:: On the Geometry of a Go-board :: by Andrzej Trybulec :: :: Received July 9, 1995 :: Copyright (c) 1995-2018 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies NUMBERS, SUBSET_1, REAL_1, PRE_TOPC, EUCLID, GOBOARD1, ARYTM_3, MCART_1, ARYTM_1, RELAT_1, XBOOLE_0, METRIC_1, XXREAL_0, CARD_1, PCOMPS_1, RCOMP_1, STRUCT_0, CONNSP_2, TOPS_1, TARSKI, SQUARE_1, FUNCT_1, FINSEQ_1, TREES_1, GOBOARD5, COMPLEX1, RLTOPSP1, SUPINF_2, NAT_1; notations TARSKI, XBOOLE_0, ORDINAL1, SUBSET_1, NUMBERS, XXREAL_0, XREAL_0, XCMPLX_0, COMPLEX1, REAL_1, NAT_1, SQUARE_1, NAT_D, BINOP_1, FINSEQ_1, MATRIX_0, STRUCT_0, METRIC_1, PRE_TOPC, TOPS_1, CONNSP_2, PCOMPS_1, RLVECT_1, RLTOPSP1, EUCLID, GOBOARD1, GOBOARD5; constructors REAL_1, SQUARE_1, NAT_1, COMPLEX1, NAT_D, TOPS_1, CONNSP_2, PCOMPS_1, GOBOARD1, GOBOARD5, FUNCSDOM, BINOP_2; registrations RELSET_1, XXREAL_0, XREAL_0, STRUCT_0, METRIC_1, PCOMPS_1, EUCLID, TOPS_1, PRE_TOPC, SQUARE_1, ORDINAL1, NAT_1; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; definitions TARSKI, XBOOLE_0; equalities EUCLID, SQUARE_1, XCMPLX_0, RLTOPSP1, RLVECT_1; expansions TARSKI; theorems GOBOARD5, TOPS_1, SPPOL_2, TOPMETR, METRIC_1, PCOMPS_1, PRE_TOPC, EUCLID, TOPREAL3, SQUARE_1, ABSVALUE, TARSKI, TOPREAL1, NAT_1, XBOOLE_0, XBOOLE_1, XCMPLX_1, COMPLEX1, XREAL_1, CONNSP_2, XXREAL_0, RLTOPSP1, MATRIX_0, RLVECT_1; begin reserve n for Nat, i,j for Nat, r,s,r1,s1,r2,s2,r9,s9 for Real, p,q for Point of TOP-REAL 2, G for Go-board, x,y for set, v for Point of Euclid 2; Lm1: (p+q)`1 = p`1+q`1 & (p+q)`2 = p`2+q`2 proof p + q = |[ p`1 + q`1, p`2 + q`2]| by EUCLID:55; hence thesis by EUCLID:52; end; Lm2: (p-q)`1 = p`1-q`1 & (p-q)`2 = p`2-q`2 proof p - q = |[ p`1 - q`1, p`2 - q`2]| by EUCLID:61; hence thesis by EUCLID:52; end; Lm3: (r*p)`1 = r*(p`1) & (r*p)`2 = r*(p`2) proof r*p = |[ r*p`1 ,r*p`2 ]| by EUCLID:57; hence thesis by EUCLID:52; end; theorem Th1: for M being non empty Reflexive MetrStruct, u being Point of M, r being Real holds r > 0 implies u in Ball(u,r) proof let M be non empty Reflexive MetrStruct, u be Point of M, r be Real; A1: Ball(u,r) = {q where q is Point of M:dist(u,q) 0; hence thesis by A1; end; Lm4: for M being MetrSpace, B being Subset of TopSpaceMetr(M),r being Real , u being Point of M st B = Ball(u,r) holds B is open proof let M be MetrSpace, B be Subset of TopSpaceMetr(M), r be Real, u be Point of M; A1: TopSpaceMetr M = TopStruct (#the carrier of M,Family_open_set M#) & Ball (u,r ) in Family_open_set M by PCOMPS_1:29,def 5; assume B = Ball(u,r); hence thesis by A1,PRE_TOPC:def 2; end; theorem Th2: for p being Point of Euclid n, q being Point of TOP-REAL n, r being Real st p = q & r > 0 holds Ball (p, r) is a_neighborhood of q proof let p be Point of Euclid n, q be Point of TOP-REAL n, r be Real; reconsider A = Ball (p, r) as Subset of TOP-REAL n by TOPREAL3:8; A1: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8; then reconsider AA = A as Subset of TopSpaceMetr Euclid n; AA is open by TOPMETR:14; then A2: A is open by A1,PRE_TOPC:30; assume p = q & r > 0; hence thesis by A2,Th1,CONNSP_2:3; end; theorem Th3: for B being Subset of TOP-REAL n, u being Point of Euclid n st B = Ball(u,r) holds B is open proof let B be Subset of TOP-REAL n, u be Point of Euclid n; A1: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8; then reconsider BB = B as Subset of TopSpaceMetr Euclid n; assume B = Ball(u,r); then BB is open by Lm4; hence thesis by A1,PRE_TOPC:30; end; theorem Th4: for M being non empty MetrSpace, u being Point of M, P being Subset of TopSpaceMetr(M) holds u in Int P iff ex r being Real st r > 0 & Ball(u,r) c= P proof let M be non empty MetrSpace, u be Point of M, P be Subset of TopSpaceMetr(M ); hereby assume u in Int P; then consider r be Real such that A1: r > 0 and A2: Ball(u,r) c= Int P by TOPMETR:15; take r; thus r > 0 by A1; Int P c= P by TOPS_1:16; hence Ball(u,r) c= P by A2; end; given r being Real such that A3: r > 0 and A4: Ball(u,r) c= P; TopSpaceMetr M = TopStruct (#the carrier of M,Family_open_set M#) by PCOMPS_1:def 5; then reconsider B = Ball(u,r) as Subset of TopSpaceMetr(M); A5: B is open by Lm4; u in Ball(u,r) by A3,Th1; hence thesis by A4,A5,TOPS_1:22; end; Lm5: for T being TopSpace, A being Subset of T, B being Subset of the TopStruct of T st A = B holds Int A = Int B proof let T be TopSpace, A be Subset of T, B be Subset of the TopStruct of T such that A1: A = B; reconsider AA = Int A as Subset of the TopStruct of T; AA is open by PRE_TOPC:30; hence Int A c= Int B by A1,TOPS_1:16,24; reconsider BB = Int B as Subset of T; BB is open by PRE_TOPC:30; hence Int B c= Int A by A1,TOPS_1:16,24; end; theorem Th5: for u being Point of Euclid n, P being Subset of TOP-REAL n holds u in Int P iff ex r being Real st r > 0 & Ball(u,r) c= P proof let u be Point of Euclid n, P be Subset of TOP-REAL n; A1: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8; then reconsider PP = P as Subset of TopSpaceMetr Euclid n; u in Int PP iff ex r being Real st r > 0 & Ball(u,r) c= PP by Th4; hence thesis by A1,Lm5; end; theorem Th6: :: TOPREAL3:12 for u,v being Point of Euclid 2 st u = |[r1,s1]| & v = |[r2,s2]| holds dist(u,v) =sqrt ((r1 - r2)^2 + (s1 - s2)^2) proof let u,v be Point of Euclid 2 such that A1: u = |[r1,s1]| & v = |[r2,s2]|; A2: |[r1,s1]|`1 = r1 & |[r1,s1]|`2 = s1 by EUCLID:52; A3: |[r2,s2]|`1 = r2 & |[r2,s2]|`2 = s2 by EUCLID:52; thus dist(u,v) = (Pitag_dist 2).(u,v) by METRIC_1:def 1 .= sqrt ((r1 - r2)^2 + (s1 - s2)^2) by A1,A2,A3,TOPREAL3:7; end; theorem Th7: for u being Point of Euclid 2 st u = |[r,s]| holds 0 <= r2 & r2 < r1 implies |[r+r2,s]| in Ball(u,r1) proof let u be Point of Euclid 2 such that A1: u = |[r,s]| and A2: 0 <= r2 and A3: r2 < r1; reconsider v = |[r+r2,s]| as Point of Euclid 2 by TOPREAL3:8; dist(u,v) = sqrt ((r - (r+r2))^2 + (s - s)^2) by A1,Th6 .= sqrt ((-(r - (r+r2)))^2) .= r2 by A2,SQUARE_1:22; hence thesis by A3,METRIC_1:11; end; theorem Th8: for u being Point of Euclid 2 st u = |[r,s]| holds 0 <= s2 & s2 < s1 implies |[r,s+s2]| in Ball(u,s1) proof let u be Point of Euclid 2 such that A1: u = |[r,s]| and A2: 0 <= s2 and A3: s2 < s1; reconsider v = |[r,s+s2]| as Point of Euclid 2 by TOPREAL3:8; dist(u,v) = sqrt ((r - r)^2 + (s - (s+s2))^2) by A1,Th6 .= sqrt ((-(s - (s+s2)))^2) .= s2 by A2,SQUARE_1:22; hence thesis by A3,METRIC_1:11; end; theorem Th9: for u being Point of Euclid 2 st u = |[r,s]| holds 0 <= r2 & r2 < r1 implies |[r-r2,s]| in Ball(u,r1) proof let u be Point of Euclid 2 such that A1: u = |[r,s]| and A2: 0 <= r2 and A3: r2 < r1; reconsider v = |[r-r2,s]| as Point of Euclid 2 by TOPREAL3:8; dist(u,v) = sqrt ((r - (r-r2))^2 + (s - s)^2) by A1,Th6 .= r2 by A2,SQUARE_1:22; hence thesis by A3,METRIC_1:11; end; theorem Th10: for u being Point of Euclid 2 st u = |[r,s]| holds 0 <= s2 & s2 < s1 implies |[r,s-s2]| in Ball(u,s1) proof let u be Point of Euclid 2 such that A1: u = |[r,s]| and A2: 0 <= s2 and A3: s2 < s1; reconsider v = |[r,s-s2]| as Point of Euclid 2 by TOPREAL3:8; dist(u,v) = sqrt ((s - (s-s2))^2 + (r - r)^2) by A1,Th6 .= s2 by A2,SQUARE_1:22; hence thesis by A3,METRIC_1:11; end; theorem Th11: 1 <= i & i < len G & 1 <= j & j < width G implies G*(i,j)+G*(i+1 ,j+1) = G*(i,j+1)+G*(i+1,j) proof assume that A1: 1 <= i & i < len G and A2: 1 <= j & j < width G; A3: 1 <= j+1 & j+1 <= width G by A2,NAT_1:13; A4: 1 <= i+1 & i+1 <= len G by A1,NAT_1:13; then A5: G*(i+1,j+1)`1 = G*(i+1,1)`1 by A3,GOBOARD5:2 .= G*(i+1,j)`1 by A2,A4,GOBOARD5:2; A6: G*(i+1,j+1)`2 = G*(1,j+1)`2 by A4,A3,GOBOARD5:1 .= G*(i,j+1)`2 by A1,A3,GOBOARD5:1; A7: G*(i,j)`2 = G*(1,j)`2 by A1,A2,GOBOARD5:1 .= G*(i+1,j)`2 by A2,A4,GOBOARD5:1; A8: (G*(i,j)+G*(i+1,j+1))`2 = G*(i,j)`2+G*(i+1,j+1)`2 by Lm1 .= (G*(i,j+1)+G*(i+1,j))`2 by A7,A6,Lm1; A9: G*(i,j)`1 = G*(i,1)`1 by A1,A2,GOBOARD5:2 .= G*(i,j+1)`1 by A1,A3,GOBOARD5:2; (G*(i,j)+G*(i+1,j+1))`1 = G*(i,j)`1+G*(i+1,j+1)`1 by Lm1 .= (G*(i,j+1)+G*(i+1,j))`1 by A9,A5,Lm1; hence G*(i,j)+G*(i+1,j+1) = |[(G*(i,j+1)+G*(i+1,j))`1,(G*(i,j+1)+G*(i+1,j))`2 ]| by A8,EUCLID:53 .= G*(i,j+1)+G*(i+1,j) by EUCLID:53; end; Lm6: the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8 .= TopStruct (#the carrier of Euclid 2,Family_open_set Euclid 2#) by PCOMPS_1:def 5; theorem Th12: Int v_strip(G,0) = { |[r,s]| : r < G*(1,1)`1 } proof 0 <> width G by MATRIX_0:def 10; then 1 <= width G by NAT_1:14; then A1: v_strip(G,0) = { |[r,s]| : r <= G*(1,1)`1 } by GOBOARD5:10; thus Int v_strip(G,0) c= { |[r,s]| : r < G*(1,1)`1 } proof let x be object; assume A2: x in Int v_strip(G,0); then reconsider u = x as Point of Euclid 2 by Lm6; consider r1 being Real such that A3: r1 > 0 and A4: Ball(u,r1) c= v_strip(G,0) by A2,Th5; reconsider p = u as Point of TOP-REAL 2 by Lm6; A5: p = |[p`1,p`2]| by EUCLID:53; set q = |[p`1+r1/2,p`2+0]|; r1/2 < r1 by A3,XREAL_1:216; then q in Ball(u,r1) by A3,A5,Th7; then q in v_strip(G,0) by A4; then ex r2,s2 st q = |[r2,s2]| & r2 <= G*(1,1)`1 by A1; then A6: p`1+r1/2 <= G*(1,1)`1 by SPPOL_2:1; p`1 < p`1 + r1/2 by A3,XREAL_1:29,215; then p`1 < G*(1,1)`1 by A6,XXREAL_0:2; hence thesis by A5; end; let x be object; assume x in { |[r,s]| : r < G*(1,1)`1 }; then consider r,s such that A7: x = |[r,s]| and A8: r < G*(1,1)`1; reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8; A9: Ball(u,G*(1,1)`1-r) c= v_strip(G,0) proof let y be object; A10: Ball(u,G*(1,1)`1-r) = { v : dist(u,v)= 0 & (r - q`1)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by XREAL_1:6,63; then A13: sqrt (r - q`1)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26; A14: q = |[q`1,q`2]| by EUCLID:53; then sqrt ((r - q`1)^2 + (s - q`2)^2) < G*(1,1)`1-r by A12,Th6; then sqrt (r - q`1)^2 <= G*(1,1)`1-r by A13,XXREAL_0:2; then A15: |.r-q`1.| <= G*(1,1)`1-r by COMPLEX1:72; per cases; suppose r <= q`1; then A16: q`1-r >= 0 by XREAL_1:48; |.r-q`1.| = |.-(r-q`1).| by COMPLEX1:52 .= q`1 - r by A16,ABSVALUE:def 1; then q`1 <= G*(1,1)`1 by A15,XREAL_1:9; hence thesis by A1,A11,A14; end; suppose r >= q`1; then q`1 <= G*(1,1)`1 by A8,XXREAL_0:2; hence thesis by A1,A11,A14; end; end; reconsider B = Ball(u,G*(1,1)`1-r) as Subset of TOP-REAL2 by TOPREAL3:8; A17: B is open by Th3; u in Ball(u,G*(1,1)`1-r) by A8,Th1,XREAL_1:50; hence thesis by A7,A9,A17,TOPS_1:22; end; theorem Th13: Int v_strip(G,len G) = { |[r,s]| : G*(len G,1)`1 < r } proof 0 <> width G by MATRIX_0:def 10; then 1 <= width G by NAT_1:14; then A1: v_strip(G,len G) = { |[r,s]| : G*(len G,1)`1 <= r } by GOBOARD5:9; thus Int v_strip(G,len G) c= { |[r,s]| : G*(len G,1)`1 < r } proof let x be object; assume A2: x in Int v_strip(G,len G); then reconsider u = x as Point of Euclid 2 by Lm6; consider r1 being Real such that A3: r1 > 0 and A4: Ball(u,r1) c= v_strip(G,len G) by A2,Th5; reconsider p = u as Point of TOP-REAL 2 by Lm6; A5: p = |[p`1,p`2]| by EUCLID:53; set q = |[p`1-r1/2,p`2+0]|; r1/2 < r1 by A3,XREAL_1:216; then q in Ball(u,r1) by A3,A5,Th9; then q in v_strip(G,len G) by A4; then ex r2,s2 st q = |[r2,s2]| & G*(len G,1)`1 <= r2 by A1; then G*(len G,1)`1 <= p`1-r1/2 by SPPOL_2:1; then A6: G*(len G,1)`1+r1/2 <= p`1 by XREAL_1:19; G*(len G,1)`1 < G*(len G,1)`1 + r1/2 by A3,XREAL_1:29,215; then G*(len G,1)`1 < p`1 by A6,XXREAL_0:2; hence thesis by A5; end; let x be object; assume x in { |[r,s]| : G*(len G,1)`1 < r }; then consider r,s such that A7: x = |[r,s]| and A8: G*(len G,1)`1 < r; reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8; A9: Ball(u,r-G*(len G,1)`1) c= v_strip(G,len G) proof let y be object; A10: Ball(u,r-G*(len G,1)`1) = { v : dist(u,v)= 0 & (r - q`1)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by XREAL_1:6,63; then A13: sqrt (r - q`1)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26; A14: q = |[q`1,q`2]| by EUCLID:53; then sqrt ((r - q`1)^2 + (s - q`2)^2) < r-G*(len G,1)`1 by A12,Th6; then sqrt (r - q`1)^2 <= r-G*(len G,1)`1 by A13,XXREAL_0:2; then A15: |.r-q`1.| <= r-G*(len G,1)`1 by COMPLEX1:72; per cases; suppose r >= q`1; then r-q`1 >= 0 by XREAL_1:48; then |.r-q`1.| = r - q`1 by ABSVALUE:def 1; then G*(len G,1)`1 <= q`1 by A15,XREAL_1:10; hence thesis by A1,A11,A14; end; suppose r <= q`1; then G*(len G,1)`1 <= q`1 by A8,XXREAL_0:2; hence thesis by A1,A11,A14; end; end; reconsider B = Ball(u,r-G*(len G,1)`1) as Subset of TOP-REAL2 by TOPREAL3:8; A16: B is open by Th3; u in Ball(u,r-G*(len G,1)`1) by A8,Th1,XREAL_1:50; hence thesis by A7,A9,A16,TOPS_1:22; end; theorem Th14: 1 <= i & i < len G implies Int v_strip(G,i) = { |[r,s]| : G*(i,1 )`1 < r & r < G*(i+1,1)`1 } proof 0 <> width G by MATRIX_0:def 10; then A1: 1 <= width G by NAT_1:14; assume 1 <= i & i < len G; then A2: v_strip(G,i) = { |[r,s]| : G*(i,1)`1 <= r & r <= G*(i+1,1)`1 } by A1, GOBOARD5:8; thus Int v_strip(G,i) c= { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 } proof let x be object; assume A3: x in Int v_strip(G,i); then reconsider u = x as Point of Euclid 2 by Lm6; consider r1 being Real such that A4: r1 > 0 and A5: Ball(u,r1) c= v_strip(G,i) by A3,Th5; reconsider p = u as Point of TOP-REAL 2 by Lm6; A6: p = |[p`1,p`2]| by EUCLID:53; set q2 = |[p`1-r1/2,p`2+0]|; A7: r1/2 < r1 by A4,XREAL_1:216; then q2 in Ball(u,r1) by A4,A6,Th9; then q2 in v_strip(G,i) by A5; then ex r2,s2 st q2 = |[r2,s2]| & G*(i,1)`1 <= r2 & r2 <= G*(i+1,1)`1 by A2 ; then G*(i,1)`1 <= p`1-r1/2 by SPPOL_2:1; then A8: G*(i,1)`1+r1/2 <= p`1 by XREAL_1:19; set q1 = |[p`1+r1/2,p`2+0]|; q1 in Ball(u,r1) by A4,A6,A7,Th7; then q1 in v_strip(G,i) by A5; then ex r2,s2 st q1 = |[r2,s2]| & G*(i,1)`1 <= r2 & r2 <= G*(i+1,1)`1 by A2 ; then A9: p`1+r1/2 <= G*(i+1,1)`1 by SPPOL_2:1; G*(i,1)`1 < G*(i,1)`1 + r1/2 by A4,XREAL_1:29,215; then A10: G*(i,1)`1 < p`1 by A8,XXREAL_0:2; p`1 < p`1 + r1/2 by A4,XREAL_1:29,215; then p`1 < G*(i+1,1)`1 by A9,XXREAL_0:2; hence thesis by A6,A10; end; let x be object; assume x in { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 }; then consider r,s such that A11: x = |[r,s]| and A12: G*(i,1)`1 < r and A13: r < G*(i+1,1)`1; reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8; G*(i+1,1)`1-r > 0 & r - G*(i,1)`1 > 0 by A12,A13,XREAL_1:50; then min(r-G*(i,1)`1,G*(i+1,1)`1-r) > 0 by XXREAL_0:15; then A14: u in Ball(u,min(r-G*(i,1)`1,G*(i+1,1)`1-r)) by Th1; A15: Ball(u,min(r-G*(i,1)`1,G*(i+1,1)`1-r)) c= v_strip(G,i) proof let y be object; A16: Ball(u,min(r-G*(i,1)`1,G*(i+1,1)`1-r)) = { v : dist(u,v)= 0 & (r - q`1)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by XREAL_1:6,63; then A19: sqrt (r - q`1)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26; A20: q = |[q`1,q`2]| by EUCLID:53; then sqrt ((r - q`1)^2 + (s - q`2)^2) < min(r-G*(i,1)`1,G*(i+1,1 )`1-r ) by A18,Th6; then sqrt (r - q`1)^2 <= min(r-G*(i,1)`1,G* (i+1,1)`1-r) by A19,XXREAL_0:2; then A21: |.r-q`1.| <= min(r-G*(i,1)`1,G*(i+1,1)`1-r) by COMPLEX1:72; then A22: |.r-q`1.| <= r-G*(i,1)`1 by XXREAL_0:22; A23: |.r-q`1.| <= G* (i+1,1)`1-r by A21,XXREAL_0:22; per cases; suppose A24: r <= q`1; then A25: q`1-r >= 0 by XREAL_1:48; |.r-q`1.| = |.-(r-q`1).| by COMPLEX1:52 .= q`1 - r by A25,ABSVALUE:def 1; then A26: q`1 <= G*(i+1,1)`1 by A23,XREAL_1:9; G*(i,1)`1 <= q`1 by A12,A24,XXREAL_0:2; hence thesis by A2,A17,A20,A26; end; suppose A27: r >= q`1; then r-q`1 >= 0 by XREAL_1:48; then |.r-q`1.| = r - q`1 by ABSVALUE:def 1; then A28: G*(i,1)`1 <= q`1 by A22,XREAL_1:10; q`1 <= G*(i+1,1)`1 by A13,A27,XXREAL_0:2; hence thesis by A2,A17,A20,A28; end; end; reconsider B = Ball(u,min(r-G*(i,1)`1,G*(i+1,1)`1-r)) as Subset of TOP-REAL2 by TOPREAL3:8; B is open by Th3; hence thesis by A11,A14,A15,TOPS_1:22; end; theorem Th15: Int h_strip(G,0) = { |[r,s]| : s < G*(1,1)`2 } proof 0 <> len G by MATRIX_0:def 10; then 1 <= len G by NAT_1:14; then A1: h_strip(G,0) = { |[r,s]| : s <= G*(1,1)`2 } by GOBOARD5:7; thus Int h_strip(G,0) c= { |[r,s]| : s < G*(1,1)`2 } proof let x be object; assume A2: x in Int h_strip(G,0); then reconsider u = x as Point of Euclid 2 by Lm6; consider s1 being Real such that A3: s1 > 0 and A4: Ball(u,s1) c= h_strip(G,0) by A2,Th5; reconsider p = u as Point of TOP-REAL 2 by Lm6; A5: p = |[p`1,p`2]| by EUCLID:53; set q = |[p`1+0,p`2+s1/2]|; s1/2 < s1 by A3,XREAL_1:216; then q in Ball(u,s1) by A3,A5,Th8; then q in h_strip(G,0) by A4; then ex r2,s2 st q = |[r2,s2]| & s2 <= G*(1,1)`2 by A1; then A6: p`2+s1/2 <= G*(1,1)`2 by SPPOL_2:1; p`2 < p`2 + s1/2 by A3,XREAL_1:29,215; then p`2 < G*(1,1)`2 by A6,XXREAL_0:2; hence thesis by A5; end; let x be object; assume x in { |[r,s]| : s < G*(1,1)`2 }; then consider r,s such that A7: x = |[r,s]| and A8: s < G*(1,1)`2; reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8; A9: Ball(u,G*(1,1)`2-s) c= h_strip(G,0) proof let y be object; A10: Ball(u,G*(1,1)`2-s) = { v : dist(u,v)= 0 & (s - q`2)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by XREAL_1:6,63; then A13: sqrt (s - q`2)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26; A14: q = |[q`1,q`2]| by EUCLID:53; then sqrt ((r - q`1)^2 + (s - q`2)^2) < G*(1,1)`2-s by A12,Th6; then sqrt (s - q`2)^2 <= G*(1,1)`2-s by A13,XXREAL_0:2; then A15: |.s-q`2.| <= G*(1,1)`2-s by COMPLEX1:72; per cases; suppose s <= q`2; then A16: q`2-s >= 0 by XREAL_1:48; |.s-q`2.| = |.-(s-q`2).| by COMPLEX1:52 .