2-p`2)^2<=(r/2)^2 by A234,A237,A246,SQUARE_1:77; then A248:(p`1 - pa`1)^2 + (p`2 - pa`2)^2 <= (r/2)^2 +(r/2)^2 by A245,A247,REAL_1:55; A249:r^2=(r/2+r/2)^2 by XCMPLX_1:66 .=(r/2)^2+2*(r/2)*(r/2)+(r/2)^2 by SQUARE_1:63 .=(r/2)^2+(r/2)^2+2*(r/2)*(r/2) by XCMPLX_1:1; 2*(r/2)>0 by A232,REAL_2:122; then 2*(r/2)*(r/2)>0 by A232,REAL_2:122; then r^2>(r/2)^2+(r/2)^2 by A249,REAL_2:174; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A248,AXIOMS:22; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A243,SQUARE_1:def 4; then A250:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A230,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A250,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A231,A242,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; suppose A251: p`2 = t1 & p`1 = s2; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A252:Q is open and A253:p in Q; consider r be real number such that A254: r>0 and A255:Ball(q,r) c= Q by A252,A253,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A256:r/2>0 by A254,SEQ_2:3; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A257:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; A258: r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A259:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A257,AXIOMS:22; A260: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A261:0<r2 by A256,SQUARE_1:38; set pa=|[p`1-r2,p`2+r2]|; A262:pa`1=p`1-r2 & pa`2=p`2+r2 by EUCLID:56; then A263:pa`2> t1 by A251,A261,REAL_2:174; A264:pa`1< s2 by A251,A261,A262,REAL_2:174; (t2-t1)/2<t2-t1 by A260,SEQ_2:4; then r2<t2-p`2 by A251,A259,AXIOMS:22; then A265: pa`2<t2 by A262,REAL_1:86; (s2-s1)/2<s2-s1 by A260,SEQ_2:4; then r2<p`1-s1 by A251,A259,AXIOMS:22; then pa`1>s1 by A262,REAL_2:165; then A266:pa in P1 by A2,A263,A264,A265; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A267: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A268: pa`2-p`2 = r2 by A262,XCMPLX_1:26; A269:(pa`2-p`2)^2=(p`2-pa`2)^2 by Lm9; A270:r2^2<= (r/2)^2 by A258,A261,SQUARE_1:77; p`1-pa`1 =r2 by A262,XCMPLX_1:18; then A271:(p`1 - pa`1)^2 + (p`2 - pa`2)^2 <= (r/2)^2 +(r/2)^2 by A268,A269,A270,REAL_1:55; A272:r^2=(r/2+r/2)^2 by XCMPLX_1:66 .=(r/2)^2+2*(r/2)*(r/2)+(r/2)^2 by SQUARE_1:63 .=(r/2)^2+(r/2)^2+2*(r/2)*(r/2) by XCMPLX_1:1; 2*(r/2)>0 by A256,REAL_2:122; then 2*(r/2)*(r/2)>0 by A256,REAL_2:122; then r^2>(r/2)^2+(r/2)^2 by A272,REAL_2:174; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A271,AXIOMS:22; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A267,SQUARE_1:def 4; then A273:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A254,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A273,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A255,A266,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; end; hence x in Cl P1 by A77; suppose A274: p`1 = s2 & p`2 <= t2 & p`2 >= t1; now per cases by A274,REAL_1:def 5; suppose A275: p`1 = s2 & p`2 < t2 & p`2 > t1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A276:Q is open and A277:p in Q; consider r being real number such that A278: r>0 and A279:Ball(q,r) c= Q by A276,A277,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A280:r/2>0 by A278,SEQ_2:3; A281:r/2<r by A278,SEQ_2:4; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A282:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A283:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A282,AXIOMS:22; A284:r2<r by A281,A282,AXIOMS:22; A285: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A286:0<r2 by A280,SQUARE_1:38; set pa=|[p`1-r2,p`2]|; A287:pa`2=p`2 & pa`1=p`1-r2 by EUCLID:56; (s2-s1)/2<s2-s1 by A285,SEQ_2:4; then r2<p`1-s1 by A275,A283,AXIOMS:22; then s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2 by