= q`2 - s by A16,ABSVALUE:def 1; then q`2 <= G*(1,1)`2 by A15,XREAL_1:9; hence thesis by A1,A11,A14; end; suppose s >= q`2; then q`2 <= G*(1,1)`2 by A8,XXREAL_0:2; hence thesis by A1,A11,A14; end; end; reconsider B = Ball(u,G*(1,1)`2-s) as Subset of TOP-REAL2 by TOPREAL3:8; A17: B is open by Th3; u in Ball(u,G*(1,1)`2-s) by A8,Th1,XREAL_1:50; hence thesis by A7,A9,A17,TOPS_1:22; end; theorem Th16: Int h_strip(G,width G) = { |[r,s]| : G*(1,width G)`2 < s } proof 0 <> len G by MATRIX_0:def 10; then 1 <= len G by NAT_1:14; then A1: h_strip(G,width G) = { |[r,s]| : G*(1,width G)`2 <= s } by GOBOARD5:6; thus Int h_strip(G,width G) c= { |[r,s]| : G*(1,width G)`2 < s } proof let x be object; assume A2: x in Int h_strip(G,width G); then reconsider u = x as Point of Euclid 2 by Lm6; consider s1 being Real such that A3: s1 > 0 and A4: Ball(u,s1) c= h_strip(G,width G) by A2,Th5; reconsider p = u as Point of TOP-REAL 2 by Lm6; A5: p = |[p`1,p`2]| by EUCLID:53; set q = |[p`1+0,p`2-s1/2]|; s1/2 < s1 by A3,XREAL_1:216; then q in Ball(u,s1) by A3,A5,Th10; then q in h_strip(G,width G) by A4; then ex r2,s2 st q = |[r2,s2]| & G*(1,width G)`2 <= s2 by A1; then G*(1,width G)`2 <= p`2-s1/2 by SPPOL_2:1; then A6: G*(1,width G)`2+s1/2 <= p`2 by XREAL_1:19; G*(1,width G)`2 < G*(1,width G)`2 + s1/2 by A3,XREAL_1:29,215; then G*(1,width G)`2 < p`2 by A6,XXREAL_0:2; hence thesis by A5; end; let x be object; assume x in { |[r,s]| : G*(1,width G)`2 < s }; then consider r,s such that A7: x = |[r,s]| and A8: G*(1,width G)`2 < s; reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8; A9: Ball(u,s-G*(1,width G)`2) c= h_strip(G,width G) proof let y be object; A10: Ball(u,s-G*(1,width G)`2) = { v : dist(u,v)= 0 & (s - q`2)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by XREAL_1:6,63; then A13: sqrt (s - q`2)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26; A14: q = |[q`1,q`2]| by EUCLID:53; then sqrt ((r - q`1)^2 + (s - q`2)^2) < s-G*(1,width G)`2 by A12,Th6; then sqrt (s - q`2)^2 <= s-G*(1,width G)`2 by A13,XXREAL_0:2; then A15: |.s-q`2.| <= s-G*(1,width G)`2 by COMPLEX1:72; per cases; suppose s >= q`2; then s-q`2 >= 0 by XREAL_1:48; then |.s-q`2.| = s - q`2 by ABSVALUE:def 1; then G*(1,width G)`2 <= q`2 by A15,XREAL_1:10; hence thesis by A1,A11,A14; end; suppose s <= q`2; then G*(1,width G)`2 <= q`2 by A8,XXREAL_0:2; hence thesis by A1,A11,A14; end; end; reconsider B = Ball(u,s-G*(1,width G)`2) as Subset of TOP-REAL2 by TOPREAL3:8 ; A16: B is open by Th3; u in Ball(u,s-G*(1,width G)`2) by A8,Th1,XREAL_1:50; hence thesis by A7,A9,A16,TOPS_1:22; end; theorem Th17: 1 <= j & j < width G implies Int h_strip(G,j) = { |[r,s]| : G*(1 ,j)`2 < s & s < G*(1,j+1)`2 } proof 0 <> len G by MATRIX_0:def 10; then A1: 1 <= len G by NAT_1:14; assume 1 <= j & j < width G; then A2: h_strip(G,j) = { |[r,s]| : G*(1,j)`2 <= s & s <= G*(1,j+1)`2 } by A1, GOBOARD5:5; thus Int h_strip(G,j) c= { |[r,s]| : G*(1,j)`2 < s & s < G*(1,j+1)`2 } proof let x be object; assume A3: x in Int h_strip(G,j); then reconsider u = x as Point of Euclid 2 by Lm6; consider s1 being Real such that A4: s1 > 0 and A5: Ball(u,s1) c= h_strip(G,j) by A3,Th5; reconsider p = u as Point of TOP-REAL 2 by Lm6; A6: p = |[p`1,p`2]| by EUCLID:53; set q2 = |[p`1+0,p`2-s1/2]|; A7: s1/2 < s1 by A4,XREAL_1:216; then q2 in Ball(u,s1) by A4,A6,Th10; then q2 in h_strip(G,j) by A5; then ex r2,s2 st q2 = |[r2,s2]| & G*(1,j)`2 <= s2 & s2 <= G*(1,j+1)`2 by A2 ; then G*(1,j)`2 <= p`2-s1/2 by SPPOL_2:1; then A8: G*(1,j)`2+s1/2 <= p`2 by XREAL_1:19; set q1 = |[p`1+0,p`2+s1/2]|; q1 in Ball(u,s1) by A4,A6,A7,Th8; then q1 in h_strip(G,j) by A5; then ex r2,s2 st q1 = |[r2,s2]| & G*(1,j)`2 <= s2 & s2 <= G*(1,j+1)`2 by A2 ; then A9: p`2+s1/2 <= G*(1,j+1)`2 by SPPOL_2:1; G*(1,j)`2 < G*(1,j)`2 + s1/2 by A4,XREAL_1:29,215; then A10: G*(1,j)`2 < p`2 by A8,XXREAL_0:2; p`2 < p`2 + s1/2 by A4,XREAL_1:29,215; then p`2 < G*(1,j+1)`2 by A9,XXREAL_0:2; hence thesis by A6,A10; end; let x be object; assume x in { |[r,s]| : G*(1,j)`2 < s & s < G*(1,j+1)`2 }; then consider r,s such that A11: x = |[r,s]| and A12: G*(1,j)`2 < s and A13: s < G*(1,j+1)`2; reconsider u = |[r,s]| as Point of Euclid 2 by TOPREAL3:8; G*(1,j+1)`2-s > 0 & s - G*(1,j)`2 > 0 by A12,A13,XREAL_1:50; then min(s-G*(1,j)`2,G*(1,j+1)`2-s) > 0 by XXREAL_0:15; then A14: u in Ball(u,min(s-G*(1,j)`2,G*(1,j+1)`2-s)) by Th1; A15: Ball(u,min(s-G*(1,j)`2,G*(1,j+1)`2-s)) c= h_strip(G,j) proof let y be object; A16: Ball(u,min(s-G*(1,j)`2,G*(1,j+1)`2-s)) = { v : dist(u,v)= 0 & (s - q`2)^2 + 0 <= (r - q`1)^2 + (s - q`2)^2 by XREAL_1:6,63; then A19: sqrt (s - q`2)^2 <= sqrt ((r - q`1)^2 + (s - q`2)^2) by SQUARE_1:26; A20: q = |[q`1,q`2]| by EUCLID:53; then sqrt ((r - q`1)^2 + (s - q`2)^2) < min(s-G*(1,j)`2,G*(1,j+1 )`2-s ) by A18,Th6; then sqrt (s - q`2)^2 <= min(s-G*(1,j)`2,G* (1,j+1)`2-s) by A19,XXREAL_0:2; then A21: |.s-q`2.| <= min(s-G*(1,j)`2,G*(1,j+1)`2-s) by COMPLEX1:72; then A22: |.s-q`2.| <= s-G*(1,j)`2 by XXREAL_0:22; A23: |.s-q`2.| <= G* (1,j+1)`2-s by A21,XXREAL_0:22; per cases; suppose A24: s <= q`2; then A25: q`2-s >= 0 by XREAL_1:48; |.s-q`2.| = |.-(s-q`2).| by COMPLEX1:52 .= q`2 - s by A25,ABSVALUE:def 1; then A26: q`2 <= G*(1,j+1)`2 by A23,XREAL_1:9; G*(1,j)`2 <= q`2 by A12,A24,XXREAL_0:2; hence thesis by A2,A17,A20,A26; end; suppose A27: s >= q`2; then s-q`2 >= 0 by XREAL_1:48; then |.s-q`2.| = s - q`2 by ABSVALUE:def 1; then A28: G*(1,j)`2 <= q`2 by A22,XREAL_1:10; q`2 <= G*(1,j+1)`2 by A13,A27,XXREAL_0:2; hence thesis by A2,A17,A20,A28; end; end; reconsider B = Ball(u,min(s-G*(1,j)`2,G*(1,j+1)`2-s)) as Subset of TOP-REAL2 by TOPREAL3:8; B is open by Th3; hence thesis by A11,A14,A15,TOPS_1:22; end; theorem Th18: Int cell(G,0,0) = { |[r,s]| : r < G*(1,1)`1 & s < G*(1,1)`2 } proof cell(G,0,0) = v_strip(G,0) /\ h_strip(G,0) by GOBOARD5:def 3; then A1: Int cell(G,0,0) = Int v_strip(G,0) /\ Int h_strip(G,0) by TOPS_1:17; A2: Int h_strip(G,0) = { |[r,s]| : s < G*(1,1)`2 } by Th15; A3: Int v_strip(G,0) = { |[r,s]| : r < G*(1,1)`1 } by Th12; thus Int cell(G,0,0) c= { |[r,s]| : r < G*(1,1)`1 & s < G*(1,1)`2 } proof let x be object; assume A4: x in Int cell(G,0,0); then x in Int v_strip(G,0) by A1,XBOOLE_0:def 4; then consider r1,s1 such that A5: x = |[r1,s1]| and A6: r1 < G*(1,1)`1 by A3; x in Int h_strip(G,0) by A1,A4,XBOOLE_0:def 4; then consider r2,s2 such that A7: x = |[r2,s2]| and A8: s2 < G*(1,1)`2 by A2; s1 = s2 by A5,A7,SPPOL_2:1; hence thesis by A5,A6,A8; end; let x be object; assume x in { |[r,s]| : r < G*(1,1)`1 & s < G*(1,1)`2 }; then A9: ex r,s st x = |[r,s]| & r < G*(1,1)`1 & s < G*(1,1)`2; then A10: x in Int h_strip(G,0) by A2; x in Int v_strip(G,0) by A3,A9; hence thesis by A1,A10,XBOOLE_0:def 4; end; theorem Th19: Int cell(G,0,width G) = { |[r,s]| : r < G*(1,1)`1 & G*(1,width G )`2 < s } proof cell(G,0,width G) = v_strip(G,0) /\ h_strip(G,width G) by GOBOARD5:def 3; then A1: Int cell(G,0,width G) = Int v_strip(G,0) /\ Int h_strip(G,width G) by TOPS_1:17; A2: Int h_strip(G,width G) = { |[r,s]| : G*(1,width G)`2 < s } by Th16; A3: Int v_strip(G,0) = { |[r,s]| : r < G*(1,1)`1 } by Th12; thus Int cell(G,0,width G) c= { |[r,s]| : r < G*(1,1)`1 & G*(1,width G)`2 < s } proof let x be object; assume A4: x in Int cell(G,0,width G); then x in Int v_strip(G,0) by A1,XBOOLE_0:def 4; then consider r1,s1 such that A5: x = |[r1,s1]| and A6: r1 < G*(1,1)`1 by A3; x in Int h_strip(G,width G) by A1,A4,XBOOLE_0:def 4; then consider r2,s2 such that A7: x = |[r2,s2]| and A8: G*(1,width G)`2 < s2 by A2; s1 = s2 by A5,A7,SPPOL_2:1; hence thesis by A5,A6,A8; end; let x be object; assume x in { |[r,s]| : r < G*(1,1)`1 & G*(1,width G)`2 < s }; then A9: ex r,s st x = |[r,s]| & r < G*(1,1)`1 & G*(1,width G)`2 < s; then A10: x in Int h_strip(G,width G) by A2; x in Int v_strip(G,0) by A3,A9; hence thesis by A1,A10,XBOOLE_0:def 4; end; theorem Th20: 1 <= j & j < width G implies Int cell(G,0,j) = { |[r,s]| : r < G *(1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1)`2 } proof cell(G,0,j) = v_strip(G,0) /\ h_strip(G,j) by GOBOARD5:def 3; then A1: Int cell(G,0,j) = Int v_strip(G,0) /\ Int h_strip(G,j) by TOPS_1:17; assume 1 <= j & j < width G; then A2: Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G* (1,j+1)`2 } by Th17; A3: Int v_strip(G,0) = { |[r,s]| : r < G*(1,1)`1 } by Th12; thus Int cell(G,0,j) c= { |[r,s]| : r < G*(1,1)`1 & G*(1,j)`2 < s & s < G*(1 ,j+1)`2 } proof let x be object; assume A4: x in Int cell(G,0,j); then x in Int v_strip(G,0) by A1,XBOOLE_0:def 4; then consider r1,s1 such that A5: x = |[r1,s1]| and A6: r1 < G*(1,1)`1 by A3; x in Int h_strip(G,j) by A1,A4,XBOOLE_0:def 4; then consider r2,s2 such that A7: x = |[r2,s2]| and A8: G*(1,j)`2 < s2 & s2 < G*(1,j+1)`2 by A2; s1 = s2 by A5,A7,SPPOL_2:1; hence thesis by A5,A6,A8; end; let x be object; assume x in { |[r,s]| : r < G*(1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1)`2 }; then A9: ex r,s st x = |[r,s]| & r < G*(1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1)`2; then A10: x in Int h_strip(G,j) by A2; x in Int v_strip(G,0) by A3,A9; hence thesis by A1,A10,XBOOLE_0:def 4; end; theorem Th21: Int cell(G,len G,0) = { |[r,s]| : G*(len G,1)`1 < r & s < G*(1,1 )`2 } proof cell(G,len G,0) = v_strip(G,len G) /\ h_strip(G,0) by GOBOARD5:def 3; then A1: Int cell(G,len G,0) = Int v_strip(G,len G) /\ Int h_strip(G,0) by TOPS_1:17 ; A2: Int h_strip(G,0) = { |[r,s]| : s < G*(1,1)`2 } by Th15; A3: Int v_strip(G,len G) = { |[r,s]| : G*(len G,1)`1 < r } by Th13; thus Int cell(G,len G,0) c= { |[r,s]| : G*(len G,1)`1 < r & s < G*(1,1)`2 } proof let x be object; assume A4: x in Int cell(G,len G,0); then x in Int v_strip(G,len G) by A1,XBOOLE_0:def 4; then consider r1,s1 such that A5: x = |[r1,s1]| and A6: G*(len G,1)`1 < r1 by A3; x in Int h_strip(G,0) by A1,A4,XBOOLE_0:def 4; then consider r2,s2 such that A7: x = |[r2,s2]| and A8: s2 < G*(1,1)`2 by A2; s1 = s2 by A5,A7,SPPOL_2:1; hence thesis by A5,A6,A8; end; let x be object; assume x in { |[r,s]| : G*(len G,1)`1 < r & s < G*(1,1)`2 }; then A9: ex r,s st x = |[r,s]| & G*(len G,1)`1 < r & s < G*(1,1)`2; then A10: x in Int h_strip(G,0) by A2; x in Int v_strip(G,len G) by A3,A9; hence thesis by A1,A10,XBOOLE_0:def 4; end; theorem Th22: Int cell(G,len G,width G) = { |[r,s]| : G*(len G,1)`1 < r & G*(1 ,width G)`2 < s } proof cell(G,len G,width G) = v_strip(G,len G) /\ h_strip(G,width G) by GOBOARD5:def 3; then A1: Int cell(G,len G,width G) = Int v_strip(G,len G) /\ Int h_strip(G, width G) by TOPS_1:17; A2: Int h_strip(G,width G) = { |[r,s]| : G*(1,width G)`2 < s } by Th16; A3: Int v_strip(G,len G) = { |[r,s]| : G*(len G,1)`1 < r } by Th13; thus Int cell(G,len G,width G) c= { |[r,s]| : G*(len G,1)`1 < r & G*(1,width G)`2 < s } proof let x be object; assume A4: x in Int cell(G,len G,width G); then x in Int v_strip(G,len G) by A1,XBOOLE_0:def 4; then consider r1,s1 such that A5: x = |[r1,s1]| and A6: G*(len G,1)`1 < r1 by A3; x in Int h_strip(G,width G) by A1,A4,XBOOLE_0:def 4; then consider r2,s2 such that A7: x = |[r2,s2]| and A8: G*(1,width G)`2 < s2 by A2; s1 = s2 by A5,A7,SPPOL_2:1; hence thesis by A5,A6,A8; end; let x be object; assume x in { |[r,s]| : G*(len G,1)`1 < r & G*(1,width G)`2 < s }; then A9: ex r,s st x = |[r,s]| & G*(len G,1)`1 < r & G*(1,width G)`2 < s; then A10: x in Int h_strip(G,width G) by A2; x in Int v_strip(G,len G) by A3,A9; hence thesis by A1,A10,XBOOLE_0:def 4; end; theorem Th23: 1 <= j & j < width G implies Int cell(G,len G,j) = { |[r,s]| : G *(len G,1)`1 < r & G*(1,j)`2 < s & s < G*(1,j+1)`2 } proof cell(G,len G,j) = v_strip(G,len G) /\ h_strip(G,j) by GOBOARD5:def 3; then A1: Int cell(G,len G,j) = Int v_strip(G,len G) /\ Int h_strip(G,j) by TOPS_1:17 ; assume 1 <= j & j < width G; then A2: Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G* (1,j+1)`2 } by Th17; A3: Int v_strip(G,len G) = { |[r,s]| : G*(len G,1)`1 < r } by Th13; thus Int cell(G,len G,j) c= { |[r,s]| : G*(len G,1)`1 < r & G*(1,j)`2 < s & s < G*(1,j+1)`2 } proof let x be object; assume A4: x in Int cell(G,len G,j); then x in Int v_strip(G,len G) by A1,XBOOLE_0:def 4; then consider r1,s1 such that A5: x = |[r1,s1]| and A6: G*(len G,1)`1 < r1 by A3; x in Int h_strip(G,j) by A1,A4,XBOOLE_0:def 4; then consider r2,s2 such that A7: x = |[r2,s2]| and A8: G*(1,j)`2 < s2 & s2 < G*(1,j+1)`2 by A2; s1 = s2 by A5,A7,SPPOL_2:1; hence thesis by A5,A6,A8; end; let x be object; assume x in { |[r,s]| : G*(len G,1)`1 < r & G*(1,j)`2 < s & s < G* (1,j+1) `2 }; then A9: ex r,s st x = |[r,s]| & G*(len G,1)`1 < r & G*(1,j)`2 < s & s < G*(1,j+ 1)`2; then A10: x in Int h_strip(G,j) by A2; x in Int v_strip(G,len G) by A3,A9; hence thesis by A1,A10,XBOOLE_0:def 4; end; theorem Th24: 1 <= i & i < len G implies Int cell(G,i,0) = { |[r,s]| : G*(i,1) `1 < r & r < G*(i+1,1)`1 & s < G*(1,1)`2 } proof cell(G,i,0) = v_strip(G,i) /\ h_strip(G,0) by GOBOARD5:def 3; then A1: Int cell(G,i,0) = Int v_strip(G,i) /\ Int h_strip(G,0) by TOPS_1:17; assume 1 <= i & i < len G; then A2: Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G* (i+1,1)`1 } by Th14; A3: Int h_strip(G,0) = { |[r,s]| : s < G*(1,1)`2 } by Th15; thus Int cell(G,i,0) c= { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & s < G* (1,1)`2 } proof let x be object; assume A4: x in Int cell(G,i,0); then x in Int v_strip(G,i) by A1,XBOOLE_0:def 4; then consider r1,s1 such that A5: x = |[r1,s1]| and A6: G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 by A2; x in Int h_strip(G,0) by A1,A4,XBOOLE_0:def 4; then consider r2,s2 such that A7: x = |[r2,s2]| and A8: s2 < G*(1,1)`2 by A3; s1 = s2 by A5,A7,SPPOL_2:1; hence thesis by A5,A6,A8; end; let x be object; assume x in { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & s < G*(1,1)`2 }; then A9: ex r,s st x = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 & s < G*(1,1)`2; then A10: x in Int h_strip(G,0) by A3; x in Int v_strip(G,i) by A2,A9; hence thesis by A1,A10,XBOOLE_0:def 4; end; theorem Th25: 1 <= i & i < len G implies Int cell(G,i,width G) = { |[r,s]| : G *(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,width G)`2 < s } proof cell(G,i,width G) = v_strip(G,i) /\ h_strip(G,width G) by GOBOARD5:def 3; then A1: Int cell(G,i,width G) = Int v_strip(G,i) /\ Int h_strip(G,width G) by TOPS_1:17; assume 1 <= i & i < len G; then A2: Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G* (i+1,1)`1 } by Th14; A3: Int h_strip(G,width G) = { |[r,s]| : G*(1,width G)`2 < s } by Th16; thus Int cell(G,i,width G) c= { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,width G)`2 < s } proof let x be object; assume A4: x in Int cell(G,i,width G); then x in Int v_strip(G,i) by A1,XBOOLE_0:def 4; then consider r1,s1 such that A5: x = |[r1,s1]| and A6: G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 by A2; x in Int h_strip(G,width G) by A1,A4,XBOOLE_0:def 4; then consider r2,s2 such that A7: x = |[r2,s2]| and A8: G*(1,width G)`2 < s2 by A3; s1 = s2 by A5,A7,SPPOL_2:1; hence thesis by A5,A6,A8; end; let x be object; assume x in { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,width G)`2 < s }; then A9: ex r,s st x = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,width G) `2 < s; then A10: x in Int h_strip(G,width G) by A3; x in Int v_strip(G,i) by A2,A9; hence thesis by A1,A10,XBOOLE_0:def 4; end; theorem Th26: 1 <= i & i < len G & 1 <= j & j < width G implies Int cell(G,i,j ) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1) `2 } proof assume that A1: 1 <= i & i < len G and A2: 1 <= j & j < width G; A3: Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G* (1,j+1)`2 } by A2 ,Th17; cell(G,i,j) = v_strip(G,i) /\ h_strip(G,j) by GOBOARD5:def 3; then A4: Int cell(G,i,j) = Int v_strip(G,i) /\ Int h_strip(G,j) by TOPS_1:17; A5: Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G* (i+1,1)`1 } by A1 ,Th14; thus Int cell(G,i,j) c= { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,j )`2 < s & s < G*(1,j+1)`2 } proof let x be object; assume A6: x in Int cell(G,i,j); then x in Int v_strip(G,i) by A4,XBOOLE_0:def 4; then consider r1,s1 such that A7: x = |[r1,s1]| and A8: G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 by A5; x in Int h_strip(G,j) by A4,A6,XBOOLE_0:def 4; then consider r2,s2 such that A9: x = |[r2,s2]| and A10: G*(1,j)`2 < s2 & s2 < G*(1,j+1)`2 by A3; s1 = s2 by A7,A9,SPPOL_2:1; hence thesis by A7,A8,A10; end; let x be object; assume x in { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1)`2 }; then A11: ex r,s st x = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1)`2; then A12: x in Int h_strip(G,j) by A3; x in Int v_strip(G,i) by A5,A11; hence thesis by A4,A12,XBOOLE_0:def 4; end; theorem 1 <= j & j <= width G & p in Int h_strip(G,j) implies p`2 > G*(1,j)`2 proof assume that A1: 1 <= j and A2: j <= width G and A3: p in Int h_strip(G,j); per cases by A2,XXREAL_0:1; suppose j = width G; then Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s } by Th16; then ex r,s st p = |[r,s]| & G*(1,j)`2 < s by A3; hence thesis by EUCLID:52; end; suppose j < width G; then Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G*(1,j+1)`2 } by A1,Th17 ; then ex r,s st p = |[r,s]| & G*(1,j)`2 < s & s < G*(1,j+1)`2 by A3; hence thesis by EUCLID:52; end; end; theorem j < width G & p in Int h_strip(G,j) implies p`2 < G*(1,j+1)`2 proof assume that A1: j < width G and A2: p in Int h_strip(G,j); per cases by NAT_1:14; suppose j = 0; then Int h_strip(G,j) = { |[r,s]| : s < G*(1,j+1)`2 } by Th15; then ex r,s st p = |[r,s]| & G*(1,j+1)`2 > s by A2; hence thesis by EUCLID:52; end; suppose j >= 1; then Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G*(1,j+1)`2 } by A1,Th17 ; then ex r,s st p = |[r,s]| & G*(1,j)`2 < s & s < G*(1,j+1)`2 by A2; hence thesis by EUCLID:52; end; end; theorem 1 <= i & i <= len G & p in Int v_strip(G,i) implies p`1 > G*(i,1)`1 proof assume that A1: 1 <= i and A2: i <= len G and A3: p in Int v_strip(G,i); per cases by A2,XXREAL_0:1; suppose i = len G; then Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r } by Th13; then ex r,s st p = |[r,s]| & G*(i,1)`1 < r by A3; hence thesis by EUCLID:52; end; suppose i < len G; then Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 } by A1,Th14 ; then ex r,s st p = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 by A3; hence thesis by EUCLID:52; end; end; theorem i < len G & p in Int v_strip(G,i) implies p`1 < G*(i+1,1)`1 proof assume that A1: i < len G and A2: p in Int v_strip(G,i); per cases by NAT_1:14; suppose i = 0; then Int v_strip(G,i) = { |[r,s]| : r < G*(i+1,1)`1 } by Th12; then ex r,s st p = |[r,s]| & G*(i+1,1)`1 > r by A2; hence thesis by EUCLID:52; end; suppose i >= 1; then Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 } by A1,Th14 ; then ex r,s st p = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 by A2; hence thesis by EUCLID:52; end; end; theorem Th31: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies 1/2*(G*( i,j)+G*(i+1,j+1)) in Int cell(G,i,j) proof assume that A1: 1 <= i and A2: i+1 <= len G and A3: 1 <= j and A4: j+1 <= width G; A5: j < j+1 by XREAL_1:29; set r1 = G*(i,j)`1, s1 = G*(i,j)`2, r2 = G*(i+1,j+1)`1, s2 = G* (i+1,j+1)`2; A6: 1 <= i+1 & 1 <= j+1 by NAT_1:11; then A7: G*(1,j+1)`2 = s2 by A2,A4,GOBOARD5:1; i < len G & j < width G by A2,A4,NAT_1:13; then A8: Int cell(G,i,j) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G*(1,j) `2 < s & s < G*(1,j+1)`2 } by A1,A3,Th26; G*(i,j) = |[r1,s1]| & G*(i+1,j+1) = |[r2,s2]| by EUCLID:53; then G*(i,j)+G*(i+1,j+1) = |[r1+r2,s1+s2]| by EUCLID:56; then A9: 1/2*(G*(i,j)+G*(i+1,j+1)) = |[1/2*(r1+r2),1/2*(s1+s2)]| by EUCLID:58; i <= i+1 by NAT_1:11; then A10: i <= len G by A2,XXREAL_0:2; then A11: 1 <= len G by A1,XXREAL_0:2; j <= j+1 by NAT_1:11; then A12: j <= width G by A4,XXREAL_0:2; then A13: 1 <= width G by A3,XXREAL_0:2; A14: G*(i,1)`1 = r1 by A1,A3,A10,A12,GOBOARD5:2; G*(1,j)`2 = s1 by A1,A3,A10,A12,GOBOARD5:1; then A15: s1 < s2 by A3,A4,A7,A11,A5,GOBOARD5:4; then s1+s1 < s1+s2 by XREAL_1:6; then 1/2*(s1+s1) < 1/2*(s1+s2) by XREAL_1:68; then A16: G*(1,j)`2 < 1/2*(s1+s2) by A1,A3,A10,A12,GOBOARD5:1; A17: i < i+1 by XREAL_1:29; G*(i+1,1)`1 = r2 by A2,A4,A6,GOBOARD5:2; then A18: r1 < r2 by A1,A2,A14,A13,A17,GOBOARD5:3; then r1+r2 < r2+r2 by XREAL_1:6; then 1/2*(r1+r2) < 1/2*(r2+r2) by XREAL_1:68; then A19: 1/2*(r1+r2) < G*(i+1,1)`1 by A2,A4,A6,GOBOARD5:2; s1+s2 < s2+s2 by A15,XREAL_1:6; then 1/2*(s1+s2) < 1/2*(s2+s2) by XREAL_1:68; then A20: 1/2*(s1+s2) < G*(1,j+1)`2 by A2,A4,A6,GOBOARD5:1; r1+r1 < r1+r2 by A18,XREAL_1:6; then 1/2*(r1+r1) < 1/2*(r1+r2) by XREAL_1:68; hence thesis by A9,A14,A19,A16,A20,A8; end; theorem Th32: 1 <= i & i+1 <= len G implies 1/2*(G*(i,width G)+G*(i+1,width G) )+|[0,1]| in Int cell(G,i,width G) proof assume that A1: 1 <= i and A2: i+1 <= len G; set r1 = G*(i,width G)`1, s1 = G*(i,width G)`2, r2 = G*(i+1,width G)`1; width G <> 0 by MATRIX_0:def 10; then A3: 1 <= width G by NAT_1:14; i < len G by A2,NAT_1:13; then A4: Int cell(G,i,width G) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G * (1,width G)`2 < s } by A1,Th25; width G <> 0 by MATRIX_0:def 10; then A5: 1 <= width G by NAT_1:14; i < i+1 by XREAL_1:29; then A6: r1 < r2 by A1,A2,A5,GOBOARD5:3; then r1+r1 < r1+r2 by XREAL_1:6; then A7: 1/2*(r1+r1) < 1/2*(r1+r2) by XREAL_1:68; A8: i < len G by A2,NAT_1:13; then A9: G*(1,width G)`2 = s1 by A1,A3,GOBOARD5:1; then A10: G*(1,width G)`2 < s1+1 by XREAL_1:29; A11: 1 <= i+1 by NAT_1:11; then G*(1,width G)`2 = G*(i+1,width G)`2 by A2,A3,GOBOARD5:1; then G*(i,width G) = |[r1,s1]| & G*(i+1,width G) = |[r2,s1]| by A9,EUCLID:53; then 1/2*(s1+s1) = s1 & G*(i,width G)+G*(i+1,width G) = |[r1+r2,s1+s1]| by EUCLID:56; then 1/2*(G*(i,width G)+G*(i+1,width G))= |[1/2*(r1+r2),s1]| by EUCLID:58; then A12: 1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]| = |[1/2*(r1+r2)+0,s1+1 ]| by EUCLID:56; r1+r2 < r2+r2 by A6,XREAL_1:6; then 1/2*(r1+r2) < 1/2*(r2+r2) by XREAL_1:68; then A13: 1/2*(r1+r2) < G*(i+1,1)`1 by A2,A11,A3,GOBOARD5:2; G*(i,1)`1 = r1 by A1,A8,A3,GOBOARD5:2; hence thesis by A12,A7,A13,A10,A4; end; theorem Th33: 1 <= i & i+1 <= len G implies 1/2*(G*(i,1)+G*(i+1,1))-|[0,1]| in Int cell(G,i,0) proof assume that A1: 1 <= i and A2: i+1 <= len G; set r1 = G*(i,1)`1, s1 = G*(i,1)`2, r2 = G*(i+1,1)`1; width G <> 0 by MATRIX_0:def 10; then A3: 1 <= width G by NAT_1:14; width G <> 0 by MATRIX_0:def 10; then A4: 1 <= width G by NAT_1:14; i < i+1 by XREAL_1:29; then A5: r1 < r2 by A1,A2,A4,GOBOARD5:3; then r1+r1 < r1+r2 by XREAL_1:6; then A6: 1/2*(r1+r1) < 1/2*(r1+r2) by XREAL_1:68; i < len G by A2,NAT_1:13; then A7: G*(1,1)`2 = s1 by A1,A3,GOBOARD5:1; then s1 < G*(1,1)`2+1 by XREAL_1:29; then A8: s1-1 < G*(1,1)`2 by XREAL_1:19; 1 <= i+1 by NAT_1:11; then G*(1,1)`2 = G*(i+1,1)`2 by A2,A3,GOBOARD5:1; then G*(i,1) = |[r1,s1]| & G*(i+1,1) = |[r2,s1]| by A7,EUCLID:53; then 1/2*(s1+s1) = s1 & G*(i,1)+G*(i+1,1) = |[r1+r2,s1+s1]| by EUCLID:56; then 1/2*(G*(i,1)+G*(i+1,1))= |[1/2*(r1+r2),s1]| by EUCLID:58; then A9: 1/2*(G*(i,1)+G*(i+1,1))-|[0,1]| = |[1/2*(r1+r2)-0,s1-1]| by EUCLID:62 .= |[1/2*(r1+r2),s1-1]|; r1+r2 < r2+r2 by A5,XREAL_1:6; then A10: 1/2*(r1+r2) < 1/2*(r2+r2) by XREAL_1:68; i < len G by A2,NAT_1:13; then Int cell(G,i,0) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & s < G*( 1,1)`2 } by A1,Th24; hence thesis by A9,A6,A10,A8; end; theorem Th34: 1 <= j & j+1 <= width G implies 1/2*(G*(len G,j)+G*(len G,j+1))+ |[1,0]| in Int cell(G,len G,j) proof assume that A1: 1 <= j and A2: j+1 <= width G; set s1 = G*(len G,j)`2, r1 = G*(len G,j)`1, s2 = G*(len G,j+1)`2; len G <> 0 by MATRIX_0:def 10; then A3: 1 <= len G by NAT_1:14; j < width G by A2,NAT_1:13; then A4: Int cell(G,len G,j) = { |[r,s]| : G*(len G,1)`1 < r & G*(1,j)`2 < s & s < G* (1,j+1)`2 } by A1,Th23; len G <> 0 by MATRIX_0:def 10; then A5: 1 <= len G by NAT_1:14; j < j+1 by XREAL_1:29; then A6: s1 < s2 by A1,A2,A5,GOBOARD5:4; then s1+s1 < s1+s2 by XREAL_1:6; then A7: 1/2*(s1+s1) < 1/2*(s1+s2) by XREAL_1:68; A8: j < width G by A2,NAT_1:13; then A9: G*(len G,1)`1 = r1 by A1,A3,GOBOARD5:2; then A10: G*(len G,1)`1 < r1+1 by XREAL_1:29; A11: 1 <= j+1 by NAT_1:11; then G*(len G,1)`1 = G*(len G,j+1)`1 by A2,A3,GOBOARD5:2; then G*(len G,j) = |[r1,s1]| & G*(len G,j+1) = |[r1,s2]| by A9,EUCLID:53; then 1/2*(r1+r1) = r1 & G*(len G,j)+G*(len G,j+1) = |[r1+r1,s1+s2]| by EUCLID:56; then 1/2*(G*(len G,j)+G*(len G,j+1))= |[r1,1/2*(s1+s2)]| by EUCLID:58; then A12: 1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]| = |[r1+1,1/2*(s1+s2)+0]| by EUCLID:56; s1+s2 < s2+s2 by A6,XREAL_1:6; then 1/2*(s1+s2) < 1/2*(s2+s2) by XREAL_1:68; then A13: 1/2*(s1+s2) < G*(1,j+1)`2 by A2,A11,A3,GOBOARD5:1; G*(1,j)`2 = s1 by A1,A8,A3,GOBOARD5:1; hence thesis by A12,A7,A13,A10,A4; end; theorem Th35: 1 <= j & j+1 <= width G implies 1/2*(G*(1,j)+G*(1,j+1))-|[1,0]| in Int cell(G,0,j) proof assume that A1: 1 <= j and A2: j+1 <= width G; set s1 = G*(1,j)`2, r1 = G*(1,j)`1, s2 = G*(1,j+1)`2; len G <> 0 by MATRIX_0:def 10; then A3: 1 <= len G by NAT_1:14; len G <> 0 by MATRIX_0:def 10; then A4: 1 <= len G by NAT_1:14; j < j+1 by XREAL_1:29; then A5: s1 < s2 by A1,A2,A4,GOBOARD5:4; then s1+s1 < s1+s2 by XREAL_1:6; then A6: 1/2*(s1+s1) < 1/2*(s1+s2) by XREAL_1:68; j < width G by A2,NAT_1:13; then A7: G*(1,1)`1 = r1 by A1,A3,GOBOARD5:2; then r1 < G*(1,1)`1+1 by XREAL_1:29; then A8: r1-1 < G*(1,1)`1 by XREAL_1:19; 1 <= j+1 by NAT_1:11; then G*(1,1)`1 = G*(1,j+1)`1 by A2,A3,GOBOARD5:2; then G*(1,j) = |[r1,s1]| & G*(1,j+1) = |[r1,s2]| by A7,EUCLID:53; then 1/2*(r1+r1) = r1 & G*(1,j)+G*(1,j+1) = |[r1+r1,s1+s2]| by EUCLID:56; then 1/2*(G*(1,j)+G*(1,j+1))= |[r1,1/2*(s1+s2)]| by EUCLID:58; then A9: 1/2*(G*(1,j)+G*(1,j+1))-|[1,0]| = |[r1-1,1/2*(s1+s2)-0]| by EUCLID:62 .= |[r1-1,1/2*(s1+s2)]|; s1+s2 < s2+s2 by A5,XREAL_1:6; then A10: 1/2*(s1+s2) < 1/2*(s2+s2) by XREAL_1:68; j < width G by A2,NAT_1:13; then Int cell(G,0,j) = { |[r,s]| : r < G*(1,1)`1 & G*(1,j)`2 < s & s < G*(1, j+1)`2 } by A1,Th20; hence thesis by A9,A6,A10,A8; end; theorem Th36: G*(1,1)-|[1,1]| in Int cell(G,0,0) proof set s1 = G*(1,1)`2, r1 = G*(1,1)`1; G*(1,1) = |[r1,s1]| by EUCLID:53; then A1: G*(1,1)-|[1,1]| = |[r1-1,s1-1]| by EUCLID:62; s1 < G*(1,1)`2+1 by XREAL_1:29; then A2: s1-1 < G*(1,1)`2 by XREAL_1:19; r1 < G*(1,1)`1+1 by XREAL_1:29; then A3: r1-1 < G*(1,1)`1 by XREAL_1:19; Int cell(G,0,0) = { |[r,s]| : r < G*(1,1)`1 & s < G*(1,1)`2 } by Th18; hence thesis by A1,A2,A3; end; theorem Th37: G*(len G,width G)+|[1,1]| in Int cell(G,len G,width G) proof set s1 = G*(len G,width G)`2, r1 = G*(len G,width G)`1; len G <> 0 by MATRIX_0:def 10; then A1: 1 <= len G by NAT_1:14; width G <> 0 by MATRIX_0:def 10; then A2: 1 <= width G by NAT_1:14; then G*(len G,1)`1 = r1 by A1,GOBOARD5:2; then A3: r1+1 > G*(len G,1)`1 by XREAL_1:29; G*(len G,width G) = |[r1,s1]| by EUCLID:53; then A4: G*(len G,width G)+|[1,1]| = |[r1+1,s1+1]| by EUCLID:56; G*(1,width G)`2 = s1 by A2,A1,GOBOARD5:1; then A5: s1+1 > G*(1,width G)`2 by XREAL_1:29; Int cell(G,len G,width G) = { |[r,s]| : G*(len G,1)`1 < r & G*(1,width G)`2 < s } by Th22; hence thesis by A4,A5,A3; end; theorem Th38: G*(1,width G)+|[-1,1]| in Int cell(G,0,width G) proof set s1 = G*(1,width G)`2, r1 = G*(1,width G)`1; len G <> 0 by MATRIX_0:def 10; then A1: 1 <= len G by NAT_1:14; width G <> 0 by MATRIX_0:def 10; then 1 <= width G by NAT_1:14; then G*(1,1)`1 = r1 by A1,GOBOARD5:2; then r1 < G*(1,1)`1+1 by XREAL_1:29; then A2: s1+1 > G*(1,width G)`2 & r1-1 < G*(1,1)`1 by XREAL_1:19,29; G*(1,width G) = |[r1,s1]| by EUCLID:53; then A3: G*(1,width G)+|[-1,1]| = |[r1+-1,s1+1]| by EUCLID:56 .= |[r1-1,s1+1]|; Int cell(G,0,width G) = { |[r,s]| : r < G*(1,1)`1 & G*(1,width G)`2 < s } by Th19; hence thesis by A3,A2; end; theorem Th39: G*(len G,1)+|[1,-1]| in Int cell(G,len G,0) proof set s1 = G*(len G,1)`2, r1 = G*(len G,1)`1; A1: r1+1 > G*(len G,1)`1 by XREAL_1:29; len G <> 0 by MATRIX_0:def 10; then A2: 1 <= len G by NAT_1:14; width G <> 0 by MATRIX_0:def 10; then 1 <= width G by NAT_1:14; then G*(1,1)`2 = s1 by A2,GOBOARD5:1; then s1 < G*(1,1)`2+1 by XREAL_1:29; then A3: s1-1 < G*(1,1)`2 by XREAL_1:19; G*(len G,1) = |[r1,s1]| by EUCLID:53; then A4: G*(len G,1)+|[1,-1]| = |[r1+1,s1+-1]| by EUCLID:56 .= |[r1+1,s1-1]|; Int cell(G,len G,0) = { |[r,s]| : G*(len G,1)`1 < r & s < G* (1,1)`2 } by Th21; hence thesis by A4,A3,A1; end; theorem Th40: 1 <= i & i < len G & 1 <= j & j < width G implies LSeg(1/2*(G*(i ,j)+G*(i+1,j+1)),1/2*(G*(i,j)+G*(i,j+1))) c= Int cell(G,i,j) \/ { 1/2*(G*(i,j)+ G*(i,j+1)) } proof assume that A1: 1 <= i and A2: i < len G and A3: 1 <= j and A4: j < width G; let x be object; assume A5: x in LSeg(1/2*(G*(i,j)+G*(i+1,j+1)),1/2*(G*(i,j)+G*(i,j+1))); then reconsider p = x as Point of TOP-REAL 2; consider r such that A6: p = (1-r)*(1/2*(G*(i,j)+G*(i+1,j+1)))+r*(1/2*(G*(i,j)+G*(i,j+1))) and A7: 0<=r and A8: r<=1 by A5; now per cases by A8,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i,j)+G*(i,j+1))) by A6,RLVECT_1:10 .= 1*(1/2*(G*(i,j)+G*(i,j+1))) by RLVECT_1:4 .= 1/2*(G*(i,j)+G*(i,j+1)) by RLVECT_1:def 8; hence p in { 1/2*(G*(i,j)+G*(i,j+1)) } by TARSKI:def 1; end; case A9: r < 1; set r3 = (1-r)*(1/2), s3 = r*(1/2); set r1 = G*(i,1)`1, r2 = G*(i+1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2; A10: r3*(s1+s1)+s3*(s1+s1) = s1; 0 <> len G by MATRIX_0:def 10; then A11: 1 <= len G by NAT_1:14; A12: j+1 <= width G by A4,NAT_1:13; j < j+1 by XREAL_1:29; then A13: s1 < s2 by A3,A12,A11,GOBOARD5:4; then A14: s1+s1 < s1+s2 by XREAL_1:6; then A15: s3*(s1+s1) <= s3*(s1+s2) by A7,XREAL_1:64; 1 - r > 0 by A9,XREAL_1:50; then A16: r3 > (1/2)*0 by XREAL_1:68; then r3*(s1+s1) < r3*(s1+s2) by A14,XREAL_1:68; then A17: s1 < r3*(s1+s2)+s3*(s1+s2) by A15,A10,XREAL_1:8; A18: s1+s2 < s2+s2 by A13,XREAL_1:6; then A19: s3*(s1+s2) <= s3*(s2+s2) by A7,XREAL_1:64; 0 <> width G by MATRIX_0:def 10; then A20: 1 <= width G by NAT_1:14; A21: 1 <= i+1 by A1,NAT_1:13; A22: Int cell(G,i,j) = { |[r9,s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 < s2 } by A1,A2,A3,A4,Th26; A23: 1 <= j+1 by A3,NAT_1:13; A24: G*(i,j+1) = |[G*(i,j+1)`1,G*(i,j+1)`2]| by EUCLID:53 .= |[r1,G*(i,j+1)`2]| by A1,A2,A23,A12,GOBOARD5:2 .= |[r1,s2]| by A1,A2,A23,A12,GOBOARD5:1; A25: r3*(s2+s2)+s3*(s2+s2) = s2; r3*(s1+s2) < r3*(s2+s2) by A16,A18,XREAL_1:68; then A26: r3*(s1+s2)+s3*(s1+s2) < s2 by A19,A25,XREAL_1:8; A27: i+1 <= len G by A2,NAT_1:13; i < i+1 by XREAL_1:29; then A28: r1 < r2 by A1,A27,A20,GOBOARD5:3; then r1+r1 < r2+r2 by XREAL_1:8; then A29: s3*(r1+r1) <= s3*(r2+r2) by A7,XREAL_1:64; r1+r2 < r2+r2 by A28,XREAL_1:6; then A30: r3*(r1+r2) < r3*(r2+r2) by A16,XREAL_1:68; r3*(r2+r2)+s3*(r2+r2) = r2; then A31: r3*(r1+r2)+s3*(r1+r1) < r2 by A30,A29,XREAL_1:8; A32: G*(i,j) = |[G*(i,j)`1,G*(i,j)`2]| by EUCLID:53 .= |[r1,G*(i,j)`2]| by A1,A2,A3,A4,GOBOARD5:2 .= |[r1,s1]| by A1,A2,A3,A4,GOBOARD5:1; A33: G*(i+1,j+1) = |[G*(i+1,j+1)`1,G*(i+1,j+1)`2]| by EUCLID:53 .= |[r2,G*(i+1,j+1)`2]| by A23,A12,A21,A27,GOBOARD5:2 .= |[r2,s2]| by A23,A12,A21,A27,GOBOARD5:1; A34: r3*(r1+r1)+s3*(r1+r1) = r1; r1+r1 < r1+r2 by A28,XREAL_1:6; then r3*(r1+r1) < r3*(r1+r2) by A16,XREAL_1:68; then A35: r1 < r3*(r1+r2)+s3*(r1+r1) by A34,XREAL_1:6; p = r3*(G*(i,j)+G*(i+1,j+1))+r*(1/2*(G*(i,j)+G* (i,j+1))) by A6, RLVECT_1:def 7 .= r3*(G*(i,j)+G*(i+1,j+1))+s3*(G*(i,j)+G*(i,j+1)) by RLVECT_1:def 7 .= r3*|[r1+r2,s1+s2]|+s3*(G*(i,j)+G*(i,j+1)) by A32,A33,EUCLID:56 .= r3*|[r1+r2,s1+s2]|+s3*|[r1+r1,s1+s2]| by A32,A24,EUCLID:56 .= |[r3*(r1+r2),r3*(s1+s2)]|+s3*|[r1+r1,s1+s2]| by EUCLID:58 .= |[r3*(r1+r2),r3*(s1+s2)]|+|[s3*(r1+r1),s3*(s1+s2)]| by EUCLID:58 .= |[r3*(r1+r2)+s3*(r1+r1),r3*(s1+s2)+s3*(s1+s2)]| by EUCLID:56; hence p in Int cell(G,i,j) by A35,A31,A17,A26,A22; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th41: 1 <= i & i < len G & 1 <= j & j < width G implies LSeg(1/2*(G*(i ,j)+G*(i+1,j+1)),1/2*(G*(i,j+1)+G*(i+1,j+1))) c= Int cell(G,i,j) \/ { 1/2*(G*(i ,j+1)+G*(i+1,j+1)) } proof assume that A1: 1 <= i and A2: i < len G and A3: 1 <= j and A4: j < width G; let x be object; assume A5: x in LSeg(1/2*(G*(i,j)+G*(i+1,j+1)),1/2*(G*(i,j+1)+G*(i+1,j+1))); then reconsider p = x as Point of TOP-REAL 2; consider r such that A6: p = (1-r)*(1/2*(G*(i,j)+G*(i+1,j+1)))+r*(1/2*(G*(i,j+1)+G*(i+1,j+1)) ) and A7: 0<=r and A8: r<=1 by A5; now per cases by A8,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i,j+1)+G*(i+1,j+1))) by A6,RLVECT_1:10 .= 1*(1/2*(G*(i,j+1)+G*(i+1,j+1))) by RLVECT_1:4 .= 1/2*(G*(i,j+1)+G*(i+1,j+1)) by RLVECT_1:def 8; hence p in { 1/2*(G*(i,j+1)+G*(i+1,j+1)) } by TARSKI:def 1; end; case A9: r < 1; set r3 = (1-r)*(1/2), s3 = r*(1/2); set r1 = G*(i,1)`1, r2 = G*(i+1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2; A10: r3*(r1+r1)+s3*(r1+r1) = r1; 0 <> width G by MATRIX_0:def 10; then A11: 1 <= width G by NAT_1:14; A12: i+1 <= len G by A2,NAT_1:13; i < i+1 by XREAL_1:29; then A13: r1 < r2 by A1,A12,A11,GOBOARD5:3; then A14: r1+r1 < r1+r2 by XREAL_1:6; then A15: s3*(r1+r1) <= s3*(r1+r2) by A7,XREAL_1:64; 1 - r > 0 by A9,XREAL_1:50; then A16: r3 > (1/2)*0 by XREAL_1:68; then r3*(r1+r1) < r3*(r1+r2) by A14,XREAL_1:68; then A17: r1 < r3*(r1+r2)+s3*(r1+r2) by A15,A10,XREAL_1:8; 0 <> len G by MATRIX_0:def 10; then A18: 1 <= len G by NAT_1:14; A19: 1 <= i+1 by A1,NAT_1:13; r1+r2 < r2+r2 by A13,XREAL_1:8; then A20: s3*(r1+r2) <= s3*(r2+r2) by A7,XREAL_1:64; A21: j+1 <= width G by A4,NAT_1:13; r1+r2 < r2+r2 by A13,XREAL_1:6; then A22: r3*(r1+r2) < r3*(r2+r2) by A16,XREAL_1:68; r3*(r2+r2)+s3*(r2+r2) = r2; then A23: r3*(r1+r2)+s3*(r1+r2) < r2 by A22,A20,XREAL_1:8; A24: Int cell(G,i,j) = { |[r9,s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 < s2 } by A1,A2,A3,A4,Th26; A25: 1 <= j+1 by A3,NAT_1:13; j < j+1 by XREAL_1:29; then A26: s1 < s2 by A3,A21,A18,GOBOARD5:4; then A27: s1+s1 < s1+s2 by XREAL_1:6; A28: G*(i+1,j+1) = |[G*(i+1,j+1)`1,G*(i+1,j+1)`2]| by EUCLID:53 .= |[r2,G*(i+1,j+1)`2]| by A25,A21,A19,A12,GOBOARD5:2 .= |[r2,s2]| by A25,A21,A19,A12,GOBOARD5:1; s1+s2 < s2+s2 by A26,XREAL_1:6; then s1+s1 < s2+s2 by A27,XXREAL_0:2; then A29: s3*(s1+s1) <= s3*(s2+s2) by A7,XREAL_1:64; A30: G*(i,j) = |[G*(i,j)`1,G*(i,j)`2]| by EUCLID:53 .= |[r1,G*(i,j)`2]| by A1,A2,A3,A4,GOBOARD5:2 .= |[r1,s1]| by A1,A2,A3,A4,GOBOARD5:1; A31: r3*(s2+s2)+s3*(s2+s2) = s2; A32: G*(i,j+1) = |[G*(i,j+1)`1,G*(i,j+1)`2]| by EUCLID:53 .= |[r1,G*(i,j+1)`2]| by A1,A2,A25,A21,GOBOARD5:2 .= |[r1,s2]| by A1,A2,A25,A21,GOBOARD5:1; A33: r3*(s1+s1)+s3*(s1+s1) = s1; s1+s2 < s2+s2 by A26,XREAL_1:6; then r3*(s1+s2) < r3*(s2+s2) by A16,XREAL_1:68; then A34: r3*(s1+s2)+s3*(s2+s2) < s2 by A31,XREAL_1:8; r3*(s1+s1) < r3*(s1+s2) by A16,A27,XREAL_1:68; then A35: s1 < r3*(s1+s2)+s3*(s2+s2) by A29,A33,XREAL_1:8; p = r3*(G*(i,j)+G*(i+1,j+1))+r*(1/2*(G*(i,j+1)+G*(i+1,j+1))) by A6, RLVECT_1:def 7 .= r3*(G*(i,j)+G*(i+1,j+1))+s3*(G*(i,j+1)+G*(i+1,j+1)) by RLVECT_1:def 7 .= r3*|[r1+r2,s1+s2]|+s3*(G*(i,j+1)+G*(i+1,j+1)) by A30,A28,EUCLID:56 .= r3*|[r1+r2,s1+s2]|+s3*|[r1+r2,s2+s2]| by A28,A32,EUCLID:56 .= |[r3*(r1+r2),r3*(s1+s2)]|+s3*|[r1+r2,s2+s2]| by EUCLID:58 .= |[r3*(r1+r2),r3*(s1+s2)]|+|[s3*(r1+r2),s3*(s2+s2)]| by EUCLID:58 .= |[r3*(r1+r2)+s3*(r1+r2),r3*(s1+s2)+s3*(s2+s2)]| by EUCLID:56; hence p in Int cell(G,i,j) by A17,A23,A35,A34,A24; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th42: 1 <= i & i < len G & 1 <= j & j < width G implies LSeg(1/2*(G*(i ,j)+G*(i+1,j+1)),1/2*(G*(i+1,j)+G*(i+1,j+1))) c= Int cell(G,i,j) \/ { 1/2*(G*(i +1,j)+G*(i+1,j+1)) } proof assume that A1: 1 <= i and A2: i < len G and A3: 1 <= j and A4: j < width G; let x be object; assume A5: x in LSeg(1/2*(G*(i,j)+G*(i+1,j+1)),1/2*(G*(i+1,j)+G*(i+1,j+1))); then reconsider p = x as Point of TOP-REAL 2; consider r such that A6: p = (1-r)*(1/2*(G*(i,j)+G*(i+1,j+1)))+r*(1/2*(G*(i+1,j)+G*(i+1,j+1)) ) and A7: 0<=r and A8: r<=1 by A5; now per cases by A8,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i+1,j)+G*(i+1,j+1))) by A6,RLVECT_1:10 .= 1*(1/2*(G*(i+1,j)+G*(i+1,j+1))) by RLVECT_1:4 .= 1/2*(G*(i+1,j)+G*(i+1,j+1)) by RLVECT_1:def 8; hence p in { 1/2*(G*(i+1,j)+G*(i+1,j+1)) } by TARSKI:def 1; end; case A9: r < 1; set r3 = (1-r)*(1/2), s3 = r*(1/2); set r1 = G*(i,1)`1, r2 = G*(i+1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2; A10: r3*(r1+r1)+s3*(r1+r1) = r1; 0 <> width G by MATRIX_0:def 10; then A11: 1 <= width G by NAT_1:14; A12: i+1 <= len G by A2,NAT_1:13; i < i+1 by XREAL_1:29; then A13: r1 < r2 by A1,A12,A11,GOBOARD5:3; then A14: r1+r1 < r1+r2 by XREAL_1:6; r1+r2 < r2+r2 by A13,XREAL_1:6; then r1+r1 < r2+r2 by A14,XXREAL_0:2; then A15: s3*(r1+r1) <= s3*(r2+r2) by A7,XREAL_1:64; 1 - r > 0 by A9,XREAL_1:50; then A16: r3 > (1/2)*0 by XREAL_1:68; then r3*(r1+r1) < r3*(r1+r2) by A14,XREAL_1:68; then A17: r1 < r3*(r1+r2)+s3*(r2+r2) by A15,A10,XREAL_1:8; 0 <> len G by MATRIX_0:def 10; then A18: 1 <= len G by NAT_1:14; A19: 1 <= j+1 by A3,NAT_1:13; A20: Int cell(G,i,j) = { |[r9,s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 < s2 } by A1,A2,A3,A4,Th26; A21: r3*(s2+s2)+s3*(s2+s2) = s2; A22: G*(i,j) = |[G*(i,j)`1,G*(i,j)`2]| by EUCLID:53 .