A275,A286,A287, REAL_2:165,174; then A288:pa in P1 by A2; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A289: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A290:(p`2 - pa`2)^2 =0 by A287,SQUARE_1:60,XCMPLX_1:14; p`1-pa`1 =r2 by A287,XCMPLX_1:18; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A284,A286,A290,SQUARE_1:78; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A289,SQUARE_1:def 4; then A291:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A278,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A291,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A279,A288,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; suppose A292: p`1 = s2 & p`2 = t1; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A293:Q is open and A294:p in Q; consider r being real number such that A295: r>0 and A296:Ball(q,r) c= Q by A293,A294,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A297:r/2>0 by A295,SEQ_2:3; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A298:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; A299: r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A300:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A298,AXIOMS:22; A301: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A302:0<r2 by A297,SQUARE_1:38; set pa=|[p`1-r2,p`2+r2]|; A303:pa`2=p`2+r2 & pa`1=p`1-r2 by EUCLID:56; then A304:pa`2> t1 by A292,A302,REAL_2:174; A305:pa`1< s2 by A292,A302,A303,REAL_2:174; (s2-s1)/2<s2-s1 by A301,SEQ_2:4; then r2<p`1-s1 by A292,A300,AXIOMS:22; then A306: pa`1>s1 by A303,REAL_2:165; (t2-t1)/2<t2-t1 by A301,SEQ_2:4; then r2<t2-p`2 by A292,A300,AXIOMS:22; then pa`2<t2 by A303,REAL_1:86; then A307:pa in P1 by A2,A304,A305,A306; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A308: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; p`1-pa`1 = r2 by A303,XCMPLX_1:18; then A309: (p`1 - pa`1)^2 <= (r/2)^2 by A299,A302,SQUARE_1:77; A310: pa`2-p`2 = r2 by A303,XCMPLX_1:26; A311:(pa`2-p`2)^2=(p`2-pa`2)^2 by Lm9; (pa`2-p`2)^2<=(r/2)^2 by A299,A302,A310,SQUARE_1:77; then A312:(p`1 - pa`1)^2 + (p`2 - pa`2)^2 <= (r/2)^2 +(r/2)^2 by A309,A311,REAL_1:55; A313:r^2=(r/2+r/2)^2 by XCMPLX_1:66 .=(r/2)^2+2*(r/2)*(r/2)+(r/2)^2 by SQUARE_1:63 .=(r/2)^2+(r/2)^2+2*(r/2)*(r/2) by XCMPLX_1:1; 2*(r/2)>0 by A297,REAL_2:122; then 2*(r/2)*(r/2)>0 by A297,REAL_2:122; then r^2>(r/2)^2+(r/2)^2 by A313,REAL_2:174; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A312,AXIOMS:22; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A308,SQUARE_1:def 4; then A314:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A295,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A314,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A296,A307,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; suppose A315: p`1 = s2 & p`2 = t2; for Q being Subset of TOP-REAL 2 st Q is open holds p in Q implies P1 meets Q proof let Q be Subset of TOP-REAL 2;assume that A316:Q is open and A317:p in Q; consider r being real number such that A318: r>0 and A319:Ball(q,r) c= Q by A316,A317,Lm10,TOPMETR:22; reconsider r as Real by XREAL_0:def 1; A320:r/2>0 by A318,SEQ_2:3; set r2=min(r/2,min((s2-s1)/2,(t2-t1)/2)); A321:r2 <= r/2 & r2<=min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:35; A322: r2 <= r/2 & ( min((s2-s1)/2,(t2-t1)/2) <=(s2-s1)/2 & min((s2-s1)/2,(t2-t1)/2)<= (t2-t1)/2) by SQUARE_1:35; then A323:r2<= r/2 & r2<=(s2-s1)/2 & r2<= (t2-t1)/2 by A321,AXIOMS:22; A324: s2-s1>0 & t2-t1>0 by A1,SQUARE_1:11; then (s2-s1)/2 >0 & (t2-t1)/2>0 by SEQ_2:3; then 0<min((s2-s1)/2,(t2-t1)/2) by SQUARE_1:38; then A325:0<r2 by A320,SQUARE_1:38; set pa=|[p`1-r2,p`2-r2]|; A326:pa`1=p`1-r2 & pa`2=p`2-r2 by EUCLID:56; then A327:pa`1< s2 by A315,A325,REAL_2:174; A328:pa`2< t2 by A315,A325,A326,REAL_2:174; (s2-s1)/2<s2-s1 