= |[r1,G*(i,j)`2]| by A1,A2,A3,A4,GOBOARD5:2 .= |[r1,s1]| by A1,A2,A3,A4,GOBOARD5:1; A23: r3*(s1+s1)+s3*(s1+s1) = s1; A24: 1 <= i+1 by A1,NAT_1:13; A25: G*(i+1,j) = |[G*(i+1,j)`1,G*(i+1,j)`2]| by EUCLID:53 .= |[r2,G*(i+1,j)`2]| by A3,A4,A24,A12,GOBOARD5:2 .= |[r2,s1]| by A3,A4,A24,A12,GOBOARD5:1; A26: r3*(r2+r2)+s3*(r2+r2) = r2; r1+r2 < r2+r2 by A13,XREAL_1:6; then r3*(r1+r2) < r3*(r2+r2) by A16,XREAL_1:68; then A27: r3*(r1+r2)+s3*(r2+r2) < r2 by A26,XREAL_1:8; A28: j+1 <= width G by A4,NAT_1:13; A29: G*(i+1,j+1) = |[G*(i+1,j+1)`1,G*(i+1,j+1)`2]| by EUCLID:53 .= |[r2,G*(i+1,j+1)`2]| by A19,A28,A24,A12,GOBOARD5:2 .= |[r2,s2]| by A19,A28,A24,A12,GOBOARD5:1; j < j+1 by XREAL_1:29; then A30: s1 < s2 by A3,A28,A18,GOBOARD5:4; then A31: s1+s1 < s1+s2 by XREAL_1:6; then A32: s3*(s1+s1) <= s3*(s1+s2) by A7,XREAL_1:64; r3*(s1+s1) < r3*(s1+s2) by A16,A31,XREAL_1:68; then A33: s1 < r3*(s1+s2)+s3*(s1+s2) by A32,A23,XREAL_1:8; A34: s1+s2 < s2+s2 by A30,XREAL_1:6; then A35: s3*(s1+s2) <= s3*(s2+s2) by A7,XREAL_1:64; r3*(s1+s2) < r3*(s2+s2) by A16,A34,XREAL_1:68; then A36: r3*(s1+s2)+s3*(s1+s2) < s2 by A35,A21,XREAL_1:8; p = r3*(G*(i,j)+G*(i+1,j+1))+r*(1/2*(G*(i+1,j)+G*(i+1,j+1))) by A6, RLVECT_1:def 7 .= r3*(G*(i,j)+G*(i+1,j+1))+s3*(G*(i+1,j)+G*(i+1,j+1)) by RLVECT_1:def 7 .= r3*|[r1+r2,s1+s2]|+s3*(G*(i+1,j)+G*(i+1,j+1)) by A22,A29,EUCLID:56 .= r3*|[r1+r2,s1+s2]|+s3*|[r2+r2,s1+s2]| by A29,A25,EUCLID:56 .= |[r3*(r1+r2),r3*(s1+s2)]|+s3*|[r2+r2,s1+s2]| by EUCLID:58 .= |[r3*(r1+r2),r3*(s1+s2)]|+|[s3*(r2+r2),s3*(s1+s2)]| by EUCLID:58 .= |[r3*(r1+r2)+s3*(r2+r2),r3*(s1+s2)+s3*(s1+s2)]| by EUCLID:56; hence p in Int cell(G,i,j) by A17,A27,A33,A36,A20; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th43: 1 <= i & i < len G & 1 <= j & j < width G implies LSeg(1/2*(G*(i ,j)+G*(i+1,j+1)),1/2*(G*(i,j)+G*(i+1,j))) c= Int cell(G,i,j) \/ { 1/2*(G*(i,j)+ G*(i+1,j)) } proof assume that A1: 1 <= i and A2: i < len G and A3: 1 <= j and A4: j < width G; let x be object; assume A5: x in LSeg(1/2*(G*(i,j)+G*(i+1,j+1)),1/2*(G*(i,j)+G*(i+1,j))); then reconsider p = x as Point of TOP-REAL 2; consider r such that A6: p = (1-r)*(1/2*(G*(i,j)+G*(i+1,j+1)))+r*(1/2*(G*(i,j)+G*(i+1,j))) and A7: 0<=r and A8: r<=1 by A5; now per cases by A8,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i,j)+G*(i+1,j))) by A6,RLVECT_1:10 .= 1*(1/2*(G*(i,j)+G*(i+1,j))) by RLVECT_1:4 .= 1/2*(G*(i,j)+G*(i+1,j)) by RLVECT_1:def 8; hence p in { 1/2*(G*(i,j)+G*(i+1,j)) } by TARSKI:def 1; end; case A9: r < 1; set r3 = (1-r)*(1/2), s3 = r*(1/2); set r1 = G*(i,1)`1, r2 = G*(i+1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2; A10: r3*(r1+r1)+s3*(r1+r1) = r1; 0 <> width G by MATRIX_0:def 10; then A11: 1 <= width G by NAT_1:14; A12: i+1 <= len G by A2,NAT_1:13; i < i+1 by XREAL_1:29; then A13: r1 < r2 by A1,A12,A11,GOBOARD5:3; then A14: r1+r1 < r1+r2 by XREAL_1:6; then A15: s3*(r1+r1) <= s3*(r1+r2) by A7,XREAL_1:64; r1+r2 < r2+r2 by A13,XREAL_1:8; then A16: s3*(r1+r2) <= s3*(r2+r2) by A7,XREAL_1:64; A17: 1 <= i+1 by A1,NAT_1:13; 1 - r > 0 by A9,XREAL_1:50; then A18: r3 > (1/2)*0 by XREAL_1:68; then r3*(r1+r1) < r3*(r1+r2) by A14,XREAL_1:68; then A19: r1 < r3*(r1+r2)+s3*(r1+r2) by A15,A10,XREAL_1:8; r1+r2 < r2+r2 by A13,XREAL_1:6; then A20: r3*(r1+r2) < r3*(r2+r2) by A18,XREAL_1:68; r3*(r2+r2)+s3*(r2+r2) = r2; then A21: r3*(r1+r2)+s3*(r1+r2) < r2 by A20,A16,XREAL_1:8; A22: Int cell(G,i,j) = { |[r9,s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 < s2 } by A1,A2,A3,A4,Th26; A23: j+1 <= width G by A4,NAT_1:13; A24: G*(i+1,j) = |[G*(i+1,j)`1,G*(i+1,j)`2]| by EUCLID:53 .= |[r2,G*(i+1,j)`2]| by A3,A4,A17,A12,GOBOARD5:2 .= |[r2,s1]| by A3,A4,A17,A12,GOBOARD5:1; A25: 1 <= j+1 by A3,NAT_1:13; 0 <> len G by MATRIX_0:def 10; then A26: 1 <= len G by NAT_1:14; A27: G*(i,j) = |[G*(i,j)`1,G*(i,j)`2]| by EUCLID:53 .= |[r1,G*(i,j)`2]| by A1,A2,A3,A4,GOBOARD5:2 .= |[r1,s1]| by A1,A2,A3,A4,GOBOARD5:1; j < j+1 by XREAL_1:29; then A28: s1 < s2 by A3,A23,A26,GOBOARD5:4; then s1+s2 < s2+s2 by XREAL_1:6; then A29: r3*(s1+s2) < r3*(s2+s2) by A18,XREAL_1:68; A30: G*(i+1,j+1) = |[G*(i+1,j+1)`1,G*(i+1,j+1)`2]| by EUCLID:53 .= |[r2,G*(i+1,j+1)`2]| by A25,A23,A17,A12,GOBOARD5:2 .= |[r2,s2]| by A25,A23,A17,A12,GOBOARD5:1; A31: r3*(s1+s1)+s3*(s1+s1) = s1; A32: s1+s1 < s1+s2 by A28,XREAL_1:6; then r3*(s1+s1) < r3*(s1+s2) by A18,XREAL_1:68; then A33: s1 < r3*(s1+s2)+s3*(s1+s1) by A31,XREAL_1:8; s1+s2 < s2+s2 by A28,XREAL_1:6; then s1+s1 < s2+s2 by A32,XXREAL_0:2; then A34: s3*(s1+s1) <= s3*(s2+s2) by A7,XREAL_1:64; r3*(s2+s2)+s3*(s2+s2) = s2; then A35: r3*(s1+s2)+s3*(s1+s1) < s2 by A29,A34,XREAL_1:8; p = r3*(G*(i,j)+G*(i+1,j+1))+r*(1/2*(G*(i,j)+G* (i+1,j))) by A6, RLVECT_1:def 7 .= r3*(G*(i,j)+G*(i+1,j+1))+s3*(G*(i,j)+G*(i+1,j)) by RLVECT_1:def 7 .= r3*|[r1+r2,s1+s2]|+s3*(G*(i,j)+G*(i+1,j)) by A27,A30,EUCLID:56 .= r3*|[r1+r2,s1+s2]|+s3*|[r1+r2,s1+s1]| by A27,A24,EUCLID:56 .= |[r3*(r1+r2),r3*(s1+s2)]|+s3*|[r1+r2,s1+s1]| by EUCLID:58 .= |[r3*(r1+r2),r3*(s1+s2)]|+|[s3*(r1+r2),s3*(s1+s1)]| by EUCLID:58 .= |[r3*(r1+r2)+s3*(r1+r2),r3*(s1+s2)+s3*(s1+s1)]| by EUCLID:56; hence p in Int cell(G,i,j) by A19,A21,A33,A35,A22; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th44: 1 <= j & j < width G implies LSeg(1/2*(G*(1,j)+G*(1,j+1)) - |[1, 0]|,1/2*(G*(1,j)+G*(1,j+1))) c= Int cell(G,0,j) \/ { 1/2*(G*(1,j)+G*(1,j+1)) } proof assume that A1: 1 <= j and A2: j < width G; let x be object; assume A3: x in LSeg(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|,1/2*(G*(1,j)+G*(1,j+1))); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|)+r*(1/2*(G*(1,j)+G* (1,j+ 1))) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(1/2*(G*(1,j)+G*(1,j+1))) by A4,RLVECT_1:10 .= 1*(1/2*(G*(1,j)+G*(1,j+1))) by RLVECT_1:4 .= 1/2*(G*(1,j)+G*(1,j+1)) by RLVECT_1:def 8; hence p in { 1/2*(G*(1,j)+G*(1,j+1)) } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2), s3 = r*(1/2); set r2 = G*(1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2; A8: r3*(s1+s1)+s3*(s1+s1) = s1; A9: j+1 <= width G by A2,NAT_1:13; 0 <> len G by MATRIX_0:def 10; then A10: 1 <= len G by NAT_1:14; j < j+1 by XREAL_1:29; then A11: s1 < s2 by A1,A9,A10,GOBOARD5:4; then A12: s1+s1 < s1+s2 by XREAL_1:6; then A13: s3*(s1+s1) <= s3*(s1+s2) by A5,XREAL_1:64; A14: 1 - r > 0 by A7,XREAL_1:50; then A15: r3 > (1/2)*0 by XREAL_1:68; then r3*(s1+s1) < r3*(s1+s2) by A12,XREAL_1:68; then A16: s1 < r3*(s1+s2)+s3*(s1+s2) by A13,A8,XREAL_1:8; r2 < r2+(1-r) by A14,XREAL_1:29; then A17: r2-(1-r) < r2 by XREAL_1:19; A18: 1 <= j+1 by A1,NAT_1:13; A19: G*(1,j+1) = |[G*(1,j+1)`1,G*(1,j+1)`2]| by EUCLID:53 .= |[r2,s2]| by A18,A9,A10,GOBOARD5:2; A20: s1+s2 < s2+s2 by A11,XREAL_1:6; then A21: s3*(s1+s2) <= s3*(s2+s2) by A5,XREAL_1:64; A22: Int cell(G,0,j) = { |[r9,s9]| : r9 < G*(1,1)`1 & G*(1,j)`2 < s9 & s9 < G*(1,j+1)`2 } by A1,A2,Th20; A23: r3*(s2+s2)+s3*(s2+s2) = s2; r3*(s1+s2) < r3*(s2+s2) by A15,A20,XREAL_1:68; then A24: r3*(s1+s2)+s3*(s1+s2) < s2 by A21,A23,XREAL_1:8; A25: G*(1,j) = |[G*(1,j)`1,G*(1,j)`2]| by EUCLID:53 .= |[r2,s1]| by A1,A2,A10,GOBOARD5:2; p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1)))-(1-r)*|[1,0]| +r*(1/2*(G*(1,j)+ G*(1,j+1))) by A4,RLVECT_1:34 .= r3*(G*(1,j)+G*(1,j+1))-(1-r)*|[1,0]|+r*(1/2*(G*(1,j)+G*(1,j+1))) by RLVECT_1:def 7 .= r3*(G*(1,j)+G*(1,j+1))-|[(1-r)*1,(1-r)*0]|+r*(1/2*(G*(1,j)+G* (1, j+1))) by EUCLID:58 .= r3*(G*(1,j)+G*(1,j+1))-|[1-r,0]|+s3*(G*(1,j)+G* (1,j+1)) by RLVECT_1:def 7 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+s3*(G*(1,j)+G*(1,j+1)) by A19,A25, EUCLID:56 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+s3*|[r2+r2,s1+s2]| by A19,A25,EUCLID:56 .= |[r3*(r2+r2),r3*(s1+s2)]|-|[1-r,0]|+s3*|[r2+r2,s1+s2]| by EUCLID:58 .= |[r3*(r2+r2),r3*(s1+s2)]|-|[1-r,0]|+|[s3*(r2+r2),s3*(s1+s2)]| by EUCLID:58 .= |[r3*(r2+r2)-(1-r),r3*(s1+s2)-0]|+|[s3*(r2+r2),s3*(s1+s2)]| by EUCLID:62 .= |[r3*(r2+r2)-(1-r)+s3*(r2+r2),r3*(s1+s2)+s3*(s1+s2)]| by EUCLID:56; hence p in Int cell(G,0,j) by A17,A16,A24,A22; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th45: 1 <= j & j < width G implies LSeg(1/2*(G*(len G,j)+G*(len G,j+1) ) + |[1,0]|,1/2*(G*(len G,j)+G*(len G,j+1))) c= Int cell(G,len G,j) \/ { 1/2*(G *(len G,j)+G*(len G,j+1)) } proof assume that A1: 1 <= j and A2: j < width G; let x be object; assume A3: x in LSeg(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|, 1/2*(G*(len G,j)+ G*(len G,j+1))); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|) +r*(1/2*(G*(len G,j)+G*(len G,j+1))) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(1/2*(G*(len G,j)+G*(len G,j+1))) by A4, RLVECT_1:10 .= 1*(1/2*(G*(len G,j)+G*(len G,j+1))) by RLVECT_1:4 .= 1/2*(G*(len G,j)+G*(len G,j+1)) by RLVECT_1:def 8; hence p in { 1/2*(G*(len G,j)+G*(len G,j+1)) } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2), s3 = r*(1/2); set r2 = G*(len G,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2; A8: r3*(s1+s1)+s3*(s1+s1) = s1; A9: j+1 <= width G by A2,NAT_1:13; 0 <> len G by MATRIX_0:def 10; then A10: 1 <= len G by NAT_1:14; j < j+1 by XREAL_1:29; then A11: s1 < s2 by A1,A9,A10,GOBOARD5:4; then A12: s1+s1 < s1+s2 by XREAL_1:6; then A13: s3*(s1+s1) <= s3*(s1+s2) by A5,XREAL_1:64; A14: 1 - r > 0 by A7,XREAL_1:50; then A15: r3 > (1/2)*0 by XREAL_1:68; then r3*(s1+s1) < r3*(s1+s2) by A12,XREAL_1:68; then A16: s1 < r3*(s1+s2)+s3*(s1+s2) by A13,A8,XREAL_1:8; A17: r2+(1-r) > r2 by A14,XREAL_1:29; A18: 1 <= j+1 by A1,NAT_1:13; A19: s1+s2 < s2+s2 by A11,XREAL_1:6; then A20: s3*(s1+s2) <= s3*(s2+s2) by A5,XREAL_1:64; A21: Int cell(G,len G,j) = { |[r9,s9]| : G*(len G,1)`1 < r9 & G*(1,j)`2 < s9 & s9 < G*(1,j+1)`2 } by A1,A2,Th23; A22: r3*(s2+s2)+s3*(s2+s2) = s2; r3*(s1+s2) < r3*(s2+s2) by A15,A19,XREAL_1:68; then A23: r3*(s1+s2)+s3*(s1+s2) < s2 by A20,A22,XREAL_1:8; A24: G*(len G,j) = |[G*(len G,j)`1,G*(len G,j)`2]| by EUCLID:53 .= |[r2,G*(len G,j)`2]| by A1,A2,A10,GOBOARD5:2 .= |[r2,s1]| by A1,A2,A10,GOBOARD5:1; A25: G*(len G,j+1) = |[G*(len G,j+1)`1,G*(len G,j+1)`2]| by EUCLID:53 .= |[r2,G*(len G,j+1)`2]| by A18,A9,A10,GOBOARD5:2 .= |[r2,s2]| by A18,A9,A10,GOBOARD5:1; p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1)))+(1-r)*|[1,0]| +r*(1/2*( G*(len G,j)+G*(len G,j+1))) by A4,RLVECT_1:def 5 .= r3*(G*(len G,j)+G*(len G,j+1))+(1-r)*|[1,0]|+ r*(1/2*(G*(len G,j) +G*(len G,j+1))) by RLVECT_1:def 7 .= r3*(G*(len G,j)+G*(len G,j+1))+|[(1-r)*1,(1-r)*0]|+ r*(1/2*(G*( len G,j)+G*(len G,j+1))) by EUCLID:58 .= r3*(G*(len G,j)+G*(len G,j+1))+|[1-r,0]|+ s3*(G*(len G,j)+G*(len G,j+1)) by RLVECT_1:def 7 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+s3*(G*(len G,j)+G*(len G,j+1)) by A25 ,A24,EUCLID:56 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+s3*|[r2+r2,s1+s2]| by A25,A24,EUCLID:56 .= |[r3*(r2+r2),r3*(s1+s2)]|+|[1-r,0]|+s3*|[r2+r2,s1+s2]| by EUCLID:58 .= |[r3*(r2+r2),r3*(s1+s2)]|+|[1-r,0]|+|[s3*(r2+r2),s3*(s1+s2)]| by EUCLID:58 .= |[r3*(r2+r2)+(1-r),r3*(s1+s2)+0]|+|[s3*(r2+r2),s3*(s1+s2)]| by EUCLID:56 .= |[r3*(r2+r2)+(1-r)+s3*(r2+r2),r3*(s1+s2)+s3*(s1+s2)]| by EUCLID:56; hence p in Int cell(G,len G,j) by A17,A16,A23,A21; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th46: 1 <= i & i < len G implies LSeg(1/2*(G*(i,1)+G*(i+1,1)) - |[0,1 ]|,1/2*(G*(i,1)+G*(i+1,1))) c= Int cell(G,i,0) \/ { 1/2*(G*(i,1)+G*(i+1,1)) } proof assume that A1: 1 <= i and A2: i < len G; let x be object; assume A3: x in LSeg(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|,1/2*(G*(i,1)+G*(i+1,1))); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|)+r*(1/2*(G*(i,1)+G* (i+1, 1))) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i,1)+G*(i+1,1))) by A4,RLVECT_1:10 .= 1*(1/2*(G*(i,1)+G*(i+1,1))) by RLVECT_1:4 .= 1/2*(G*(i,1)+G*(i+1,1)) by RLVECT_1:def 8; hence p in { 1/2*(G*(i,1)+G*(i+1,1)) } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2), s3 = r*(1/2); set s2 = G*(1,1)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1; A8: r3*(r1+r1)+s3*(r1+r1) = r1; A9: i+1 <= len G by A2,NAT_1:13; 0 <> width G by MATRIX_0:def 10; then A10: 1 <= width G by NAT_1:14; i < i+1 by XREAL_1:29; then A11: r1 < r2 by A1,A9,A10,GOBOARD5:3; then A12: r1+r1 < r1+r2 by XREAL_1:6; then A13: s3*(r1+r1) <= s3*(r1+r2) by A5,XREAL_1:64; A14: 1 - r > 0 by A7,XREAL_1:50; then A15: r3 > (1/2)*0 by XREAL_1:68; then r3*(r1+r1) < r3*(r1+r2) by A12,XREAL_1:68; then A16: r1 < r3*(r1+r2)+s3*(r1+r2) by A13,A8,XREAL_1:8; s2 < s2+(1-r) by A14,XREAL_1:29; then A17: s2-(1-r) < s2 by XREAL_1:19; A18: 1 <= i+1 by A1,NAT_1:13; A19: G*(i+1,1) = |[G*(i+1,1)`1,G*(i+1,1)`2]| by EUCLID:53 .= |[r2,s2]| by A18,A9,A10,GOBOARD5:1; A20: r1+r2 < r2+r2 by A11,XREAL_1:6; then A21: s3*(r1+r2) <= s3*(r2+r2) by A5,XREAL_1:64; A22: Int cell(G,i,0) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1,1)`1 & s9 < G*(1,1)`2 } by A1,A2,Th24; A23: r3*(r2+r2)+s3*(r2+r2) = r2; r3*(r1+r2) < r3*(r2+r2) by A15,A20,XREAL_1:68; then A24: r3*(r1+r2)+s3*(r1+r2) < r2 by A21,A23,XREAL_1:8; A25: G*(i,1) = |[G*(i,1)`1,G*(i,1)`2]| by EUCLID:53 .= |[r1,s2]| by A1,A2,A10,GOBOARD5:1; p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1)))-(1-r)*|[0,1]| +r*(1/2*(G*(i,1)+ G*(i+1,1))) by A4,RLVECT_1:34 .= r3*(G*(i,1)+G*(i+1,1))-(1-r)*|[0,1]|+r*(1/2*(G*(i,1)+G*(i+1,1))) by RLVECT_1:def 7 .= r3*(G*(i,1)+G*(i+1,1))-|[(1-r)*0,(1-r)*1]|+r*(1/2*(G*(i,1)+G* (i+ 1,1))) by EUCLID:58 .= r3*(G*(i,1)+G*(i+1,1))-|[0,1-r]|+s3*(G*(i,1)+G* (i+1,1)) by RLVECT_1:def 7 .= r3*|[r1+r2,s2+s2]|-|[0,1-r]|+s3*(G*(i,1)+G*(i+1,1)) by A19,A25, EUCLID:56 .= r3*|[r1+r2,s2+s2]|-|[0,1-r]|+s3*|[r1+r2,s2+s2]| by A19,A25,EUCLID:56 .= |[r3*(r1+r2),r3*(s2+s2)]|-|[0,1-r]|+s3*|[r1+r2,s2+s2]| by EUCLID:58 .= |[r3*(r1+r2),r3*(s2+s2)]|-|[0,1-r]|+|[s3*(r1+r2),s3*(s2+s2)]| by EUCLID:58 .= |[r3*(r1+r2)-0,r3*(s2+s2)-(1-r)]|+|[s3*(r1+r2),s3*(s2+s2)]| by EUCLID:62 .= |[r3*(r1+r2)+s3*(r1+r2),r3*(s2+s2)-(1-r)+s3*(s2+s2)]| by EUCLID:56; hence p in Int cell(G,i,0) by A17,A16,A24,A22; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th47: 1 <= i & i < len G implies LSeg(1/2*(G*(i,width G)+G*(i+1,width G))+ |[0,1]|, 1/2*(G*(i,width G)+G*(i+1,width G))) c= Int cell(G,i,width G) \/ { 1/2*(G*(i,width G)+G*(i+1,width G)) } proof assume that A1: 1 <= i and A2: i < len G; let x be object; assume A3: x in LSeg(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|, 1/2*(G*(i, width G)+G*(i+1,width G))); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|) +r*(1/2*(G*( i,width G)+G*(i+1,width G))) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(1/2*(G*(i,width G)+G*(i+1,width G))) by A4, RLVECT_1:10 .= 1*(1/2*(G*(i,width G)+G*(i+1,width G))) by RLVECT_1:4 .= 1/2*(G*(i,width G)+G*(i+1,width G)) by RLVECT_1:def 8; hence p in { 1/2*(G*(i,width G)+G*(i+1,width G)) } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2), s3 = r*(1/2); set s2 = G*(1,width G)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1; A8: r3*(r1+r1)+s3*(r1+r1) = r1; A9: i+1 <= len G by A2,NAT_1:13; 0 <> width G by MATRIX_0:def 10; then A10: 1 <= width G by NAT_1:14; i < i+1 by XREAL_1:29; then A11: r1 < r2 by A1,A9,A10,GOBOARD5:3; then A12: r1+r1 < r1+r2 by XREAL_1:6; then A13: s3*(r1+r1) <= s3*(r1+r2) by A5,XREAL_1:64; A14: 1 - r > 0 by A7,XREAL_1:50; then A15: r3 > (1/2)*0 by XREAL_1:68; then r3*(r1+r1) < r3*(r1+r2) by A12,XREAL_1:68; then A16: r1 < r3*(r1+r2)+s3*(r1+r2) by A13,A8,XREAL_1:8; A17: s2+(1-r) > s2 by A14,XREAL_1:29; A18: 1 <= i+1 by A1,NAT_1:13; A19: r1+r2 < r2+r2 by A11,XREAL_1:6; then A20: s3*(r1+r2) <= s3*(r2+r2) by A5,XREAL_1:64; A21: Int cell(G,i,width G) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1, 1)`1 & G* (1,width G)`2 < s9 } by A1,A2,Th25; A22: r3*(r2+r2)+s3*(r2+r2) = r2; r3*(r1+r2) < r3*(r2+r2) by A15,A19,XREAL_1:68; then A23: r3*(r1+r2)+s3*(r1+r2) < r2 by A20,A22,XREAL_1:8; A24: G*(i,width G) = |[G*(i,width G)`1,G*(i,width G)`2]| by EUCLID:53 .= |[G*(i,width G)`1,s2]| by A1,A2,A10,GOBOARD5:1 .= |[r1,s2]| by A1,A2,A10,GOBOARD5:2; A25: G*(i+1,width G) = |[G*(i+1,width G)`1,G*(i+1,width G)`2]| by EUCLID:53 .= |[G*(i+1,width G)`1,s2]| by A18,A9,A10,GOBOARD5:1 .= |[r2,s2]| by A18,A9,A10,GOBOARD5:2; p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G)))+(1-r)*|[0,1]| +r*(1 /2*(G*(i,width G)+G* (i+1,width G))) by A4,RLVECT_1:def 5 .= r3*(G*(i,width G)+G*(i+1,width G))+(1-r)*|[0,1]|+ r*(1/2*(G*(i, width G)+G*(i+1,width G))) by RLVECT_1:def 7 .= r3*(G*(i,width G)+G*(i+1,width G))+|[(1-r)*0,(1-r)*1]|+ r*(1/2*(G *(i,width G)+G*(i+1,width G))) by EUCLID:58 .= r3*(G*(i,width G)+G*(i+1,width G))+|[0,1-r]|+ s3*(G*(i,width G)+G *(i+1,width G)) by RLVECT_1:def 7 .= r3*|[r1+r2,s2+s2]|+|[0,1-r]|+s3*(G*(i,width G)+G*(i+1,width G)) by A25,A24,EUCLID:56 .= r3*|[r1+r2,s2+s2]|+|[0,1-r]|+s3*|[r1+r2,s2+s2]| by A25,A24,EUCLID:56 .= |[r3*(r1+r2),r3*(s2+s2)]|+|[0,1-r]|+s3*|[r1+r2,s2+s2]| by EUCLID:58 .= |[r3*(r1+r2),r3*(s2+s2)]|+|[0,1-r]|+|[s3*(r1+r2),s3*(s2+s2)]| by EUCLID:58 .= |[r3*(r1+r2)+0,r3*(s2+s2)+(1-r)]|+|[s3*(r1+r2),s3*(s2+s2)]| by EUCLID:56 .= |[r3*(r1+r2)+s3*(r1+r2),r3*(s2+s2)+(1-r)+s3*(s2+s2)]| by EUCLID:56; hence p in Int cell(G,i,width G) by A17,A16,A23,A21; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th48: 1 <= j & j < width G implies LSeg(1/2*(G*(1,j)+G*(1,j+1)) - |[1, 0]|,G*(1,j) - |[1,0]|) c= Int cell(G,0,j) \/ { G*(1,j) - |[1,0]| } proof assume that A1: 1 <= j and A2: j < width G; let x be object; assume A3: x in LSeg(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|,G*(1,j) - |[1,0]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|)+r*(G*(1,j) - |[1,0]|) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(1,j) - |[1,0]|) by A4,RLVECT_1:10 .= 1*(G*(1,j) - |[1,0]|) by RLVECT_1:4 .= (G*(1,j) - |[1,0]|) by RLVECT_1:def 8; hence p in { G*(1,j) - |[1,0]| } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2); 1 - r > 0 by A7,XREAL_1:50; then A8: r3 > (1/2)*0 by XREAL_1:68; set r2 = G*(1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2; A9: r3*(s1+s1)+r*s1 = s1; A10: j+1 <= width G by A2,NAT_1:13; r2 < r2+1 by XREAL_1:29; then A11: r2-1 < r2 by XREAL_1:19; A12: Int cell(G,0,j) = { |[r9,s9]| : r9 < G*(1,1)`1 & G*(1,j)`2 < s9 & s9 < G*(1,j+1)`2 } by A1,A2,Th20; 0 <> len G by MATRIX_0:def 10; then A13: 1 <= len G by NAT_1:14; j < j+1 by XREAL_1:29; then A14: s1 < s2 by A1,A10,A13,GOBOARD5:4; then s1+s2 < s2+s2 by XREAL_1:6; then A15: r3*(s1+s2) < r3*(s2+s2) by A8,XREAL_1:68; s1+s1 < s1+s2 by A14,XREAL_1:6; then r3*(s1+s1) < r3*(s1+s2) by A8,XREAL_1:68; then A16: s1 < r3*(s1+s2)+r*s1 by A9,XREAL_1:6; A17: r3*(s2+s2)+r*s2 = s2; r*s1 <= r*s2 by A5,A14,XREAL_1:64; then A18: r3*(s1+s2)+r*s1 < s2 by A15,A17,XREAL_1:8; A19: G*(1,j) = |[G*(1,j)`1,G*(1,j)`2]| by EUCLID:53 .= |[r2,s1]| by A1,A2,A13,GOBOARD5:2; A20: 1 <= j+1 by A1,NAT_1:13; A21: G*(1,j+1) = |[G*(1,j+1)`1,G*(1,j+1)`2]| by EUCLID:53 .= |[r2,s2]| by A20,A10,A13,GOBOARD5:2; p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1)))-(1-r)*|[1,0]| +r*(G*(1,j) - |[1 ,0]|) by A4,RLVECT_1:34 .= r3*(G*(1,j)+G*(1,j+1))-(1-r)*|[1,0]|+r*(G* (1,j) - |[1,0]|) by RLVECT_1:def 7 .= r3*(G*(1,j)+G*(1,j+1))-|[(1-r)*1,(1-r)*0]|+r*(G*(1,j) - |[1,0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+r*(|[r2,s1]| - |[1,0]|) by A21,A19, EUCLID:56 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(r*|[r2,s1]| - r*|[1,0]|) by RLVECT_1:34 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(|[r*r2,r*s1]| - r*|[1,0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(|[r*r2,r*s1]| - |[r*1,r*0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+|[r*r2-r,r*s1-0]| by EUCLID:62 .= |[r3*(r2+r2),r3*(s1+s2)]|-|[1-r,0]|+|[r*r2-r,r*s1-0]| by EUCLID:58 .= |[r3*(r2+r2)-(1-r),r3*(s1+s2)-0]|+|[r*r2-r,r*s1-0]| by EUCLID:62 .= |[r3*(r2+r2)-(1-r)+(r*r2-r),r3*(s1+s2)+r*s1]| by EUCLID:56; hence p in Int cell(G,0,j) by A11,A16,A18,A12; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th49: 1 <= j & j < width G implies LSeg(1/2*(G*(1,j)+G*(1,j+1)) - |[1, 0]|,G*(1,j+1) - |[1,0]|) c= Int cell(G,0,j) \/ { G*(1,j+1) - |[1,0]| } proof assume that A1: 1 <= j and A2: j < width G; let x be object; assume A3: x in LSeg(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|,G*(1,j+1) - |[1,0]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|)+r*(G*(1,j+1) - |[1,0]|) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(1,j+1) - |[1,0]|) by A4,RLVECT_1:10 .= 1*(G*(1,j+1) - |[1,0]|) by RLVECT_1:4 .= (G*(1,j+1) - |[1,0]|) by RLVECT_1:def 8; hence p in { G*(1,j+1) - |[1,0]| } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2); 1 - r > 0 by A7,XREAL_1:50; then A8: r3 > (1/2)*0 by XREAL_1:68; set r2 = G*(1,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2; A9: r3*(s1+s1)+r*s1 = s1; A10: j+1 <= width G by A2,NAT_1:13; 0 <> len G by MATRIX_0:def 10; then A11: 1 <= len G by NAT_1:14; j < j+1 by XREAL_1:29; then A12: s1 < s2 by A1,A10,A11,GOBOARD5:4; then s1+s1 < s1+s2 by XREAL_1:6; then A13: r3*(s1+s1) < r3*(s1+s2) by A8,XREAL_1:68; r*s1 <= r*s2 by A5,A12,XREAL_1:64; then A14: s1 < r3*(s1+s2)+r*s2 by A13,A9,XREAL_1:8; A15: 1 <= j+1 by A1,NAT_1:13; A16: G*(1,j) = |[G*(1,j)`1,G*(1,j)`2]| by EUCLID:53 .= |[r2,s1]| by A1,A2,A11,GOBOARD5:2; r2 < r2+1 by XREAL_1:29; then A17: r2-1 < r2 by XREAL_1:19; A18: r3*(s2+s2)+r*s2 = s2; s1+s2 < s2+s2 by A12,XREAL_1:6; then r3*(s1+s2) < r3*(s2+s2) by A8,XREAL_1:68; then A19: r3*(s1+s2)+r*s2 < s2 by A18,XREAL_1:8; A20: Int cell(G,0,j) = { |[r9,s9]| : r9 < G*(1,1)`1 & G*(1,j)`2 < s9 & s9 < G*(1,j+1)`2 } by A1,A2,Th20; A21: G*(1,j+1) = |[G*(1,j+1)`1,G*(1,j+1)`2]| by EUCLID:53 .= |[r2,s2]| by A15,A10,A11,GOBOARD5:2; p = (1-r)*(1/2*(G*(1,j)+G*(1,j+1)))-(1-r)*|[1,0]| +r*(G*(1,j+1) - |[1,0]|) by A4,RLVECT_1:34 .= r3*(G*(1,j)+G*(1,j+1))-(1-r)*|[1,0]|+r*(G*(1,j+1) - |[1,0]|) by RLVECT_1:def 7 .= r3*(G*(1,j)+G*(1,j+1))-|[(1-r)*1,(1-r)*0]|+r*(G*(1,j+1) - |[1,0]| ) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+r*(|[r2,s2]| - |[1,0]|) by A21,A16, EUCLID:56 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(r*|[r2,s2]| - r*|[1,0]|) by RLVECT_1:34 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(|[r*r2,r*s2]| - r*|[1,0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+(|[r*r2,r*s2]| - |[r*1,r*0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|-|[1-r,0]|+|[r*r2-r,r*s2-0]| by EUCLID:62 .= |[r3*(r2+r2),r3*(s1+s2)]|-|[1-r,0]|+|[r*r2-r,r*s2-0]| by EUCLID:58 .= |[r3*(r2+r2)-(1-r),r3*(s1+s2)-0]|+|[r*r2-r,r*s2-0]| by EUCLID:62 .= |[r3*(r2+r2)-(1-r)+(r*r2-r),r3*(s1+s2)+r*s2]| by EUCLID:56; hence p in Int cell(G,0,j) by A17,A14,A19,A20; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th50: 1 <= j & j < width G implies LSeg(1/2*(G*(len G,j)+G*(len G,j+1) ) + |[1,0]|,G*(len G,j) + |[1,0]|) c= Int cell(G,len G,j) \/ { G*(len G,j) + |[ 1,0]| } proof assume that A1: 1 <= j and A2: j < width G; let x be object; assume A3: x in LSeg(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|,G* (len G,j) + |[1 ,0]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|)+ r*(G*(len G,j) + |[1,0]|) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(len G,j) + |[1,0]|) by A4,RLVECT_1:10 .= 1*(G*(len G,j) + |[1,0]|) by RLVECT_1:4 .= (G*(len G,j) + |[1,0]|) by RLVECT_1:def 8; hence p in { G*(len G,j) + |[1,0]| } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2); 1 - r > 0 by A7,XREAL_1:50; then A8: r3 > (1/2)*0 by XREAL_1:68; set r2 = G*(len G,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2; A9: r3*(s1+s1)+r*s1 = s1; A10: j+1 <= width G by A2,NAT_1:13; 0 <> len G by MATRIX_0:def 10; then A11: 1 <= len G by NAT_1:14; A12: G*(len G,j) = |[G*(len G,j)`1,G*(len G,j)`2]| by EUCLID:53 .= |[r2,G*(len G,j)`2]| by A1,A2,A11,GOBOARD5:2 .= |[r2,s1]| by A1,A2,A11,GOBOARD5:1; A13: 1 <= j+1 by A1,NAT_1:13; j < j+1 by XREAL_1:29; then A14: s1 < s2 by A1,A10,A11,GOBOARD5:4; then s1+s2 < s2+s2 by XREAL_1:6; then A15: r3*(s1+s2) < r3*(s2+s2) by A8,XREAL_1:68; s1+s1 < s1+s2 by A14,XREAL_1:6; then r3*(s1+s1) < r3*(s1+s2) by A8,XREAL_1:68; then A16: r2 < r2+1 & s1 < r3*(s1+s2)+r*s1 by A9,XREAL_1:6,29; A17: Int cell(G,len G,j) = { |[r9,s9]| : G*(len G,1)`1 < r9 & G*(1,j)`2 < s9 & s9 < G*(1,j+1)`2 } by A1,A2,Th23; A18: G*(len G,j+1) = |[G*(len G,j+1)`1,G*(len G,j+1)`2]| by EUCLID:53 .= |[r2,G*(len G,j+1)`2]| by A13,A10,A11,GOBOARD5:2 .= |[r2,s2]| by A13,A10,A11,GOBOARD5:1; A19: r3*(s2+s2)+r*s2 = s2; r*s1 <= r*s2 by A5,A14,XREAL_1:64; then A20: r3*(s1+s2)+r*s1 < s2 by A15,A19,XREAL_1:8; p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1)))+(1-r)*|[1,0]| +r*(G*( len G,j) + |[1,0]|) by A4,RLVECT_1:def 5 .= r3*(G*(len G,j)+G*(len G,j+1))+(1-r)*|[1,0]|+r*(G* (len G,j) + |[ 1,0]|) by RLVECT_1:def 7 .= r3*(G*(len G,j)+G*(len G,j+1))+|[(1-r)*1,(1-r)*0]|+ r*(G*(len G,j ) + |[1,0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+r*(|[r2,s1]| + |[1,0]|) by A18,A12, EUCLID:56 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(r*|[r2,s1]| + r*|[1,0]|) by RLVECT_1:def 5 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(|[r*r2,r*s1]| + r*|[1,0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(|[r*r2,r*s1]| + |[r*1,r*0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+|[r*r2+r,r*s1+0]| by EUCLID:56 .= |[r3*(r2+r2),r3*(s1+s2)]|+|[1-r,0]|+|[r*r2+r,r*s1+0]| by EUCLID:58 .= |[r3*(r2+r2)+(1-r),r3*(s1+s2)+0]|+|[r*r2+r,r*s1+0]| by EUCLID:56 .= |[r3*(r2+r2)+(1-r)+(r*r2+r),r3*(s1+s2)+r*s1]| by EUCLID:56; hence p in Int cell(G,len G,j) by A16,A20,A17; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th51: 1 <= j & j < width G implies LSeg(1/2*(G*(len G,j)+G*(len G,j+1) ) + |[1,0]|,G*(len G,j+1) + |[1,0]|) c= Int cell(G,len G,j) \/ { G*(len G,j+1) + |[1,0]| } proof assume that A1: 1 <= j and A2: j < width G; let x be object; assume A3: x in LSeg(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|,G* (len G,j+1) + |[1,0]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|)+ r*(G*(len G,j+1 ) + |[1,0]|) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(len G,j+1) + |[1,0]|) by A4,RLVECT_1:10 .= 1*(G*(len G,j+1) + |[1,0]|) by RLVECT_1:4 .= (G*(len G,j+1) + |[1,0]|) by RLVECT_1:def 8; hence p in { G*(len G,j+1) + |[1,0]| } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2); 1 - r > 0 by A7,XREAL_1:50; then A8: r3 > (1/2)*0 by XREAL_1:68; set r2 = G*(len G,1)`1, s1 = G*(1,j)`2, s2 = G*(1,j+1)`2; A9: r3*(s1+s1)+r*s1 = s1; A10: j+1 <= width G by A2,NAT_1:13; 0 <> len G by MATRIX_0:def 10; then A11: 1 <= len G by NAT_1:14; j < j+1 by XREAL_1:29; then A12: s1 < s2 by A1,A10,A11,GOBOARD5:4; then s1+s1 < s1+s2 by XREAL_1:6; then A13: r3*(s1+s1) < r3*(s1+s2) by A8,XREAL_1:68; A14: r3*(s2+s2)+r*s2 = s2; s1+s2 < s2+s2 by A12,XREAL_1:6; then r3*(s1+s2) < r3*(s2+s2) by A8,XREAL_1:68; then A15: r3*(s1+s2)+r*s2 < s2 by A14,XREAL_1:8; A16: G*(len G,j) = |[G*(len G,j)`1,G*(len G,j)`2]| by EUCLID:53 .= |[r2,G*(len G,j)`2]| by A1,A2,A11,GOBOARD5:2 .= |[r2,s1]| by A1,A2,A11,GOBOARD5:1; A17: 1 <= j+1 by A1,NAT_1:13; r*s1 <= r*s2 by A5,A12,XREAL_1:64; then A18: r2+1 > r2 & s1 < r3*(s1+s2)+r*s2 by A13,A9,XREAL_1:8,29; A19: Int cell(G,len G,j) = { |[r9,s9]| : G*(len G,1)`1 < r9 & G*(1,j)`2 < s9 & s9 < G*(1,j+1)`2 } by A1,A2,Th23; A20: G*(len G,j+1) = |[G*(len G,j+1)`1,G*(len G,j+1)`2]| by EUCLID:53 .= |[r2,G*(len G,j+1)`2]| by A17,A10,A11,GOBOARD5:2 .= |[r2,s2]| by A17,A10,A11,GOBOARD5:1; p = (1-r)*(1/2*(G*(len G,j)+G*(len G,j+1)))+(1-r)*|[1,0]| +r*(G*( len G,j+1) + |[1,0]|) by A4,RLVECT_1:def 5 .= r3*(G*(len G,j)+G*(len G,j+1))+(1-r)*|[1,0]|+ r*(G*(len G,j+1) + |[1,0]|) by RLVECT_1:def 7 .= r3*(G*(len G,j)+G*(len G,j+1))+|[(1-r)*1,(1-r)*0]|+ r*(G*(len G,j +1) + |[1,0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+r*(|[r2,s2]| + |[1,0]|) by A20,A16, EUCLID:56 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(r*|[r2,s2]| + r*|[1,0]|) by RLVECT_1:def 5 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(|[r*r2,r*s2]| + r*|[1,0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+(|[r*r2,r*s2]| + |[r*1,r*0]|) by EUCLID:58 .= r3*|[r2+r2,s1+s2]|+|[1-r,0]|+|[r*r2+r,r*s2+0]| by EUCLID:56 .= |[r3*(r2+r2),r3*(s1+s2)]|+|[1-r,0]|+|[r*r2+r,r*s2+0]| by EUCLID:58 .= |[r3*(r2+r2)+(1-r),r3*(s1+s2)+0]|+|[r*r2+r,r*s2+0]| by EUCLID:56 .= |[r3*(r2+r2)+(1-r)+(r*r2+r),r3*(s1+s2)+r*s2]| by EUCLID:56; hence p in Int cell(G,len G,j) by A18,A15,A19; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th52: 1 <= i & i < len G implies LSeg(1/2*(G*(i,1)+G*(i+1,1)) - |[0,1 ]|,G*(i,1) - |[0,1]|) c= Int cell(G,i,0) \/ { G*(i,1) - |[0,1]| } proof assume that A1: 1 <= i and A2: i < len G; let x be object; assume A3: x in LSeg(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|,G*(i,1) - |[0,1]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|)+r*(G*(i,1) - |[0,1]|) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(i,1) - |[0,1]|) by A4,RLVECT_1:10 .= 1*(G*(i,1) - |[0,1]|) by RLVECT_1:4 .= (G*(i,1) - |[0,1]|) by RLVECT_1:def 8; hence p in { G*(i,1) - |[0,1]| } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2); 1 - r > 0 by A7,XREAL_1:50; then A8: r3 > (1/2)*0 by XREAL_1:68; set s1 = G*(1,1)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1; A9: r3*(r1+r1)+r*r1 = r1; A10: i+1 <= len G by A2,NAT_1:13; s1 < s1+1 by XREAL_1:29; then A11: s1-1 < s1 by XREAL_1:19; A12: Int cell(G,i,0) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1,1)`1 & s9 < G*(1,1)`2 } by A1,A2,Th24; 0 <> width G by MATRIX_0:def 10; then A13: 1 <= width G by NAT_1:14; i < i+1 by XREAL_1:29; then A14: r1 < r2 by A1,A10,A13,GOBOARD5:3; then r1+r2 < r2+r2 by XREAL_1:6; then A15: r3*(r1+r2) < r3*(r2+r2) by A8,XREAL_1:68; r1+r1 < r1+r2 by A14,XREAL_1:6; then r3*(r1+r1) < r3*(r1+r2) by A8,XREAL_1:68; then A16: r1 < r3*(r1+r2)+r*r1 by A9,XREAL_1:6; A17: r3*(r2+r2)+r*r2 = r2; r*r1 <= r*r2 by A5,A14,XREAL_1:64; then A18: r3*(r1+r2)+r*r1 < r2 by A15,A17,XREAL_1:8; A19: G*(i,1) = |[G*(i,1)`1,G*(i,1)`2]| by EUCLID:53 .= |[r1,s1]| by A1,A2,A13,GOBOARD5:1; A20: 1 <= i+1 by A1,NAT_1:13; A21: G*(i+1,1) = |[G*(i+1,1)`1,G*(i+1,1)`2]| by EUCLID:53 .= |[r2,s1]| by A20,A10,A13,GOBOARD5:1; p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1)))-(1-r)*|[0,1]| +r*(G*(i,1) - |[0 ,1]|) by A4,RLVECT_1:34 .= r3*(G*(i,1)+G*(i+1,1))-(1-r)*|[0,1]|+r*(G* (i,1) - |[0,1]|) by RLVECT_1:def 7 .= r3*(G*(i,1)+G*(i+1,1))-|[(1-r)*0,(1-r)*1]|+r*(G*(i,1) - |[0,1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+r*(|[r1,s1]| - |[0,1]|) by A21,A19, EUCLID:56 .= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(r*|[r1,s1]| - r*|[0,1]|) by RLVECT_1:34 .= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(|[r*r1,r*s1]| - r*|[0,1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(|[r*r1,r*s1]| - |[r*0,r*1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+|[r*r1-0,r*s1-r]| by EUCLID:62 .= |[r3*(r1+r2),r3*(s1+s1)]|-|[0,1-r]|+|[r*r1-0,r*s1-r]| by EUCLID:58 .= |[r3*(r1+r2)-0,r3*(s1+s1)-(1-r)]|+|[r*r1-0,r*s1-r]| by EUCLID:62 .= |[r3*(r1+r2)+r*r1,r3*(s1+s1)-(1-r)+(r*s1-r)]| by EUCLID:56; hence p in Int cell(G,i,0) by A11,A16,A18,A12; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th53: 1 <= i & i < len G implies LSeg(1/2*(G*(i,1)+G*(i+1,1)) - |[0,1 ]|,G*(i+1,1) - |[0,1]|) c= Int cell(G,i,0) \/ { G*(i+1,1) - |[0,1]| } proof assume that A1: 1 <= i and A2: i < len G; let x be object; assume A3: x in LSeg(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|,G*(i+1,1) - |[0,1]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|)+r*(G*(i+1,1) - |[0,1]|) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(i+1,1) - |[0,1]|) by A4,RLVECT_1:10 .= 1*(G*(i+1,1) - |[0,1]|) by RLVECT_1:4 .= (G*(i+1,1) - |[0,1]|) by RLVECT_1:def 8; hence p in { G*(i+1,1) - |[0,1]| } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2); 1 - r > 0 by A7,XREAL_1:50; then A8: r3 > (1/2)*0 by XREAL_1:68; set s1 = G*(1,1)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1; A9: r3*(r1+r1)+r*r1 = r1; A10: i+1 <= len G by A2,NAT_1:13; 0 <> width G by MATRIX_0:def 10; then A11: 1 <= width G by NAT_1:14; i < i+1 by XREAL_1:29; then A12: r1 < r2 by A1,A10,A11,GOBOARD5:3; then r1+r1 < r1+r2 by XREAL_1:6; then A13: r3*(r1+r1) < r3*(r1+r2) by A8,XREAL_1:68; r*r1 <= r*r2 by A5,A12,XREAL_1:64; then A14: r1 < r3*(r1+r2)+r*r2 by A13,A9,XREAL_1:8; A15: 1 <= i+1 by A1,NAT_1:13; A16: G*(i,1) = |[G*(i,1)`1,G*(i,1)`2]| by EUCLID:53 .= |[r1,s1]| by A1,A2,A11,GOBOARD5:1; s1 < s1+1 by XREAL_1:29; then A17: s1-1 < s1 by XREAL_1:19; A18: r3*(r2+r2)+r*r2 = r2; r1+r2 < r2+r2 by A12,XREAL_1:6; then r3*(r1+r2) < r3*(r2+r2) by A8,XREAL_1:68; then A19: r3*(r1+r2)+r*r2 < r2 by A18,XREAL_1:8; A20: Int cell(G,i,0) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1,1)`1 & s9 < G*(1,1)`2 } by A1,A2,Th24; A21: G*(i+1,1) = |[G*(i+1,1)`1,G*(i+1,1)`2]| by EUCLID:53 .= |[r2,s1]| by A15,A10,A11,GOBOARD5:1; p = (1-r)*(1/2*(G*(i,1)+G*(i+1,1)))-(1-r)*|[0,1]| +r*(G*(i+1,1) - |[0,1]|) by A4,RLVECT_1:34 .= r3*(G*(i,1)+G*(i+1,1))-(1-r)*|[0,1]|+r*(G*(i+1,1) - |[0,1]|) by RLVECT_1:def 7 .= r3*(G*(i,1)+G*(i+1,1))-|[(1-r)*0,(1-r)*1]|+r*(G*(i+1,1) - |[0,1]| ) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+r*(|[r2,s1]| - |[0,1]|) by A21,A16, EUCLID:56 .= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(r*|[r2,s1]| - r*|[0,1]|) by RLVECT_1:34 .= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(|[r*r2,r*s1]| - r*|[0,1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+(|[r*r2,r*s1]| - |[r*0,r*1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|-|[0,1-r]|+|[r*r2-0,r*s1-r]| by EUCLID:62 .= |[r3*(r1+r2),r3*(s1+s1)]|-|[0,1-r]|+|[r*r2-0,r*s1-r]| by EUCLID:58 .= |[r3*(r1+r2)-0,r3*(s1+s1)-(1-r)]|+|[r*r2-0,r*s1-r]| by EUCLID:62 .= |[r3*(r1+r2)+r*r2,r3*(s1+s1)-(1-r)+(r*s1-r)]| by EUCLID:56; hence p in Int cell(G,i,0) by A17,A14,A19,A20; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th54: 1 <= i & i < len G implies LSeg(1/2*(G*(i,width G)+G*(i+1,width G)) + |[0,1]|,G* (i,width G) + |[0,1]|) c= Int cell(G,i,width G) \/ { G*(i, width G) + |[0,1]| } proof assume that A1: 1 <= i and A2: i < len G; let x be object; assume A3: x in LSeg(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|, G*(i,width G) + |[0,1]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|)+ r*(G*(i, width G) + |[0,1]|) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(i,width G) + |[0,1]|) by A4,RLVECT_1:10 .= 1*(G*(i,width G) + |[0,1]|) by RLVECT_1:4 .= (G*(i,width G) + |[0,1]|) by RLVECT_1:def 8; hence p in { G*(i,width G) + |[0,1]| } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2); 1 - r > 0 by A7,XREAL_1:50; then A8: r3 > (1/2)*0 by XREAL_1:68; set s1 = G*(1,width G)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1; A9: r3*(r1+r1)+r*r1 = r1; A10: i+1 <= len G by A2,NAT_1:13; 0 <> width G by MATRIX_0:def 10; then A11: 1 <= width G by NAT_1:14; A12: G*(i,width G) = |[G*(i,width G)`1,G*(i,width G)`2]| by EUCLID:53 .= |[G*(i,width G)`1,s1]| by A1,A2,A11,GOBOARD5:1 .= |[r1,s1]| by A1,A2,A11,GOBOARD5:2; A13: 1 <= i+1 by A1,NAT_1:13; i < i+1 by XREAL_1:29; then A14: r1 < r2 by A1,A10,A11,GOBOARD5:3; then r1+r2 < r2+r2 by XREAL_1:6; then A15: r3*(r1+r2) < r3*(r2+r2) by A8,XREAL_1:68; r1+r1 < r1+r2 by A14,XREAL_1:6; then r3*(r1+r1) < r3*(r1+r2) by A8,XREAL_1:68; then A16: s1 < s1+1 & r1 < r3*(r1+r2)+r*r1 by A9,XREAL_1:6,29; A17: Int cell(G,i,width G) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1, 1)`1 & G*(1,width G)`2 < s9 } by A1,A2,Th25; A18: G*(i+1,width G) = |[G*(i+1,width G)`1,G*(i+1,width G)`2]| by EUCLID:53 .= |[G*(i+1,width G)`1,s1]| by A13,A10,A11,GOBOARD5:1 .= |[r2,s1]| by A13,A10,A11,GOBOARD5:2; A19: r3*(r2+r2)+r*r2 = r2; r*r1 <= r*r2 by A5,A14,XREAL_1:64; then A20: r3*(r1+r2)+r*r1 < r2 by A15,A19,XREAL_1:8; p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G)))+(1-r)*|[0,1]| +r*(G *(i,width G) + |[0,1]|) by A4,RLVECT_1:def 5 .= r3*(G*(i,width G)+G*(i+1,width G))+(1-r)*|[0,1]|+ r*(G*(i,width G ) + |[0,1]|) by RLVECT_1:def 7 .