by A324,SEQ_2:4; then r2<p`1-s1 by A315,A323,AXIOMS:22; then A329: pa`1>s1 by A326,REAL_2:165; (t2-t1)/2<t2-t1 by A324,SEQ_2:4; then r2<p`2-t1 by A315,A323,AXIOMS:22; then pa`2>t1 by A326,REAL_2:165; then A330:pa in P1 by A2,A327,A328,A329; reconsider qa=pa as Point of Euclid 2 by TOPREAL3:13; 0 <= (p`1 - pa`1)^2 & 0 <= (p`2 - pa`2)^2 by SQUARE_1:72; then A331: 0+0 <= (p`1 - pa`1)^2 + (p`2 - pa`2)^2 by REAL_1:55; A332: p`1-pa`1 = r2 by A326,XCMPLX_1:18; A333:r2^2<= (r/2)^2 by A322,A325,SQUARE_1:77; p`2-pa`2 =r2 by A326,XCMPLX_1:18; then A334:(p`1 - pa`1)^2 + (p`2 - pa`2)^2 <= (r/2)^2 +(r/2)^2 by A332,A333,REAL_1:55; A335:r^2=(r/2+r/2)^2 by XCMPLX_1:66 .=(r/2)^2+2*(r/2)*(r/2)+(r/2)^2 by SQUARE_1:63 .=(r/2)^2+(r/2)^2+2*(r/2)*(r/2) by XCMPLX_1:1; 2*(r/2)>0 by A320,REAL_2:122; then 2*(r/2)*(r/2)>0 by A320,REAL_2:122; then r^2>(r/2)^2+(r/2)^2 by A335,REAL_2:174; then (p`1 - pa`1)^2 + (p`2 - pa`2)^2 < r^2 by A334,AXIOMS:22; then (sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2))^2<r^2 by A331,SQUARE_1:def 4; then A336:sqrt ((p`1 - pa`1)^2 + (p`2 - pa`2)^2) <r by A318,SQUARE_1:77; Euclid 2=MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; then (Pitag_dist 2).(q,qa)=dist(q,qa) by METRIC_1:def 1; then dist(q,qa)<r by A336,TOPREAL3:12; then pa in Ball(q,r) by METRIC_1:12; hence thesis by A319,A330,XBOOLE_0:3; end; hence p in Cl P1 by PRE_TOPC:def 13; end; hence x in Cl P1 by A77; end; hence x in Cl P1; end; then P c= Cl P1 /\ P1` by A68,XBOOLE_1:19; hence thesis by SUBSET_1:32; end; hence thesis by A14,A15,A76,XBOOLE_0:def 10; end; theorem Th43: for s1,s2,t1,t2,P,P1 st P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} holds P1 c= [#] ((TOP-REAL 2)|P`) proof let s1,s2,t1,t2,P,P1; assume that A1:P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2}; let x be set;assume A2: x in P1; then A3: ex pa being Point of TOP-REAL 2 st pa=x & s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2 by A1; now assume x in { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1}; then ex p being Point of TOP-REAL 2 st p=x & ( p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1); hence contradiction by A3; end; then x in P` by A1,A2,SUBSET_1:50; hence thesis by PRE_TOPC:def 10; end; theorem for s1,s2,t1,t2,P,P1 st P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} holds P1 is Subset of (TOP-REAL 2)|P` proof let s1,s2,t1,t2,P,P1;assume that A1:P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2}; P1 c= [#]((TOP-REAL 2)|P`)by A1,Th43; hence thesis by PRE_TOPC:16; end; theorem Th45: for s1,s2,t1,t2,P,P2 st s1<s2 & t1<t2 & P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)} holds P2 c= [#]((TOP-REAL 2)|P`) proof let s1,s2,t1,t2,P,P2;assume that A1:s1<s2 & t1<t2 and A2: P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)}; let x be set;assume A3: x in P2; then A4: ex pa being Point of TOP-REAL 2 st pa=x & not( s1<=pa`1 & pa`1<=s2 & t1<=pa`2 & pa`2<=t2) by A2; now assume x in { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1}; then ex p being Point of TOP-REAL 2 st p=x & ( p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1); hence contradiction by A1,A4; end; then x in P` by A2,A3,SUBSET_1:50; hence thesis by PRE_TOPC:def 10; end; theorem for s1,s2,t1,t2,P,P2 st s1<s2 & t1< t2 & P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)} holds P2 is Subset of (TOP-REAL 2)|P` proof let s1,s2,t1,t2,P,P2;assume that A1:s1<s2 & t1<t2 and A2:P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)}; P2 c= [#]((TOP-REAL 2)|P`) by A1,A2,Th45; hence thesis by PRE_TOPC:16; end; begin :: :: The Jordan's property :: definition let S be Subset of TOP-REAL 2; attr S is Jordan means :Def2: S`<> {} & ex A1,A2 being Subset of TOP-REAL 2 st S` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & for C1,C2 being Subset of (TOP-REAL 2)|S` st C1 = A1 & C2 = A2 holds C1 is_a_component_of (TOP-REAL 2)|S` & C2 is_a_component_of (TOP-REAL 2)|S`; synonym S has_property_J; end; theorem for S being Subset of TOP-REAL 2 st S has_property_J holds S`<> {} & ex A1,A2 being Subset of TOP-REAL 2 st ex C1,C2 being Subset of (TOP-REAL 2)|S` st S` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & C1 = A1 & C2 = A2 & C1 is_a_component_of (TOP-REAL 2)|S` & C2 is_a_component_of (TOP-REAL 2)|S` & for C3 being Subset of (TOP-REAL 2)|S` st C3 is_a_component_of (TOP-REAL 2)|S` holds C3 = C1 or C3 = C2 proof let S be Subset of TOP-REAL 2; assume A1: S has_property_J; then reconsider S' = S` as non empty Subset of TOP-REAL 2 by Def2; consider A1,A2 being Subset of TOP-REAL 2 such that A2: S`=A1 \/ A2 and A3: A1 misses A2 and A4: (Cl A1)\A1=(Cl A2)\A2 and A5: (for C1,C2 being Subset of (TOP-REAL 2)|S` st C1=A1 & C2=A2 holds C1 is_a_component_of (TOP-REAL 2)|S` & C2 is_a_component_of (TOP-REAL 2)|S`) by A1,Def2; A6:A1 c= S` & A2 c= S` by A2,XBOOLE_1:7; A7:[#]((TOP-REAL 2)|S`)=S` by PRE_TOPC:def 10; A1 c= [#]((TOP-REAL 2)|S`) & A2 c= [#] ((TOP-REAL 2)|S`) by A6,PRE_TOPC:def 10; then reconsider G0A=A1,G0B=A2 as Subset of (TOP-REAL 2)|S' by PRE_TOPC:16; G0A=A1 & G0B=A2; then A8:G0A is_a_component_of (TOP-REAL 2)|S' & G0B is_a_component_of (TOP-REAL 2)|S' by A5; now let C3 be Subset of (TOP-REAL 2)|S'; assume A9: C3 is_a_component_of (TOP-REAL 2)|S'; then A10: C3 <>{}((TOP-REAL 2)|S') by CONNSP_1:34; C3 /\(G0A \/ G0B)=C3 by A2,A7,PRE_TOPC:15; then A11: (C3 /\ G0A) \/ (C3 /\ G0B) <>{} by A10,XBOOLE_1:23; now per cases by A11; suppose C3 /\ G0A<>{}; then C3 meets G0A by XBOOLE_0:def 7; then A12:not (C3,G0A are_separated) by CONNSP_1:2; A13:C3 is connected & G0A is connected by A8,A9,CONNSP_1:def 5; then A14:C3 \/ G0A is connected by A12,CONNSP_1:18; G0A c= C3 \/ G0A by XBOOLE_1:7; then G0A=C3 \/ G0A by A8,A14,CONNSP_1:def 5; then C3 c= G0A by XBOOLE_1:7; hence C3=G0A or C3=G0B by A9,A13,CONNSP_1:def 5; suppose C3 /\ A2<>{}; then C3 meets G0B by XBOOLE_0:def 7; then A15:not (C3,G0B are_separated) by CONNSP_1:2; A16:C3 is connected & G0B is connected by A8,A9,CONNSP_1:def 5; then A17:C3 \/ G0B is connected by A15,CONNSP_1:18; G0B c= C3 \/ G0B by XBOOLE_1:7; then G0B=C3 \/ G0B by A8,A17,CONNSP_1:def 5; then C3 c= G0B by XBOOLE_1:7; hence C3=G0A or C3=G0B by A9,A16,CONNSP_1:def 5; end; hence C3=G0A or C3=G0B; end; hence thesis by A2,A3,A4,A8; end; theorem for s1,s2,t1,t2 for P being Subset of TOP-REAL 2 st s1<s2 & t1<t2 & P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} holds P has_property_J proof let s1,s2,t1,t2; let P be Subset of TOP-REAL 2;assume that A1: s1<s2 & t1<t2 and A2: P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1}; {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} is Subset of TOP-REAL 2 & {pc where pc is Point of TOP-REAL 2: not( s1<=pc`1 & pc`1<=s2 & t1<=pc`2 & pc`2<=t2)} is Subset of TOP-REAL 2 by Th34,Th35; then reconsider P1= {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2}, P2= {pc where pc is Point of TOP-REAL 2: not( s1<=pc`1 & pc`1<=s2 & t1<=pc`2 & pc`2<=t2)} as Subset of TOP-REAL 2; reconsider PC = P` as Subset of TOP-REAL 2; A3:P1= {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2}& P2= {pc where pc is Point of TOP-REAL 2: not( s1<=pc`1 & pc`1<=s2 & t1<=pc`2 & pc`2<=t2)}; then A4:PC=P1 \/ P2 & PC<>{} & P1 misses P2 by A1,A2,Th41; A5:P=(Cl P1)\P1 & P=(Cl P2)\P2 by A1,A2,A3,Th42; for P1A,P2B be Subset of (TOP-REAL 2)|P` holds P1A=P1 & P2B=P2 implies P1A is_a_component_of (TOP-REAL 2)|P` & P2B is_a_component_of (TOP-REAL 2)|P` by A1,A2,Th41; hence thesis by A4,A5,Def2; end;

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