= r3*(G*(i,width G)+G*(i+1,width G))+|[(1-r)*0,(1-r)*1]|+ r*(G*(i, width G) + |[0,1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+r*(|[r1,s1]| + |[0,1]|) by A18,A12, EUCLID:56 .= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(r*|[r1,s1]| + r*|[0,1]|) by RLVECT_1:def 5 .= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(|[r*r1,r*s1]| + r*|[0,1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(|[r*r1,r*s1]| + |[r*0,r*1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+|[r*r1+0,r*s1+r]| by EUCLID:56 .= |[r3*(r1+r2),r3*(s1+s1)]|+|[0,1-r]|+|[r*r1+0,r*s1+r]| by EUCLID:58 .= |[r3*(r1+r2)+0,r3*(s1+s1)+(1-r)]|+|[r*r1+0,r*s1+r]| by EUCLID:56 .= |[r3*(r1+r2)+r*r1,r3*(s1+s1)+(1-r)+(r*s1+r)]| by EUCLID:56; hence p in Int cell(G,i,width G) by A16,A20,A17; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th55: 1 <= i & i < len G implies LSeg(1/2*(G*(i,width G)+G*(i+1,width G)) + |[0,1]|,G* (i+1,width G) + |[0,1]|) c= Int cell(G,i,width G) \/ { G*(i+1, width G) + |[0,1]| } proof assume that A1: 1 <= i and A2: i < len G; let x be object; assume A3: x in LSeg(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|, G*(i+1,width G) + |[0,1]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A4: p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|)+ r*(G*(i+1, width G) + |[0,1]|) and A5: 0<=r and A6: r<=1 by A3; now per cases by A6,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(i+1,width G) + |[0,1]|) by A4,RLVECT_1:10 .= 1*(G*(i+1,width G) + |[0,1]|) by RLVECT_1:4 .= (G*(i+1,width G) + |[0,1]|) by RLVECT_1:def 8; hence p in { G*(i+1,width G) + |[0,1]| } by TARSKI:def 1; end; case A7: r < 1; set r3 = (1-r)*(1/2); 1 - r > 0 by A7,XREAL_1:50; then A8: r3 > (1/2)*0 by XREAL_1:68; set s1 = G*(1,width G)`2, r1 = G*(i,1)`1, r2 = G*(i+1,1)`1; A9: r3*(r1+r1)+r*r1 = r1; A10: i+1 <= len G by A2,NAT_1:13; 0 <> width G by MATRIX_0:def 10; then A11: 1 <= width G by NAT_1:14; i < i+1 by XREAL_1:29; then A12: r1 < r2 by A1,A10,A11,GOBOARD5:3; then r1+r1 < r1+r2 by XREAL_1:6; then A13: r3*(r1+r1) < r3*(r1+r2) by A8,XREAL_1:68; A14: r3*(r2+r2)+r*r2 = r2; r1+r2 < r2+r2 by A12,XREAL_1:6; then r3*(r1+r2) < r3*(r2+r2) by A8,XREAL_1:68; then A15: r3*(r1+r2)+r*r2 < r2 by A14,XREAL_1:8; A16: G*(i,width G) = |[G*(i,width G)`1,G*(i,width G)`2]| by EUCLID:53 .= |[G*(i,width G)`1,s1]| by A1,A2,A11,GOBOARD5:1 .= |[r1,s1]| by A1,A2,A11,GOBOARD5:2; A17: 1 <= i+1 by A1,NAT_1:13; r*r1 <= r*r2 by A5,A12,XREAL_1:64; then A18: s1+1 > s1 & r1 < r3*(r1+r2)+r*r2 by A13,A9,XREAL_1:8,29; A19: Int cell(G,i,width G) = { |[r9,s9]| : G*(i,1)`1 < r9 & r9 < G*(i+1, 1)`1 & G*(1,width G)`2 < s9 } by A1,A2,Th25; A20: G*(i+1,width G) = |[G*(i+1,width G)`1,G*(i+1,width G)`2]| by EUCLID:53 .= |[G*(i+1,width G)`1,s1]| by A17,A10,A11,GOBOARD5:1 .= |[r2,s1]| by A17,A10,A11,GOBOARD5:2; p = (1-r)*(1/2*(G*(i,width G)+G*(i+1,width G)))+(1-r)*|[0,1]| +r*(G *(i+1,width G) + |[0,1]|) by A4,RLVECT_1:def 5 .= r3*(G*(i,width G)+G*(i+1,width G))+(1-r)*|[0,1]|+ r*(G*(i+1,width G) + |[0,1]|) by RLVECT_1:def 7 .= r3*(G*(i,width G)+G*(i+1,width G))+|[(1-r)*0,(1-r)*1]|+ r*(G*(i+1 ,width G) + |[0,1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+r*(|[r2,s1]| + |[0,1]|) by A20,A16, EUCLID:56 .= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(r*|[r2,s1]| + r*|[0,1]|) by RLVECT_1:def 5 .= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(|[r*r2,r*s1]| + r*|[0,1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+(|[r*r2,r*s1]| + |[r*0,r*1]|) by EUCLID:58 .= r3*|[r1+r2,s1+s1]|+|[0,1-r]|+|[r*r2+0,r*s1+r]| by EUCLID:56 .= |[r3*(r1+r2),r3*(s1+s1)]|+|[0,1-r]|+|[r*r2+0,r*s1+r]| by EUCLID:58 .= |[r3*(r1+r2)+0,r3*(s1+s1)+(1-r)]|+|[r*r2+0,r*s1+r]| by EUCLID:56 .= |[r3*(r1+r2)+r*r2,r3*(s1+s1)+(1-r)+(r*s1+r)]| by EUCLID:56; hence p in Int cell(G,i,width G) by A18,A15,A19; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th56: LSeg(G*(1,1) - |[1,1]|,G*(1,1) - |[1,0]|) c= Int cell(G,0,0) \/ { G*(1,1) - |[1,0]| } proof let x be object; set r1 = G*(1,1)`1, s1 = G*(1,1)`2; assume A1: x in LSeg(G*(1,1)-|[1,1]|,G*(1,1) - |[1,0]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A2: p = (1-r)*(G*(1,1)-|[1,1]|)+r*(G*(1,1) - |[1,0]|) and 0<=r and A3: r<=1 by A1; now per cases by A3,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(1,1) - |[1,0]|) by A2,RLVECT_1:10 .= 1*(G*(1,1) - |[1,0]|) by RLVECT_1:4 .= G*(1,1) - |[1,0]| by RLVECT_1:def 8; hence p in { G*(1,1) - |[1,0]| } by TARSKI:def 1; end; case r < 1; then 1 - r > 0 by XREAL_1:50; then s1 < s1 +(1-r) by XREAL_1:29; then A4: s1-(1-r) < s1 by XREAL_1:19; A5: G*(1,1) = |[r1,s1]| by EUCLID:53; r1 < r1+1 by XREAL_1:29; then A6: r1-1 < r1 by XREAL_1:19; A7: Int cell(G,0,0) = { |[r9,s9]| : r9 < G*(1,1)`1 & s9 < G* (1,1)`2 } by Th18; p = (1-r)*(G*(1,1))-(1-r)*|[1,1]|+r*(G*(1,1) - |[1,0]|) by A2,RLVECT_1:34 .= (1-r)*(G*(1,1))-(1-r)*|[1,1]|+(r*(G*(1,1)) - r*|[1,0]|) by RLVECT_1:34 .= r*(G*(1,1)) + ((1-r)*(G*(1,1))-(1-r)*|[1,1]|) - r*|[1,0]| by RLVECT_1:def 3 .= r*(G*(1,1)) + (1-r)*(G*(1,1))-(1-r)*|[1,1]| - r*|[1,0]| by RLVECT_1:def 3 .= (r+(1-r))*(G*(1,1)) -(1-r)*|[1,1]| - r*|[1,0]| by RLVECT_1:def 6 .= G*(1,1) -(1-r)*|[1,1]| - r*|[1,0]| by RLVECT_1:def 8 .= G*(1,1)-|[(1-r)*1,(1-r)*1]| - r*|[1,0]| by EUCLID:58 .= G*(1,1)-|[1-r,1-r]| - |[r*1,r*0]| by EUCLID:58 .= |[r1-(1-r),s1-(1-r)]| - |[r,0]| by A5,EUCLID:62 .= |[r1-(1-r)-r,s1-(1-r)-0]| by EUCLID:62 .= |[r1-1,s1-(1-r)]|; hence p in Int cell(G,0,0) by A4,A6,A7; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th57: LSeg(G*(len G,1) + |[1,-1]|,G*(len G,1) + |[1,0]|) c= Int cell(G ,len G,0) \/ { G*(len G,1) + |[1,0]| } proof let x be object; set r1 = G*(len G,1)`1, s1 = G*(1,1)`2; assume A1: x in LSeg(G*(len G,1)+|[1,-1]|,G*(len G,1) + |[1,0]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A2: p = (1-r)*(G*(len G,1)+|[1,-1]|)+r*(G*(len G,1) + |[1,0]|) and 0<=r and A3: r<=1 by A1; now per cases by A3,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(len G,1) + |[1,0]|) by A2,RLVECT_1:10 .= 1*(G*(len G,1) + |[1,0]|) by RLVECT_1:4 .= G*(len G,1) + |[1,0]| by RLVECT_1:def 8; hence p in { G*(len G,1) + |[1,0]| } by TARSKI:def 1; end; case r < 1; then 1 - r > 0 by XREAL_1:50; then A4: s1 < s1 +(1-r) by XREAL_1:29; s1+(r-1) = s1-(1-r); then A5: s1+(r-1) < s1 by A4,XREAL_1:19; A6: r1 < r1+1 by XREAL_1:29; 0 <> len G by MATRIX_0:def 10; then A7: 1 <= len G by NAT_1:14; 0 <> width G by MATRIX_0:def 10; then A8: 1 <= width G by NAT_1:14; A9: G*(len G,1) = |[r1,G*(len G,1)`2]| by EUCLID:53 .= |[r1,s1]| by A8,A7,GOBOARD5:1; A10: Int cell(G,len G,0) = { |[r9,s9]| : G*(len G,1)`1 < r9 & s9 < G*(1, 1)`2 } by Th21; p = (1-r)*(G*(len G,1))+(1-r)*|[1,-1]|+r*(G*(len G,1) + |[1,0]|) by A2, RLVECT_1:def 5 .= (1-r)*(G*(len G,1))+(1-r)*|[1,-1]|+(r*(G*(len G,1)) + r*|[1,0]|) by RLVECT_1:def 5 .= r*(G*(len G,1)) + ((1-r)*(G*(len G,1))+(1-r)*|[1,-1]|) + r*|[1,0 ]| by RLVECT_1:def 3 .= r*(G*(len G,1)) + (1-r)*(G*(len G,1))+(1-r)*|[1,-1]| + r*|[1,0]| by RLVECT_1:def 3 .= (r+(1-r))*(G*(len G,1)) +(1-r)*|[1,-1]| + r*|[1,0]| by RLVECT_1:def 6 .= G*(len G,1) +(1-r)*|[1,-1]| + r*|[1,0]| by RLVECT_1:def 8 .= G*(len G,1)+|[(1-r)*1,(1-r)*(-1)]| + r*|[1,0]| by EUCLID:58 .= G*(len G,1)+|[1-r,r-1]| + |[r*1,r*0]| by EUCLID:58 .= |[r1+(1-r),s1+(r-1)]| + |[r,0]| by A9,EUCLID:56 .= |[r1+(1-r)+r,s1+(r-1)+0]| by EUCLID:56 .= |[r1+1,s1+(r-1)]|; hence p in Int cell(G,len G,0) by A5,A6,A10; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th58: LSeg(G*(1,width G) + |[-1,1]|,G*(1,width G) - |[1,0]|) c= Int cell(G,0,width G) \/ { G*(1,width G) - |[1,0]| } proof let x be object; set r1 = G*(1,1)`1, s1 = G*(1,width G)`2; assume A1: x in LSeg(G*(1,width G)+|[-1,1]|,G*(1,width G) - |[1,0]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A2: p = (1-r)*(G*(1,width G)+|[-1,1]|)+r*(G*(1,width G) - |[1,0]|) and 0<=r and A3: r<=1 by A1; now per cases by A3,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(1,width G) - |[1,0]|) by A2,RLVECT_1:10 .= 1*(G*(1,width G) - |[1,0]|) by RLVECT_1:4 .= G*(1,width G) - |[1,0]| by RLVECT_1:def 8; hence p in { G*(1,width G) - |[1,0]| } by TARSKI:def 1; end; case r < 1; then 1 - r > 0 by XREAL_1:50; then A4: s1 < s1 +(1-r) by XREAL_1:29; 0 <> width G by MATRIX_0:def 10; then A5: 1 <= width G by NAT_1:14; 0 <> len G by MATRIX_0:def 10; then A6: 1 <= len G by NAT_1:14; A7: G*(1,width G) = |[G*(1,width G)`1,s1]| by EUCLID:53 .= |[r1,s1]| by A5,A6,GOBOARD5:2; A8: Int cell(G,0,width G) = { |[r9,s9]| : r9 < G*(1,1)`1 & G*(1,width G )`2 < s9 } by Th19; r1 < r1+1 by XREAL_1:29; then A9: r1-1 < r1 by XREAL_1:19; p = (1-r)*(G*(1,width G))+(1-r)*|[-1,1]|+r*(G*(1,width G) - |[1,0]| ) by A2,RLVECT_1:def 5 .= (1-r)*(G*(1,width G))+(1-r)*|[-1,1]|+(r*(G*(1,width G)) - r*|[1,0 ]|) by RLVECT_1:34 .= r*(G*(1,width G)) + ((1-r)*(G* (1,width G))+(1-r)*|[-1,1]|) - r* |[1,0]| by RLVECT_1:def 3 .= r*(G*(1,width G)) + (1-r)*(G*(1,width G))+(1-r)*|[-1,1]| - r*|[1, 0]| by RLVECT_1:def 3 .= (r+(1-r))*(G*(1,width G)) +(1-r)*|[-1,1]| - r*|[1,0]| by RLVECT_1:def 6 .= G*(1,width G) +(1-r)*|[-1,1]| - r*|[1,0]| by RLVECT_1:def 8 .= G*(1,width G)+|[(1-r)*(-1),(1-r)*1]| - r*|[1,0]| by EUCLID:58 .= G*(1,width G)+|[r-1,1-r]| - |[r*1,r*0]| by EUCLID:58 .= |[r1+(r-1),s1+(1-r)]| - |[r,0]| by A7,EUCLID:56 .= |[r1+(r-1)-r,s1+(1-r)-0]| by EUCLID:62 .= |[r1-1,s1+(1-r)]|; hence p in Int cell(G,0,width G) by A4,A9,A8; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th59: LSeg(G*(len G,width G) + |[1,1]|,G*(len G,width G) + |[1,0]|) c= Int cell(G,len G,width G) \/ { G*(len G,width G) + |[1,0]| } proof let x be object; set r1 = G*(len G,1)`1, s1 = G*(1,width G)`2; assume A1: x in LSeg(G*(len G,width G)+|[1,1]|,G*(len G,width G) + |[1,0]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A2: p = (1-r)*(G*(len G,width G)+|[1,1]|)+r*(G*(len G,width G) + |[1,0]| ) and 0<=r and A3: r<=1 by A1; now per cases by A3,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(len G,width G) + |[1,0]|) by A2,RLVECT_1:10 .= 1*(G*(len G,width G) + |[1,0]|) by RLVECT_1:4 .= G*(len G,width G) + |[1,0]| by RLVECT_1:def 8; hence p in { G*(len G,width G) + |[1,0]| } by TARSKI:def 1; end; case r < 1; then 1 - r > 0 by XREAL_1:50; then A4: s1 < s1 +(1-r) by XREAL_1:29; A5: r1 < r1+1 by XREAL_1:29; 0 <> width G by MATRIX_0:def 10; then A6: 1 <= width G by NAT_1:14; 0 <> len G by MATRIX_0:def 10; then A7: 1 <= len G by NAT_1:14; A8: G*(len G,width G) = |[G*(len G,width G)`1,G*(len G,width G)`2]| by EUCLID:53 .= |[r1,G*(len G,width G)`2]| by A6,A7,GOBOARD5:2 .= |[r1,s1]| by A6,A7,GOBOARD5:1; A9: Int cell(G,len G,width G) = { |[r9,s9]| : G*(len G,1)`1 < r9 & G*(1 ,width G)`2 < s9 } by Th22; p = (1-r)*(G*(len G,width G))+(1-r)*|[1,1]|+r*(G* (len G,width G) + |[1,0]|) by A2,RLVECT_1:def 5 .= (1-r)*(G*(len G,width G))+(1-r)*|[1,1]|+(r*(G* (len G,width G)) + r*|[1,0]|) by RLVECT_1:def 5 .= r*(G*(len G,width G))+((1-r)*(G* (len G,width G))+(1-r)*|[1,1]|) + r*|[1,0]| by RLVECT_1:def 3 .= r*(G*(len G,width G)) + (1-r)*(G* (len G,width G))+(1-r)*|[1,1]| + r*|[1,0]| by RLVECT_1:def 3 .= (r+(1-r))*(G*(len G,width G)) +(1-r)*|[1,1]| + r*|[1,0]| by RLVECT_1:def 6 .= G*(len G,width G) +(1-r)*|[1,1]| + r*|[1,0]| by RLVECT_1:def 8 .= G*(len G,width G)+|[(1-r)*1,(1-r)*1]| + r*|[1,0]| by EUCLID:58 .= G*(len G,width G)+|[1-r,1-r]| + |[r*1,r*0]| by EUCLID:58 .= |[r1+(1-r),s1+(1-r)]| + |[r,0]| by A8,EUCLID:56 .= |[r1+(1-r)+r,s1+(1-r)+0]| by EUCLID:56 .= |[r1+1,s1+(1-r)]|; hence p in Int cell(G,len G,width G) by A4,A5,A9; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th60: LSeg(G*(1,1) - |[1,1]|,G*(1,1) - |[0,1]|) c= Int cell(G,0,0) \/ { G*(1,1) - |[0,1]| } proof let x be object; set r1 = G*(1,1)`1, s1 = G*(1,1)`2; assume A1: x in LSeg(G*(1,1)-|[1,1]|,G*(1,1) - |[0,1]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A2: p = (1-r)*(G*(1,1)-|[1,1]|)+r*(G*(1,1) - |[0,1]|) and 0<=r and A3: r<=1 by A1; now per cases by A3,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(1,1) - |[0,1]|) by A2,RLVECT_1:10 .= 1*(G*(1,1) - |[0,1]|) by RLVECT_1:4 .= G*(1,1) - |[0,1]| by RLVECT_1:def 8; hence p in { G*(1,1) - |[0,1]| } by TARSKI:def 1; end; case r < 1; then 1 - r > 0 by XREAL_1:50; then r1 < r1+(1-r) by XREAL_1:29; then A4: r1-(1-r) < r1 by XREAL_1:19; A5: G*(1,1) = |[r1,s1]| by EUCLID:53; s1 < s1 +1 by XREAL_1:29; then A6: s1-1 < s1 by XREAL_1:19; A7: Int cell(G,0,0) = { |[r9,s9]| : r9 < G*(1,1)`1 & s9 < G* (1,1)`2 } by Th18; p = (1-r)*(G*(1,1))-(1-r)*|[1,1]|+r*(G*(1,1) - |[0,1]|) by A2,RLVECT_1:34 .= (1-r)*(G*(1,1))-(1-r)*|[1,1]|+(r*(G*(1,1)) - r*|[0,1]|) by RLVECT_1:34 .= r*(G*(1,1)) + ((1-r)*(G*(1,1))-(1-r)*|[1,1]|) - r*|[0,1]| by RLVECT_1:def 3 .= r*(G*(1,1)) + (1-r)*(G*(1,1))-(1-r)*|[1,1]| - r*|[0,1]| by RLVECT_1:def 3 .= (r+(1-r))*(G*(1,1)) -(1-r)*|[1,1]| - r*|[0,1]| by RLVECT_1:def 6 .= G*(1,1) -(1-r)*|[1,1]| - r*|[0,1]| by RLVECT_1:def 8 .= G*(1,1)-|[(1-r)*1,(1-r)*1]| - r*|[0,1]| by EUCLID:58 .= G*(1,1)-|[1-r,1-r]| - |[r*0,r*1]| by EUCLID:58 .= |[r1-(1-r),s1-(1-r)]| - |[0,r]| by A5,EUCLID:62 .= |[r1-(1-r)-0,s1-(1-r)-r]| by EUCLID:62 .= |[r1-(1-r),s1-1]|; hence p in Int cell(G,0,0) by A6,A4,A7; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th61: LSeg(G*(len G,1) + |[1,-1]|,G*(len G,1) - |[0,1]|) c= Int cell(G ,len G,0) \/ { G*(len G,1) - |[0,1]| } proof let x be object; set r1 = G*(len G,1)`1, s1 = G*(1,1)`2; assume A1: x in LSeg(G*(len G,1)+|[1,-1]|,G*(len G,1) - |[0,1]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A2: p = (1-r)*(G*(len G,1)+|[1,-1]|)+r*(G*(len G,1) - |[0,1]|) and 0<=r and A3: r<=1 by A1; now per cases by A3,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(len G,1) - |[0,1]|) by A2,RLVECT_1:10 .= 1*(G*(len G,1) - |[0,1]|) by RLVECT_1:4 .= G*(len G,1) - |[0,1]| by RLVECT_1:def 8; hence p in { G*(len G,1) - |[0,1]| } by TARSKI:def 1; end; case r < 1; then 1 - r > 0 by XREAL_1:50; then A4: r1 < r1+(1-r) by XREAL_1:29; s1 < s1+1 by XREAL_1:29; then A5: s1-1 < s1 by XREAL_1:19; 0 <> len G by MATRIX_0:def 10; then A6: 1 <= len G by NAT_1:14; 0 <> width G by MATRIX_0:def 10; then A7: 1 <= width G by NAT_1:14; A8: G*(len G,1) = |[r1,G*(len G,1)`2]| by EUCLID:53 .= |[r1,s1]| by A7,A6,GOBOARD5:1; A9: Int cell(G,len G,0) = { |[r9,s9]| : G*(len G,1)`1 < r9 & s9 < G*(1, 1)`2 } by Th21; p = (1-r)*(G*(len G,1))+(1-r)*|[1,-1]|+r*(G*(len G,1) - |[0,1]|) by A2, RLVECT_1:def 5 .= (1-r)*(G*(len G,1))+(1-r)*|[1,-1]|+(r*(G*(len G,1)) - r*|[0,1]|) by RLVECT_1:34 .= r*(G*(len G,1)) + ((1-r)*(G*(len G,1))+(1-r)*|[1,-1]|) - r*|[0,1 ]| by RLVECT_1:def 3 .= r*(G*(len G,1)) + (1-r)*(G*(len G,1))+(1-r)*|[1,-1]| - r*|[0,1]| by RLVECT_1:def 3 .= (r+(1-r))*(G*(len G,1)) +(1-r)*|[1,-1]| - r*|[0,1]| by RLVECT_1:def 6 .= G*(len G,1) +(1-r)*|[1,-1]| - r*|[0,1]| by RLVECT_1:def 8 .= G*(len G,1)+|[(1-r)*1,(1-r)*(-1)]| - r*|[0,1]| by EUCLID:58 .= G*(len G,1)+|[1-r,r-1]| - |[r*0,r*1]| by EUCLID:58 .= |[r1+(1-r),s1+(r-1)]| - |[0,r]| by A8,EUCLID:56 .= |[r1+(1-r)-0,s1+(r-1)-r]| by EUCLID:62 .= |[r1+(1-r),s1-1]|; hence p in Int cell(G,len G,0) by A5,A4,A9; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th62: LSeg(G*(1,width G) + |[-1,1]|,G*(1,width G) + |[0,1]|) c= Int cell(G,0,width G) \/ { G*(1,width G) + |[0,1]| } proof let x be object; set r1 = G*(1,1)`1, s1 = G*(1,width G)`2; assume A1: x in LSeg(G*(1,width G)+|[-1,1]|,G*(1,width G) + |[0,1]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A2: p = (1-r)*(G*(1,width G)+|[-1,1]|)+r*(G*(1,width G) + |[0,1]|) and 0<=r and A3: r<=1 by A1; now per cases by A3,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(1,width G) + |[0,1]|) by A2,RLVECT_1:10 .= 1*(G*(1,width G) + |[0,1]|) by RLVECT_1:4 .= G*(1,width G) + |[0,1]| by RLVECT_1:def 8; hence p in { G*(1,width G) + |[0,1]| } by TARSKI:def 1; end; case r < 1; then 1 - r > 0 by XREAL_1:50; then r1 < r1+(1-r) by XREAL_1:29; then A4: s1 < s1 +1 & r1-(1-r) < r1 by XREAL_1:19,29; 0 <> width G by MATRIX_0:def 10; then A5: 1 <= width G by NAT_1:14; 0 <> len G by MATRIX_0:def 10; then A6: 1 <= len G by NAT_1:14; A7: G*(1,width G) = |[G*(1,width G)`1,s1]| by EUCLID:53 .= |[r1,s1]| by A5,A6,GOBOARD5:2; A8: Int cell(G,0,width G) = { |[r9,s9]| : r9 < G*(1,1)`1 & G*(1,width G )`2 < s9 } by Th19; p = (1-r)*(G*(1,width G))+(1-r)*|[-1,1]|+r*(G*(1,width G) + |[0,1]| ) by A2,RLVECT_1:def 5 .= (1-r)*(G*(1,width G))+(1-r)*|[-1,1]|+(r*(G*(1,width G)) + r*|[0,1 ]|) by RLVECT_1:def 5 .= r*(G*(1,width G)) + ((1-r)*(G* (1,width G))+(1-r)*|[-1,1]|) + r* |[0,1]| by RLVECT_1:def 3 .= r*(G*(1,width G)) + (1-r)*(G*(1,width G))+(1-r)*|[-1,1]| + r*|[0, 1]| by RLVECT_1:def 3 .= (r+(1-r))*(G*(1,width G)) +(1-r)*|[-1,1]| + r*|[0,1]| by RLVECT_1:def 6 .= G*(1,width G) +(1-r)*|[-1,1]| + r*|[0,1]| by RLVECT_1:def 8 .= G*(1,width G)+|[(1-r)*(-1),(1-r)*1]| + r*|[0,1]| by EUCLID:58 .= G*(1,width G)+|[r-1,1-r]| + |[r*0,r*1]| by EUCLID:58 .= |[r1+(r-1),s1+(1-r)]| + |[0,r]| by A7,EUCLID:56 .= |[r1+(r-1)+0,s1+(1-r)+r]| by EUCLID:56 .= |[r1-(1-r),s1+1]|; hence p in Int cell(G,0,width G) by A4,A8; end; end; hence thesis by XBOOLE_0:def 3; end; theorem Th63: LSeg(G*(len G,width G) + |[1,1]|,G*(len G,width G) + |[0,1]|) c= Int cell(G,len G,width G) \/ { G*(len G,width G) + |[0,1]| } proof let x be object; set r1 = G*(len G,1)`1, s1 = G*(1,width G)`2; assume A1: x in LSeg(G*(len G,width G)+|[1,1]|,G*(len G,width G) + |[0,1]|); then reconsider p = x as Point of TOP-REAL 2; consider r such that A2: p = (1-r)*(G*(len G,width G)+|[1,1]|)+r*(G*(len G,width G) + |[0,1]| ) and 0<=r and A3: r<=1 by A1; now per cases by A3,XXREAL_0:1; case r = 1; then p = 0.TOP-REAL 2 + 1*(G*(len G,width G) + |[0,1]|) by A2,RLVECT_1:10 .= 1*(G*(len G,width G) + |[0,1]|) by RLVECT_1:4 .= G*(len G,width G) + |[0,1]| by RLVECT_1:def 8; hence p in { G*(len G,width G) + |[0,1]| } by TARSKI:def 1; end; case r < 1; then 1 - r > 0 by XREAL_1:50; then A4: s1 < s1 +1 & r1 < r1+(1-r) by XREAL_1:29; 0 <> width G by MATRIX_0:def 10; then A5: 1 <= width G by NAT_1:14; 0 <> len G by MATRIX_0:def 10; then A6: 1 <= len G by NAT_1:14; A7: G*(len G,width G) = |[G*(len G,width G)`1,G*(len G,width G)`2]| by EUCLID:53 .= |[r1,G*(len G,width G)`2]| by A5,A6,GOBOARD5:2 .= |[r1,s1]| by A5,A6,GOBOARD5:1; A8: Int cell(G,len G,width G) = { |[r9,s9]| : G*(len G,1)`1 < r9 & G*(1 ,width G)`2 < s9 } by Th22; p = (1-r)*(G*(len G,width G))+(1-r)*|[1,1]|+r*(G* (len G,width G) + |[0,1]|) by A2,RLVECT_1:def 5 .= (1-r)*(G*(len G,width G))+(1-r)*|[1,1]|+(r*(G* (len G,width G)) + r*|[0,1]|) by RLVECT_1:def 5 .= r*(G*(len G,width G))+((1-r)*(G* (len G,width G))+(1-r)*|[1,1]|) + r*|[0,1]| by RLVECT_1:def 3 .= r*(G*(len G,width G)) + (1-r)*(G* (len G,width G))+(1-r)*|[1,1]| + r*|[0,1]| by RLVECT_1:def 3 .= (r+(1-r))*(G*(len G,width G)) +(1-r)*|[1,1]| + r*|[0,1]| by RLVECT_1:def 6 .= G*(len G,width G) +(1-r)*|[1,1]| + r*|[0,1]| by RLVECT_1:def 8 .= G*(len G,width G)+|[(1-r)*1,(1-r)*1]| + r*|[0,1]| by EUCLID:58 .= G*(len G,width G)+|[1-r,1-r]| + |[r*0,r*1]| by EUCLID:58 .= |[r1+(1-r),s1+(1-r)]| + |[0,r]| by A7,EUCLID:56 .= |[r1+(1-r)+0,s1+(1-r)+r]| by EUCLID:56 .= |[r1+(1-r),s1+1]|; hence p in Int cell(G,len G,width G) by A4,A8; end; end; hence thesis by XBOOLE_0:def 3; end; theorem 1 <= i & i < len G & 1 <= j & j+1 < width G implies LSeg(1/2*(G*(i,j)+ G*(i+1,j+1)),1/2*(G*(i,j+1)+G*(i+1,j+2))) c= Int cell(G,i,j) \/ Int cell(G,i,j+ 1) \/ { 1/2*(G*(i,j+1)+G*(i+1,j+1)) } proof assume that A1: 1 <= i and A2: i < len G and A3: 1 <= j and A4: j+1 < width G; set p1 = G*(i,j), p2 = G*(i,j+1), q2 = G*(i+1,j+1), q3 = G*(i+1,j+2), r = ( p2`2-p1`2)/(q3`2-p1`2); A5: j+1 >= 1 by NAT_1:11; set I1 = Int cell(G,i,j), I2 = Int cell(G,i,j+1); j <= j+1 by NAT_1:11; then A6: j < width G by A4,XXREAL_0:2; then A7: LSeg(1/2*(p1+q2),1/2*(p2+q2)) c= I1 \/ { 1/2*(p2+q2) } by A1,A2,A3,Th41; j < j+1 by XREAL_1:29; then p1`2 < p2`2 by A1,A2,A3,A4,GOBOARD5:4; then A8: p2`2-p1`2 > 0 by XREAL_1:50; A9: j+1+1 = j+(1+1); then A10: j+2 >= 1 by NAT_1:11; A11: j+(1+1) <= width G by A4,A9,NAT_1:13; A12: i+1 >= 1 & i+1 <= len G by A2,NAT_1:11,13; then A13: q2`1 = G*(i+1,1)`1 by A4,A5,GOBOARD5:2 .= q3`1 by A11,A10,A12,GOBOARD5:2; A14: q2`2 = G*(1,j+1)`2 by A4,A5,A12,GOBOARD5:1 .= p2`2 by A1,A2,A4,A5,GOBOARD5:1; j+1 < j+2 by XREAL_1:6; then q2`2 < q3`2 by A5,A11,A12,GOBOARD5:4; then A15: p2`2-p1`2 < q3`2-p1`2 by A14,XREAL_1:9; then A16: r*(q3`2-p1`2) = p2`2-p1`2 by A8,XCMPLX_1:87; p1`1 = G*(i,1)`1 by A1,A2,A3,A6,GOBOARD5:2 .= p2`1 by A1,A2,A4,A5,GOBOARD5:2; then A17: (p2+q2)`1 = (1-r)*(p1`1+q2`1)+r*(p2`1+q3`1) by A13,Lm1 .= (1-r)*(p1+q2)`1+r*(p2`1+q3`1) by Lm1 .= (1-r)*(p1+q2)`1+r*(p2+q3)`1 by Lm1 .= (1-r)*(p1+q2)`1+(r*(p2+q3))`1 by Lm3 .= ((1-r)*(p1+q2))`1+(r*(p2+q3))`1 by Lm3 .= ((1-r)*(p1+q2)+r*(p2+q3))`1 by Lm1; (p2+q2)`2 = p2`2+(r+(1-r))*q2`2 by Lm1 .= (1-r)*(p1`2+q2`2)+r*(p2`2+q3`2) by A14,A16 .= (1-r)*(p1`2+q2`2)+r*(p2+q3)`2 by Lm1 .= (1-r)*(p1+q2)`2+r*(p2+q3)`2 by Lm1 .= (1-r)*(p1+q2)`2+(r*(p2+q3))`2 by Lm3 .= ((1-r)*(p1+q2))`2+(r*(p2+q3))`2 by Lm3 .= ((1-r)*(p1+q2)+r*(p2+q3))`2 by Lm1; then (1-r)*(p1+q2)+r*(p2+q3) = |[(p2+q2)`1,(p2+q2)`2]| by A17,EUCLID:53 .= p2+q2 by EUCLID:53; then A18: 1/2*(p2+q2) = (1/2)*((1-r)*(p1+q2))+(1/2)*(r*(p2+q3)) by RLVECT_1:def 5 .= 1/2*(1-r)*(p1+q2)+1/2*(r*(p2+q3)) by RLVECT_1:def 7 .= (1-r)*(1/2*(p1+q2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7 .= (1-r)*(1/2*(p1+q2))+1/2*r*(p2+q3) by RLVECT_1:def 7 .= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7; r < 1 by A15,A8,XREAL_1:189; then 1/2*(p2+q2) in LSeg(1/2*(p1+q2),1/2*(p2+q3)) by A15,A8,A18; then A19: LSeg(1/2*(p1+q2),1/2*(p2+q3)) = LSeg(1/2*(p1+q2),1/2*(p2+q2)) \/ LSeg(1 /2*(p2+q2),1/2*(p2+q3)) by TOPREAL1:5; A20: I1 \/ I2 \/ { 1/2*(p2+q2) } = I1 \/ (I2 \/ ({ 1/2*(p2+q2) } \/ { 1/2*( p2+q2) })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { 1/2*(p2+q2) } \/ { 1/2*(p2+q2) }) by XBOOLE_1:4 .= I1 \/ { 1/2*(p2+q2) } \/ (I2 \/ { 1/2*(p2+q2) }) by XBOOLE_1:4; LSeg(1/2*(p2+q2),1/2*(p2+q3)) c= I2 \/ { 1/2*(p2+q2) } by A1,A2,A4,A5,A9,Th43 ; hence thesis by A19,A7,A20,XBOOLE_1:13; end; theorem 1 <= j & j < width G & 1 <= i & i+1 < len G implies LSeg(1/2*(G*(i,j)+ G*(i+1,j+1)),1/2*(G*(i+1,j)+G*(i+2,j+1))) c= Int cell(G,i,j) \/ Int cell(G,i+1, j) \/ { 1/2*(G*(i+1,j)+G*(i+1,j+1)) } proof assume that A1: 1 <= j and A2: j < width G and A3: 1 <= i and A4: i+1 < len G; set p1 = G*(i,j), p2 = G*(i+1,j), q2 = G*(i+1,j+1), q3 = G*(i+2,j+1), r = ( p2`1-p1`1)/(q3`1-p1`1); A5: i+1 >= 1 by NAT_1:11; set I1 = Int cell(G,i,j), I2 = Int cell(G,i+1,j); i <= i+1 by NAT_1:11; then A6: i < len G by A4,XXREAL_0:2; then A7: LSeg(1/2*(p1+q2),1/2*(p2+q2)) c= I1 \/ { 1/2*(p2+q2) } by A1,A2,A3,Th42; i < i+1 by XREAL_1:29; then p1`1 < p2`1 by A1,A2,A3,A4,GOBOARD5:3; then A8: p2`1-p1`1 > 0 by XREAL_1:50; A9: i+1+1 = i+(1+1); then A10: i+2 >= 1 by NAT_1:11; A11: i+(1+1) <= len G by A4,A9,NAT_1:13; A12: j+1 >= 1 & j+1 <= width G by A2,NAT_1:11,13; then A13: q2`2 = G*(1,j+1)`2 by A4,A5,GOBOARD5:1 .= q3`2 by A11,A10,A12,GOBOARD5:1; A14: q2`1 = G*(i+1,1)`1 by A4,A5,A12,GOBOARD5:2 .= p2`1 by A1,A2,A4,A5,GOBOARD5:2; i+1 < i+2 by XREAL_1:6; then q2`1 < q3`1 by A5,A11,A12,GOBOARD5:3; then A15: p2`1-p1`1 < q3`1-p1`1 by A14,XREAL_1:9; then A16: r*(q3`1-p1`1) = p2`1-p1`1 by A8,XCMPLX_1:87; p1`2 = G*(1,j)`2 by A1,A2,A3,A6,GOBOARD5:1 .= p2`2 by A1,A2,A4,A5,GOBOARD5:1; then A17: (p2+q2)`2 = (1-r)*(p1`2+q2`2)+r*(p2`2+q3`2) by A13,Lm1 .= (1-r)*(p1+q2)`2+r*(p2`2+q3`2) by Lm1 .= (1-r)*(p1+q2)`2+r*(p2+q3)`2 by Lm1 .= (1-r)*(p1+q2)`2+(r*(p2+q3))`2 by Lm3 .= ((1-r)*(p1+q2))`2+(r*(p2+q3))`2 by Lm3 .= ((1-r)*(p1+q2)+r*(p2+q3))`2 by Lm1; (p2+q2)`1 = p2`1+(r+(1-r))*q2`1 by Lm1 .= (1-r)*(p1`1+q2`1)+r*(p2`1+q3`1) by A14,A16 .= (1-r)*(p1`1+q2`1)+r*(p2+q3)`1 by Lm1 .= (1-r)*(p1+q2)`1+r*(p2+q3)`1 by Lm1 .= (1-r)*(p1+q2)`1+(r*(p2+q3))`1 by Lm3 .= ((1-r)*(p1+q2))`1+(r*(p2+q3))`1 by Lm3 .= ((1-r)*(p1+q2)+r*(p2+q3))`1 by Lm1; then (1-r)*(p1+q2)+r*(p2+q3) = |[(p2+q2)`1,(p2+q2)`2]| by A17,EUCLID:53 .= p2+q2 by EUCLID:53; then A18: 1/2*(p2+q2) = (1/2)*((1-r)*(p1+q2))+(1/2)*(r*(p2+q3)) by RLVECT_1:def 5 .= 1/2*(1-r)*(p1+q2)+1/2*(r*(p2+q3)) by RLVECT_1:def 7 .= (1-r)*(1/2*(p1+q2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7 .= (1-r)*(1/2*(p1+q2))+1/2*r*(p2+q3) by RLVECT_1:def 7 .= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7; r < 1 by A15,A8,XREAL_1:189; then 1/2*(p2+q2) in LSeg(1/2*(p1+q2),1/2*(p2+q3)) by A15,A8,A18; then A19: LSeg(1/2*(p1+q2),1/2*(p2+q3)) = LSeg(1/2*(p1+q2),1/2*(p2+q2)) \/ LSeg(1 /2*(p2+q2),1/2*(p2+q3)) by TOPREAL1:5; A20: I1 \/ I2 \/ { 1/2*(p2+q2) } = I1 \/ (I2 \/ ({ 1/2*(p2+q2) } \/ { 1/2*( p2+q2) })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { 1/2*(p2+q2) } \/ { 1/2*(p2+q2) }) by XBOOLE_1:4 .= I1 \/ { 1/2*(p2+q2) } \/ (I2 \/ { 1/2*(p2+q2) }) by XBOOLE_1:4; LSeg(1/2*(p2+q2),1/2*(p2+q3)) c= I2 \/ { 1/2*(p2+q2) } by A1,A2,A4,A5,A9,Th40 ; hence thesis by A19,A7,A20,XBOOLE_1:13; end; theorem 1 <= i & i < len G & 1 < width G implies LSeg(1/2*(G*(i,1)+G*(i+1,1))- |[0,1]|,1/2*(G*(i,1)+G*(i+1,2))) c= Int cell(G,i,0) \/ Int cell(G,i,1) \/ { 1/2 *(G*(i,1)+G*(i+1,1)) } proof assume that A1: 1 <= i and A2: i < len G and A3: 1 < width G; set p1 = G*(i,1), q2 = G*(i+1,1), q3 = G*(i+1,2), r = 1/(1/2*(q3`2-p1`2)+1); A4: i+1 >= 1 & i+1 <= len G by A2,NAT_1:11,13; A5: 0+(1+1) <= width G by A3,NAT_1:13; then A6: q2`1 = q3`1 by A4,GOBOARD5:2; A7: q2`2 = G*(1,0+1)`2 by A3,A4,GOBOARD5:1 .= p1`2 by A1,A2,A3,GOBOARD5:1; then p1`2 < q3`2 by A5,A4,GOBOARD5:4; then A8: q3`2-p1`2 > 0 by XREAL_1:50; then 1 < 1/2*(q3`2-p1`2)+1 by XREAL_1:29,129; then A9: r < 1 by XREAL_1:212; set I1 = Int cell(G,i,0), I2 = Int cell(G,i,1); A10: LSeg(1/2*(p1+q2)-|[0,1]|,1/2*(p1+q2)) c= I1 \/ { 1/2*(p1+q2) } by A1,A2 ,Th46; A11: I1 \/ I2 \/ { 1/2*(p1+q2) } = I1 \/ (I2 \/ ({ 1/2*(p1+q2) } \/ { 1/2*( p1+q2) })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { 1/2*(p1+q2) } \/ { 1/2*(p1+q2) }) by XBOOLE_1:4 .= I1 \/ { 1/2*(p1+q2) } \/ (I2 \/ { 1/2*(p1+q2) }) by XBOOLE_1:4; A12: ((1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[0,1]|)`1 = ((1-r)*((1/2)*q2)+r*( (1/2)*q3))`1-((1-r)*|[0,1]|)`1 by Lm2 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1-|[(1-r)*0,(1-r)*1]|`1 by EUCLID:58 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1-0 by EUCLID:52 .= ((1-r)*((1/2)*q2))`1+(r*((1/2)*q3))`1 by Lm1 .= (1-r)*((1/2)*q2)`1+(r*((1/2)*q3))`1 by Lm3 .= (1-r)*((1/2)*q2)`1+r*((1/2)*q3)`1 by Lm3 .= (1-r)*((1/2)*q2`1)+r*((1/2)*q3)`1 by Lm3 .= (1-r)*((1/2)*q2`1)+r*((1/2)*q2`1) by A6,Lm3 .= ((1/2)*q2)`1 by Lm3; A13: (1-r)*((1/2)*p1)+r*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*q3) = (1-r)*((1 /2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1)+r*((1/2)*q3) by RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+(r*((1/2)*p1)+r*((1/2)*q3)) by RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5 .= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5 .= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5 .= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p1+q3)) by RLVECT_1:def 5; A14: r*((1/2)*q3`2)-r*((1/2)*q2`2)+r = r*((1/2)*(q3`2-q2`2)+1) .= 1 by A7,A8,XCMPLX_1:106; ((1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[0,1]|)`2 = ((1-r)*((1/2)*q2)+r*( (1/2)*q3))`2-((1-r)*|[0,1]|)`2 by Lm2 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2-|[(1-r)*0,(1-r)*1]|`2 by EUCLID:58 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2-(1-r) by EUCLID:52 .= ((1-r)*((1/2)*q2))`2+(r*((1/2)*q3))`2-(1-r) by Lm1 .= (1-r)*((1/2)*q2)`2+(r*((1/2)*q3))`2-(1-r) by Lm3 .= (1-r)*((1/2)*q2)`2+r*((1/2)*q3)`2-(1-r) by Lm3 .= (1-r)*((1/2)*q2`2)+r*((1/2)*q3)`2-(1-r) by Lm3 .= (1-r)*((1/2)*q2`2)+r*((1/2)*q3`2)-(1-r) by Lm3 .= ((1/2)*q2)`2 by A14,Lm3; then A15: (1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[0,1]| = |[(1/2*q2)`1,(1/2* q2 )`2 ]| by A12,EUCLID:53 .= 1/2*q2 by EUCLID:53; 1/2*(p1+q2) = 1/2*p1+1/2*q2 by RLVECT_1:def 5 .= (1-r+r)*((1/2)*p1)+1/2*q2 by RLVECT_1:def 8 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+1/2*q2 by RLVECT_1:def 6 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ ((1-r)*((1/2)*q2)+r*((1/2)*q3))-(1-r)* |[0,1]| by A15,RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ (1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[ 0,1]| by RLVECT_1:def 3 .= (1-r)*((1/2)*(p1+q2))+(r*((1/2)*(p1+q3))-(1-r)*|[0,1]|) by A13, RLVECT_1:def 3 .= (1-r)*((1/2)*(p1+q2))+-((1-r)*|[0,1]|-r*((1/2)*(p1+q3))) by RLVECT_1:33 .= (1-r)*((1/2)*(p1+q2))-((1-r)*|[0,1]|-r*((1/2)*(p1+q3))) .= (1-r)*((1/2)*(p1+q2))-(1-r)*|[0,1]|+r*((1/2)*(p1+q3)) by RLVECT_1:29 .= (1-r)*((1/2)*(p1+q2)-|[0,1]|)+r*((1/2)*(p1+q3)) by RLVECT_1:34; then 1/2*(p1+q2) in LSeg(1/2*(p1+q2)-|[0,1]|,1/2*(p1+q3)) by A8,A9; then A16: LSeg(1/2*(p1+q2)-|[0,1]|,1/2*(p1+q3)) = LSeg(1/2*(p1+q2)-|[0,1]|,1/2*( p1+q2)) \/ LSeg(1/2*(p1+q2),1/2*(p1+q3)) by TOPREAL1:5; 0+1+1 = 0+(1+1); then LSeg(1/2*(p1+q2),1/2*(p1+q3)) c= I2 \/ { 1/2*(p1+q2) } by A1,A2,A3,Th43; hence thesis by A16,A10,A11,XBOOLE_1:13; end; theorem 1 <= i & i < len G & 1 < width G implies LSeg(1/2*(G*(i,width G)+G*(i+ 1,width G))+|[0,1]|, 1/2*(G*(i,width G)+G*(i+1,width G -' 1))) c= Int cell(G,i, width G -'1) \/ Int cell(G,i,width G) \/ { 1/2*(G*(i,width G)+G*(i+1,width G)) } proof assume that A1: 1 <= i and A2: i < len G and A3: 1 < width G; set I1 = Int cell(G,i,width G -'1), I2 = Int cell(G,i,width G); set p1 = G*(i,width G), q2 = G*(i+1,width G), q3 = G*(i+1,width G -' 1), r = 1/(1/2*(p1`2-q3`2)+1); A4: width G -'1 + 1 = width G by A3,XREAL_1:235; then A5: 1 <= width G -'1 by A3,NAT_1:13; A6: width G -'1 < width G by A4,NAT_1:13; then G*(i,width G)+G*(i+1,width G -' 1) = G*(i,width G -' 1)+G*(i+1,width G) by A1,A2,A4,A5,Th11; then A7: LSeg(1/2*(p1+q2),1/2*(p1+q3)) c= I1 \/ { 1/2*(p1+q2) } by A1,A2,A4,A5,A6 ,Th41; A8: i+1 >= 1 & i+1 <= len G by A2,NAT_1:11,13; then A9: q2`1 = G*(i+1,1)`1 by A3,GOBOARD5:2 .= q3`1 by A5,A6,A8,GOBOARD5:2; A10: q2`2 = G*(1,width G)`2 by A3,A8,GOBOARD5:1 .= p1`2 by A1,A2,A3,GOBOARD5:1; then q3`2 < p1`2 by A5,A6,A8,GOBOARD5:4; then A11: p1`2-q3`2 > 0 by XREAL_1:50; then 1 < 1/2*(p1`2-q3`2)+1 by XREAL_1:29,129; then A12: r < 1 by XREAL_1:212; A13: ((1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[0,1]|)`1 = ((1-r)*((1/2)*q2)+r*( (1/2)*q3))`1+((1-r)*|[0,1]|)`1 by Lm1 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1+|[(1-r)*0,(1-r)*1]|`1 by EUCLID:58 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1+0 by EUCLID:52 .= ((1-r)*((1/2)*q2))`1+(r*((1/2)*q3))`1 by Lm1 .= (1-r)*((1/2)*q2)`1+(r*((1/2)*q3))`1 by Lm3 .= (1-r)*((1/2)*q2)`1+r*((1/2)*q3)`1 by Lm3 .= (1-r)*((1/2)*q2`1)+r*((1/2)*q3)`1 by Lm3 .= (1-r)*((1/2)*q2`1)+r*((1/2)*q2`1) by A9,Lm3 .= ((1/2)*q2)`1 by Lm3; A14: I1 \/ I2 \/ { 1/2*(p1+q2) } = I1 \/ (I2 \/ ({ 1/2*(p1+q2) } \/ { 1/2*( p1+q2) })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { 1/2*(p1+q2) } \/ { 1/2*(p1+q2) }) by XBOOLE_1:4 .= I1 \/ { 1/2*(p1+q2) } \/ (I2 \/ { 1/2*(p1+q2) }) by XBOOLE_1:4; A15: (1-r)*((1/2)*p1)+r*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*q3) = (1-r)*((1 /2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1)+r*((1/2)*q3) by RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+(r*((1/2)*p1)+r*((1/2)*q3)) by RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5 .= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5 .= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5 .= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p1+q3)) by RLVECT_1:def 5; A16: r*((1/2)*q2`2)-r*((1/2)*q3`2)+r = r*((1/2)*(q2`2-q3`2)+1) .= 1 by A10,A11,XCMPLX_1:106; ((1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[0,1]|)`2 = ((1-r)*((1/2)*q2)+r*( (1/2)*q3))`2+((1-r)*|[0,1]|)`2 by Lm1 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2+|[(1-r)*0,(1-r)*1]|`2 by EUCLID:58 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2+(1-r) by EUCLID:52 .= ((1-r)*((1/2)*q2))`2+(r*((1/2)*q3))`2+(1-r) by Lm1 .= (1-r)*((1/2)*q2)`2+(r*((1/2)*q3))`2+(1-r) by Lm3 .= (1-r)*((1/2)*q2)`2+r*((1/2)*q3)`2+(1-r) by Lm3 .= (1-r)*((1/2)*q2`2)+r*((1/2)*q3)`2+(1-r) by Lm3 .= (1-r)*((1/2)*q2`2)+r*((1/2)*q3`2)+(1-r) by Lm3 .= ((1/2)*q2)`2 by A16,Lm3; then A17: (1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[0,1]| = |[(1/2*q2)`1,(1/2* q2 )`2 ]| by A13,EUCLID:53 .= 1/2*q2 by EUCLID:53; 1/2*(p1+q2) = 1/2*p1+1/2*q2 by RLVECT_1:def 5 .= (1-r+r)*((1/2)*p1)+1/2*q2 by RLVECT_1:def 8 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+1/2*q2 by RLVECT_1:def 6 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ ((1-r)*((1/2)*q2)+r*((1/2)*q3))+(1-r)* |[0,1]| by A17,RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ (1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[ 0,1]| by RLVECT_1:def 3 .= (1-r)*((1/2)*(p1+q2))+(1-r)*|[0,1]|+r*((1/2)*(p1+q3)) by A15, RLVECT_1:def 3 .= (1-r)*((1/2)*(p1+q2)+|[0,1]|)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5; then 1/2*(p1+q2) in LSeg(1/2*(p1+q2)+|[0,1]|,1/2*(p1+q3)) by A11,A12; then A18: LSeg(1/2*(p1+q2)+|[0,1]|,1/2*(p1+q3)) = LSeg(1/2*(p1+q2)+|[0,1]|,1/2*( p1+q2)) \/ LSeg(1/2*(p1+q2),1/2*(p1+q3)) by TOPREAL1:5; LSeg(1/2*(p1+q2)+|[0,1]|,1/2*(p1+q2)) c= I2 \/ { 1/2*(p1+q2) } by A1,A2,Th47; hence thesis by A18,A7,A14,XBOOLE_1:13; end; theorem 1 <= j & j < width G & 1 < len G implies LSeg(1/2*(G*(1,j)+G*(1,j+1))- |[1,0]|,1/2*(G*(1,j)+G*(2,j+1))) c= Int cell(G,0,j) \/ Int cell(G,1,j) \/ { 1/2 *(G*(1,j)+G*(1,j+1)) } proof assume that A1: 1 <= j and A2: j < width G and A3: 1 < len G; set p1 = G*(1,j), q2 = G*(1,j+1), q3 = G*(2,j+1), r = 1/(1/2*(q3`1-p1`1)+1); A4: j+1 >= 1 & j+1 <= width G by A2,NAT_1:11,13; A5: 0+(1+1) <= len G by A3,NAT_1:13; then A6: q2`2 = q3`2 by A4,GOBOARD5:1; A7: q2`1 = G*(1,1)`1 by A3,A4,GOBOARD5:2 .= p1`1 by A1,A2,A3,GOBOARD5:2; then p1`1 < q3`1 by A5,A4,GOBOARD5:3; then A8: q3`1-p1`1 > 0 by XREAL_1:50; then 1 < 1/2*(q3`1-p1`1)+1 by XREAL_1:29,129; then A9: r < 1 by XREAL_1:212; set I1 = Int cell(G,0,j), I2 = Int cell(G,1,j); A10: LSeg(1/2*(p1+q2)-|[1,0]|,1/2*(p1+q2)) c= I1 \/ { 1/2*(p1+q2) } by A1,A2 ,Th44; A11: I1 \/ I2 \/ { 1/2*(p1+q2) } = I1 \/ (I2 \/ ({ 1/2*(p1+q2) } \/ { 1/2*( p1+q2) })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { 1/2*(p1+q2) } \/ { 1/2*(p1+q2) }) by XBOOLE_1:4 .= I1 \/ { 1/2*(p1+q2) } \/ (I2 \/ { 1/2*(p1+q2) }) by XBOOLE_1:4; A12: ((1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[1,0]|)`2 = ((1-r)*((1/2)*q2)+r*( (1/2)*q3))`2-((1-r)*|[1,0]|)`2 by Lm2 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2-|[(1-r)*1,(1-r)*0]|`2 by EUCLID:58 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2-0 by EUCLID:52 .= ((1-r)*((1/2)*q2))`2+(r*((1/2)*q3))`2 by Lm1 .= (1-r)*((1/2)*q2)`2+(r*((1/2)*q3))`2 by Lm3 .= (1-r)*((1/2)*q2)`2+r*((1/2)*q3)`2 by Lm3 .= (1-r)*((1/2)*q2`2)+r*((1/2)*q3)`2 by Lm3 .= (1-r)*((1/2)*q2`2)+r*((1/2)*q2`2) by A6,Lm3 .= ((1/2)*q2)`2 by Lm3; A13: (1-r)*((1/2)*p1)+r*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*q3) = (1-r)*((1 /2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1)+r*((1/2)*q3) by RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+(r*((1/2)*p1)+r*((1/2)*q3)) by RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5 .= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5 .= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5 .= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p1+q3)) by RLVECT_1:def 5; A14: r*((1/2)*q3`1)-r*((1/2)*q2`1)+r = r*((1/2)*(q3`1-q2`1)+1) .= 1 by A7,A8,XCMPLX_1:106; ((1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[1,0]|)`1 = ((1-r)*((1/2)*q2)+r*( (1/2)*q3))`1-((1-r)*|[1,0]|)`1 by Lm2 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1-|[(1-r)*1,(1-r)*0]|`1 by EUCLID:58 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1-(1-r) by EUCLID:52 .= ((1-r)*((1/2)*q2))`1+(r*((1/2)*q3))`1-(1-r) by Lm1 .= (1-r)*((1/2)*q2)`1+(r*((1/2)*q3))`1-(1-r) by Lm3 .= (1-r)*((1/2)*q2)`1+r*((1/2)*q3)`1-(1-r) by Lm3 .= (1-r)*((1/2)*q2`1)+r*((1/2)*q3)`1-(1-r) by Lm3 .= (1-r)*((1/2)*q2`1)+r*((1/2)*q3`1)-(1-r) by Lm3 .= ((1/2)*q2)`1 by A14,Lm3; then A15: (1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[1,0]| = |[(1/2*q2)`1,(1/2* q2 )`2 ]| by A12,EUCLID:53 .= 1/2*q2 by EUCLID:53; 1/2*(p1+q2) = 1/2*p1+1/2*q2 by RLVECT_1:def 5 .= (1-r+r)*((1/2)*p1)+1/2*q2 by RLVECT_1:def 8 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+1/2*q2 by RLVECT_1:def 6 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ ((1-r)*((1/2)*q2)+r*((1/2)*q3))-(1-r)* |[1,0]| by A15,RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ (1-r)*((1/2)*q2)+r*((1/2)*q3)-(1-r)*|[ 1,0]| by RLVECT_1:def 3 .= (1-r)*((1/2)*(p1+q2))+(r*((1/2)*(p1+q3))-(1-r)*|[1,0]|) by A13, RLVECT_1:def 3 .= (1-r)*((1/2)*(p1+q2))+-((1-r)*|[1,0]|-r*((1/2)*(p1+q3))) by RLVECT_1:33 .= (1-r)*((1/2)*(p1+q2))-((1-r)*|[1,0]|-r*((1/2)*(p1+q3))) .= (1-r)*((1/2)*(p1+q2))-(1-r)*|[1,0]|+r*((1/2)*(p1+q3)) by RLVECT_1:29 .= (1-r)*((1/2)*(p1+q2)-|[1,0]|)+r*((1/2)*(p1+q3)) by RLVECT_1:34; then 1/2*(p1+q2) in LSeg(1/2*(p1+q2)-|[1,0]|,1/2*(p1+q3)) by A8,A9; then A16: LSeg(1/2*(p1+q2)-|[1,0]|,1/2*(p1+q3)) = LSeg(1/2*(p1+q2)-|[1,0]|,1/2*( p1+q2)) \/ LSeg(1/2*(p1+q2),1/2*(p1+q3)) by TOPREAL1:5; 0+1+1 = 0+(1+1); then LSeg(1/2*(p1+q2),1/2*(p1+q3)) c= I2 \/ { 1/2*(p1+q2) } by A1,A2,A3,Th40; hence thesis by A16,A10,A11,XBOOLE_1:13; end; theorem 1 <= j & j < width G & 1 < len G implies LSeg(1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|, 1/2*(G*(len G,j)+G*(len G -' 1,j+1))) c= Int cell(G,len G -' 1 ,j) \/ Int cell(G,len G,j) \/ { 1/2*(G*(len G,j)+G*(len G,j+1)) } proof assume that A1: 1 <= j and A2: j < width G and A3: 1 < len G; set I1 = Int cell(G,len G -' 1,j), I2 = Int cell(G,len G,j); set p1 = G*(len G,j), q2 = G*(len G,j+1), q3 = G*(len G -' 1,j+1), r = 1/(1/ 2*(p1`1-q3`1)+1); A4: len G -'1 + 1 = len G by A3,XREAL_1:235; then A5: 1 <= len G -'1 by A3,NAT_1:13; A6: len G -'1 < len G by A4,NAT_1:13; then G*(len G -' 1,j)+G*(len G,j+1) = G*(len G,j)+G* (len G -' 1,j+1) by A1,A2,A4 ,A5,Th11; then A7: LSeg(1/2*(p1+q2),1/2*(p1+q3)) c= I1 \/ { 1/2*(p1+q2) } by A1,A2,A4,A5,A6 ,Th42; A8: j+1 >= 1 & j+1 <= width G by A2,NAT_1:11,13; then A9: q2`2 = G*(1,j+1)`2 by A3,GOBOARD5:1 .= q3`2 by A5,A6,A8,GOBOARD5:1; A10: q2`1 = G*(len G,1)`1 by A3,A8,GOBOARD5:2 .= p1`1 by A1,A2,A3,GOBOARD5:2; then q3`1 < p1`1 by A5,A6,A8,GOBOARD5:3; then A11: p1`1-q3`1 > 0 by XREAL_1:50; then 1 < 1/2*(p1`1-q3`1)+1 by XREAL_1:29,129; then A12: r < 1 by XREAL_1:212; A13: ((1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[1,0]|)`2 = ((1-r)*((1/2)*q2)+r*( (1/2)*q3))`2+((1-r)*|[1,0]|)`2 by Lm1 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2+|[(1-r)*1,(1-r)*0]|`2 by EUCLID:58 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`2+0 by EUCLID:52 .= ((1-r)*((1/2)*q2))`2+(r*((1/2)*q3))`2 by Lm1 .= (1-r)*((1/2)*q2)`2+(r*((1/2)*q3))`2 by Lm3 .= (1-r)*((1/2)*q2)`2+r*((1/2)*q3)`2 by Lm3 .= (1-r)*((1/2)*q2`2)+r*((1/2)*q3)`2 by Lm3 .= (1-r)*((1/2)*q2`2)+r*((1/2)*q2`2) by A9,Lm3 .= ((1/2)*q2)`2 by Lm3; A14: I1 \/ I2 \/ { 1/2*(p1+q2) } = I1 \/ (I2 \/ ({ 1/2*(p1+q2) } \/ { 1/2*( p1+q2) })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { 1/2*(p1+q2) } \/ { 1/2*(p1+q2) }) by XBOOLE_1:4 .= I1 \/ { 1/2*(p1+q2) } \/ (I2 \/ { 1/2*(p1+q2) }) by XBOOLE_1:4; A15: (1-r)*((1/2)*p1)+r*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*q3) = (1-r)*((1 /2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1)+r*((1/2)*q3) by RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+(r*((1/2)*p1)+r*((1/2)*q3)) by RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+(1-r)*((1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5 .= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*p1+(1/2)*q3) by RLVECT_1:def 5 .= (1-r)*((1/2)*p1+(1/2)*q2)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5 .= (1-r)*((1/2)*(p1+q2))+r*((1/2)*(p1+q3)) by RLVECT_1:def 5; A16: r*((1/2)*q2`1)-r*((1/2)*q3`1)+r = r*((1/2)*(q2`1-q3`1)+1) .= 1 by A10,A11,XCMPLX_1:106; ((1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[1,0]|)`1 = ((1-r)*((1/2)*q2)+r*( (1/2)*q3))`1+((1-r)*|[1,0]|)`1 by Lm1 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1+|[(1-r)*1,(1-r)*0]|`1 by EUCLID:58 .= ((1-r)*((1/2)*q2)+r*((1/2)*q3))`1+(1-r) by EUCLID:52 .= ((1-r)*((1/2)*q2))`1+(r*((1/2)*q3))`1+(1-r) by Lm1 .= (1-r)*((1/2)*q2)`1+(r*((1/2)*q3))`1+(1-r) by Lm3 .= (1-r)*((1/2)*q2)`1+r*((1/2)*q3)`1+(1-r) by Lm3 .= (1-r)*((1/2)*q2`1)+r*((1/2)*q3)`1+(1-r) by Lm3 .= (1-r)*((1/2)*q2`1)+r*((1/2)*q3`1)+(1-r) by Lm3 .= ((1/2)*q2)`1 by A16,Lm3; then A17: (1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[1,0]| = |[(1/2*q2)`1,(1/2* q2 )`2 ]| by A13,EUCLID:53 .= 1/2*q2 by EUCLID:53; 1/2*(p1+q2) = 1/2*p1+1/2*q2 by RLVECT_1:def 5 .= (1-r+r)*((1/2)*p1)+1/2*q2 by RLVECT_1:def 8 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+1/2*q2 by RLVECT_1:def 6 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ ((1-r)*((1/2)*q2)+r*((1/2)*q3))+(1-r)* |[1,0]| by A17,RLVECT_1:def 3 .= (1-r)*((1/2)*p1)+r*((1/2)*p1)+ (1-r)*((1/2)*q2)+r*((1/2)*q3)+(1-r)*|[ 1,0]| by RLVECT_1:def 3 .= (1-r)*((1/2)*(p1+q2))+(1-r)*|[1,0]|+r*((1/2)*(p1+q3)) by A15, RLVECT_1:def 3 .= (1-r)*((1/2)*(p1+q2)+|[1,0]|)+r*((1/2)*(p1+q3)) by RLVECT_1:def 5; then 1/2*(p1+q2) in LSeg(1/2*(p1+q2)+|[1,0]|,1/2*(p1+q3)) by A11,A12; then A18: LSeg(1/2*(p1+q2)+|[1,0]|,1/2*(p1+q3)) = LSeg(1/2*(p1+q2)+|[1,0]|,1/2*( p1+q2)) \/ LSeg(1/2*(p1+q2),1/2*(p1+q3)) by TOPREAL1:5; LSeg(1/2*(p1+q2)+|[1,0]|,1/2*(p1+q2)) c= I2 \/ { 1/2*(p1+q2) } by A1,A2,Th45; hence thesis by A18,A7,A14,XBOOLE_1:13; end; Lm7: 1/2+1/2 = 1; theorem 1 < len G & 1 <= j & j+1 < width G implies LSeg(1/2*(G*(1,j)+G*(1,j+1) )-|[1,0]|,1/2*(G*(1,j+1)+G*(1,j+2))-|[1,0]|) c= Int cell(G,0,j) \/ Int cell(G,0 ,j+1) \/ { G*(1,j+1)-|[1,0]| } proof assume that A1: 1 < len G and A2: 1 <= j and A3: j+1 < width G; set p1 = G*(1,j), p2 = G*(1,j+1), q3 = G*(1,j+2), r = (p2`2-p1`2)/(q3`2-p1`2 ); A4: j+1+1 = j+(1+1); then A5: j+2 >= 1 by NAT_1:11; A6: j+(1+1) <= width G by A3,A4,NAT_1:13; set I1 = Int cell(G,0,j), I2 = Int cell(G,0,j+1); A7: I1 \/ I2 \/ { p2-|[1,0]| } = I1 \/ (I2 \/ ({ p2-|[1,0]| } \/ { p2-|[1,0 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { p2-|[1,0]| } \/ { p2-|[1,0]| }) by XBOOLE_1:4 .= I1 \/ { p2-|[1,0]| } \/ (I2 \/ { p2-|[1,0]| }) by XBOOLE_1:4; A8: LSeg(1/2*(p2+q3)-|[1,0]|,p2-|[1,0]|) c= I2 \/ { p2-|[1,0]| } by A3,A4,Th48, NAT_1:11; j < j+1 by XREAL_1:29; then p1`2 < p2`2 by A1,A2,A3,GOBOARD5:4; then A9: p2`2-p1`2 > 0 by XREAL_1:50; A10: j+1 >= 1 by NAT_1:11; then A11: p2`1 = G*(1,1)`1 by A1,A3,GOBOARD5:2 .= q3`1 by A1,A6,A5,GOBOARD5:2; j <= j+1 by NAT_1:11; then A12: j < width G by A3,XXREAL_0:2; then p1`1 = G*(1,1)`1 by A1,A2,GOBOARD5:2 .= p2`1 by A1,A3,A10,GOBOARD5:2; then A13: 1*p2`1 = (1-r)*p1`1+r*q3`1 by A11 .= ((1-r)*p1)`1+r*q3`1 by Lm3 .= ((1-r)*p1)`1+(r*q3)`1 by Lm3 .= ((1-r)*p1+r*q3)`1 by Lm1; j+1 < j+2 by XREAL_1:6; then p2`2 < q3`2 by A1,A10,A6,GOBOARD5:4; then A14: p2`2-p1`2 < q3`2-p1`2 by XREAL_1:9; then r*(q3`2-p1`2) = p2`2-p1`2 by A9,XCMPLX_1:87; then p2`2 = (1-r)*p1`2 +r*q3`2; then 1*p2`2 = ((1-r)*p1)`2+r*q3`2 by Lm3 .= ((1-r)*p1)`2+(r*q3)`2 by Lm3 .= ((1-r)*p1+(r*q3))`2 by Lm1; then A15: (1-r)*p1+r*q3 = |[p2`1,p2`2]| by A13,EUCLID:53 .= p2 by EUCLID:53; p2 = 1*p2 by RLVECT_1:def 8 .= 1/2*p2+1/2*p2 by Lm7,RLVECT_1:def 6 .= 1/2*(((1-r)+r)*p2) + 1/2*((1-r)*p1+r*q3) by A15,RLVECT_1:def 8 .= 1/2*((1-r)*p2+r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 6 .= 1/2*((1-r)*p2)+1/2*(r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 5 .= 1/2*((1-r)*p2)+1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3)) by RLVECT_1:def 5 .= 1/2*((1-r)*p2)+(1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3))) by RLVECT_1:def 3 .= 1/2*((1-r)*p2)+(1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3))) by RLVECT_1:def 3 .= 1/2*((1-r)*p2)+1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3 .= (1/2*((1-r)*p2)+1/2*((1-r)*p1))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 3 .= 1/2*((1-r)*p2+(1-r)*p1)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5 .= 1/2*((1-r)*(p1+p2))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5 .= 1/2*(1-r)*(p1+p2)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 7 .= 1/2*(1-r)*(p1+p2)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3 .= 1/2*(1-r)*(p1+p2)+1/2*(r*p2+r*q3) by RLVECT_1:def 5 .= 1/2*(1-r)*(p1+p2)+1/2*(r*(p2+q3)) by RLVECT_1:def 5; then A16: p2 = (1-r)*(1/2*(p1+p2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7 .= (1-r)*(1/2*(p1+p2))+1/2*r*(p2+q3) by RLVECT_1:def 7 .= (1-r)*((1/2)*(p1+p2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7; A17: (1-r)*(1/2*(p1+p2)-|[1,0]|)+r*(1/2*(p2+q3)-|[1,0]|) = (1-r)*(1/2*(p1+p2 ))-(1-r)*|[1,0]|+r*(1/2*(p2+q3)-|[1,0]|) by RLVECT_1:34 .= (1-r)*(1/2*(p1+p2))-(1-r)*|[1,0]|+(r*(1/2*(p2+q3))-r*|[1,0]|) by RLVECT_1:34 .= r*(1/2*(p2+q3))+((1-r)*(1/2*(p1+p2))-(1-r)*|[1,0]|)-r*|[1,0]| by RLVECT_1:def 3 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-(1-r)*|[1,0]|-r*|[1,0]| by RLVECT_1:def 3 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-((1-r)*|[1,0]|+r*|[1,0]|) by RLVECT_1:27 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-((1-r)+r)*|[1,0]| by RLVECT_1:def 6 .= p2-|[1,0]| by A16,RLVECT_1:def 8; r < 1 by A14,A9,XREAL_1:189; then p2-|[1,0]| in LSeg(1/2*(p1+p2)-|[1,0]|,1/2*(p2+q3)-|[1,0]|) by A14,A9 ,A17; then A18: LSeg(1/2*(p1+p2)-|[1,0]|,1/2*(p2+q3)-|[1,0]|) = LSeg(1/2*(p1+p2)-|[1,0 ]|,p2-|[1,0]|) \/ LSeg(p2-|[1,0]|,1/2*(p2+q3)-|[1,0]|) by TOPREAL1:5; LSeg(1/2*(p1+p2)-|[1,0]|,p2-|[1,0]|) c= I1 \/ { p2-|[1,0]| } by A2,A12,Th49; hence thesis by A18,A8,A7,XBOOLE_1:13; end; theorem 1 < len G & 1 <= j & j+1 < width G implies LSeg(1/2*(G*(len G,j)+G*( len G,j+1))+|[1,0]|, 1/2*(G*(len G,j+1)+G*(len G,j+2))+|[1,0]|) c= Int cell(G, len G,j) \/ Int cell(G,len G,j+1) \/ { G*(len G,j+1)+|[1,0]| } proof assume that A1: 1 < len G and A2: 1 <= j and A3: j+1 < width G; set p1 = G*(len G,j), p2 = G*(len G,j+1), q3 = G*(len G,j+2), r = (p2`2-p1`2 )/(q3`2-p1`2); A4: j+1+1 = j+(1+1); then A5: j+2 >= 1 by NAT_1:11; A6: j+(1+1) <= width G by A3,A4,NAT_1:13; set I1 = Int cell(G,len G,j), I2 = Int cell(G,len G,j+1); A7: I1 \/ I2 \/ { p2+|[1,0]| } = I1 \/ (I2 \/ ({ p2+|[1,0]| } \/ { p2+|[1,0 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { p2+|[1,0]| } \/ { p2+|[1,0]| }) by XBOOLE_1:4 .= I1 \/ { p2+|[1,0]| } \/ (I2 \/ { p2+|[1,0]| }) by XBOOLE_1:4; A8: LSeg(1/2*(p2+q3)+|[1,0]|,p2+|[1,0]|) c= I2 \/ { p2+|[1,0]| } by A3,A4,Th50, NAT_1:11; j < j+1 by XREAL_1:29; then p1`2 < p2`2 by A1,A2,A3,GOBOARD5:4; then A9: p2`2-p1`2 > 0 by XREAL_1:50; A10: j+1 >= 1 by NAT_1:11; then A11: p2`1 = G*(len G,1)`1 by A1,A3,GOBOARD5:2 .= q3`1 by A1,A6,A5,GOBOARD5:2; j <= j+1 by NAT_1:11; then A12: j < width G by A3,XXREAL_0:2; then p1`1 = G*(len G,1)`1 by A1,A2,GOBOARD5:2 .= p2`1 by A1,A3,A10,GOBOARD5:2; then A13: 1*p2`1 = (1-r)*p1`1+r*q3`1 by A11 .= ((1-r)*p1)`1+r*q3`1 by Lm3 .= ((1-r)*p1)`1+(r*q3)`1 by Lm3 .= ((1-r)*p1+r*q3)`1 by Lm1; j+1 < j+2 by XREAL_1:6; then p2`2 < q3`2 by A1,A10,A6,GOBOARD5:4; then A14: p2`2-p1`2 < q3`2-p1`2 by XREAL_1:9; then r*(q3`2-p1`2) = p2`2-p1`2 by A9,XCMPLX_1:87; then p2`2 = (1-r)*p1`2 +r*q3`2; then 1*p2`2 = ((1-r)*p1)`2+r*q3`2 by Lm3 .= ((1-r)*p1)`2+(r*q3)`2 by Lm3 .= ((1-r)*p1+(r*q3))`2 by Lm1; then A15: (1-r)*p1+r*q3 = |[p2`1,p2`2]| by A13,EUCLID:53 .= p2 by EUCLID:53; p2 = 1*p2 by RLVECT_1:def 8 .= 1/2*p2+1/2*p2 by Lm7,RLVECT_1:def 6 .= 1/2*(((1-r)+r)*p2) + 1/2*((1-r)*p1+r*q3) by A15,RLVECT_1:def 8 .= 1/2*((1-r)*p2+r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 6 .= 1/2*((1-r)*p2)+1/2*(r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 5 .= 1/2*((1-r)*p2)+1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3)) by RLVECT_1:def 5 .= 1/2*((1-r)*p2)+(1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3))) by RLVECT_1:def 3 .= 1/2*((1-r)*p2)+(1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3))) by RLVECT_1:def 3 .= 1/2*((1-r)*p2)+1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3 .= (1/2*((1-r)*p2)+1/2*((1-r)*p1))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 3 .= 1/2*((1-r)*p2+(1-r)*p1)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5 .= 1/2*((1-r)*(p1+p2))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5 .= 1/2*(1-r)*(p1+p2)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 7; then A16: p2 = 1/2*(1-r)*(p1+p2)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3 .= 1/2*(1-r)*(p1+p2)+1/2*(r*p2+r*q3) by RLVECT_1:def 5 .= 1/2*(1-r)*(p1+p2)+1/2*(r*(p2+q3)) by RLVECT_1:def 5 .= (1-r)*(1/2*(p1+p2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7 .= (1-r)*(1/2*(p1+p2))+1/2*r*(p2+q3) by RLVECT_1:def 7 .= (1-r)*((1/2)*(p1+p2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7; A17: (1-r)*(1/2*(p1+p2)+|[1,0]|)+r*(1/2*(p2+q3)+|[1,0]|) = (1-r)*(1/2*(p1+p2 ))+(1-r)*|[1,0]|+r*(1/2*(p2+q3)+|[1,0]|) by RLVECT_1:def 5 .= (1-r)*(1/2*(p1+p2))+(1-r)*|[1,0]|+(r*(1/2*(p2+q3))+r*|[1,0]|) by RLVECT_1:def 5 .= r*(1/2*(p2+q3))+((1-r)*(1/2*(p1+p2))+(1-r)*|[1,0]|)+r*|[1,0]| by RLVECT_1:def 3 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+(1-r)*|[1,0]|+r*|[1,0]| by RLVECT_1:def 3 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+((1-r)*|[1,0]|+r*|[1,0]|) by RLVECT_1:def 3 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+((1-r)+r)*|[1,0]| by RLVECT_1:def 6 .= p2+|[1,0]| by A16,RLVECT_1:def 8; r < 1 by A14,A9,XREAL_1:189; then p2+|[1,0]| in LSeg(1/2*(p1+p2)+|[1,0]|,1/2*(p2+q3)+|[1,0]|) by A14,A9 ,A17; then A18: LSeg(1/2*(p1+p2)+|[1,0]|,1/2*(p2+q3)+|[1,0]|) = LSeg(1/2*(p1+p2)+|[1,0 ]|,p2+|[1,0]|) \/ LSeg(p2+|[1,0]|,1/2*(p2+q3)+|[1,0]|) by TOPREAL1:5; LSeg(1/2*(p1+p2)+|[1,0]|,p2+|[1,0]|) c= I1 \/ { p2+|[1,0]| } by A2,A12,Th51; hence thesis by A18,A8,A7,XBOOLE_1:13; end; theorem 1 < width G & 1 <= i & i+1 < len G implies LSeg(1/2*(G*(i,1)+G*(i+1,1) )-|[0,1]|,1/2*(G*(i+1,1)+G*(i+2,1))-|[0,1]|) c= Int cell(G,i,0) \/ Int cell(G,i +1,0) \/ { G*(i+1,1)-|[0,1]| } proof assume that A1: 1 < width G and A2: 1 <= i and A3: i+1 < len G; set p1 = G*(i,1), p2 = G*(i+1,1), q3 = G*(i+2,1), r = (p2`1-p1`1)/(q3`1-p1`1 ); A4: i+1+1 = i+(1+1); then A5: i+2 >= 1 by NAT_1:11; A6: i+(1+1) <= len G by A3,A4,NAT_1:13; set I1 = Int cell(G,i,0), I2 = Int cell(G,i+1,0); A7: I1 \/ I2 \/ { p2-|[0,1]| } = I1 \/ (I2 \/ ({ p2-|[0,1]| } \/ { p2-|[0,1 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { p2-|[0,1]| } \/ { p2-|[0,1]| }) by XBOOLE_1:4 .= I1 \/ { p2-|[0,1]| } \/ (I2 \/ { p2-|[0,1]| }) by XBOOLE_1:4; A8: LSeg(1/2*(p2+q3)-|[0,1]|,p2-|[0,1]|) c= I2 \/ { p2-|[0,1]| } by A3,A4,Th52, NAT_1:11; i < i+1 by XREAL_1:29; then p1`1 < p2`1 by A1,A2,A3,GOBOARD5:3; then A9: p2`1-p1`1 > 0 by XREAL_1:50; A10: i+1 >= 1 by NAT_1:11; then A11: p2`2 = G*(1,1)`2 by A1,A3,GOBOARD5:1 .= q3`2 by A1,A6,A5,GOBOARD5:1; i <= i+1 by NAT_1:11; then A12: i < len G by A3,XXREAL_0:2; then p1`2 = G*(1,1)`2 by A1,A2,GOBOARD5:1 .= p2`2 by A1,A3,A10,GOBOARD5:1; then A13: 1*p2`2 = (1-r)*p1`2+r*q3`2 by A11 .= ((1-r)*p1)`2+r*q3`2 by Lm3 .= ((1-r)*p1)`2+(r*q3)`2 by Lm3 .= ((1-r)*p1+r*q3)`2 by Lm1; i+1 < i+2 by XREAL_1:6; then p2`1 < q3`1 by A1,A10,A6,GOBOARD5:3; then A14: p2`1-p1`1 < q3`1-p1`1 by XREAL_1:9; then r*(q3`1-p1`1) = p2`1-p1`1 by A9,XCMPLX_1:87; then p2`1 = (1-r)*p1`1 +r*q3`1; then 1*p2`1 = ((1-r)*p1)`1+r*q3`1 by Lm3 .= ((1-r)*p1)`1+(r*q3)`1 by Lm3 .= ((1-r)*p1+(r*q3))`1 by Lm1; then A15: (1-r)*p1+r*q3 = |[p2`1,p2`2]| by A13,EUCLID:53 .= p2 by EUCLID:53; p2 = 1*p2 by RLVECT_1:def 8 .= 1/2*p2+1/2*p2 by Lm7,RLVECT_1:def 6 .= 1/2*(((1-r)+r)*p2) + 1/2*((1-r)*p1+r*q3) by A15,RLVECT_1:def 8 .= 1/2*((1-r)*p2+r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 6 .= 1/2*((1-r)*p2)+1/2*(r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 5 .= 1/2*((1-r)*p2)+1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3)) by RLVECT_1:def 5 .= 1/2*((1-r)*p2)+(1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3))) by RLVECT_1:def 3 .= 1/2*((1-r)*p2)+(1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3))) by RLVECT_1:def 3 .= 1/2*((1-r)*p2)+1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3 .= (1/2*((1-r)*p2)+1/2*((1-r)*p1))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 3 .= 1/2*((1-r)*p2+(1-r)*p1)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5 .= 1/2*((1-r)*(p1+p2))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5 .= 1/2*(1-r)*(p1+p2)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 7 .= 1/2*(1-r)*(p1+p2)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3 .= 1/2*(1-r)*(p1+p2)+1/2*(r*p2+r*q3) by RLVECT_1:def 5; then A16: p2 = 1/2*(1-r)*(p1+p2)+1/2*(r*(p2+q3)) by RLVECT_1:def 5 .= (1-r)*(1/2*(p1+p2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7 .= (1-r)*(1/2*(p1+p2))+1/2*r*(p2+q3) by RLVECT_1:def 7 .= (1-r)*((1/2)*(p1+p2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7; A17: (1-r)*(1/2*(p1+p2)-|[0,1]|)+r*(1/2*(p2+q3)-|[0,1]|) = (1-r)*(1/2*(p1+p2 ))-(1-r)*|[0,1]|+r*(1/2*(p2+q3)-|[0,1]|) by RLVECT_1:34 .= (1-r)*(1/2*(p1+p2))-(1-r)*|[0,1]|+(r*(1/2*(p2+q3))-r*|[0,1]|) by RLVECT_1:34 .= r*(1/2*(p2+q3))+((1-r)*(1/2*(p1+p2))-(1-r)*|[0,1]|)-r*|[0,1]| by RLVECT_1:def 3 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-(1-r)*|[0,1]|-r*|[0,1]| by RLVECT_1:def 3 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-((1-r)*|[0,1]|+r*|[0,1]|) by RLVECT_1:27 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))-((1-r)+r)*|[0,1]| by RLVECT_1:def 6 .= p2-|[0,1]| by A16,RLVECT_1:def 8; r < 1 by A14,A9,XREAL_1:189; then p2-|[0,1]| in LSeg(1/2*(p1+p2)-|[0,1]|,1/2*(p2+q3)-|[0,1]|) by A14,A9 ,A17; then A18: LSeg(1/2*(p1+p2)-|[0,1]|,1/2*(p2+q3)-|[0,1]|) = LSeg(1/2*(p1+p2)-|[0,1 ]|,p2-|[0,1]|) \/ LSeg(p2-|[0,1]|,1/2*(p2+q3)-|[0,1]|) by TOPREAL1:5; LSeg(1/2*(p1+p2)-|[0,1]|,p2-|[0,1]|) c= I1 \/ { p2-|[0,1]| } by A2,A12,Th53; hence thesis by A18,A8,A7,XBOOLE_1:13; end; theorem 1 < width G & 1 <= i & i+1 < len G implies LSeg(1/2*(G*(i,width G)+G*( i+1,width G))+|[0,1]|, 1/2*(G*(i+1,width G)+G*(i+2,width G))+|[0,1]|) c= Int cell(G,i,width G) \/ Int cell(G,i+1,width G) \/ { G*(i+1,width G)+|[0,1]| } proof assume that A1: 1 < width G and A2: 1 <= i and A3: i+1 < len G; set p1 = G*(i,width G), p2 = G*(i+1,width G), q3 = G*(i+2,width G), r = (p2 `1-p1`1)/(q3`1-p1`1); A4: i+1+1 = i+(1+1); then A5: i+2 >= 1 by NAT_1:11; A6: i+(1+1) <= len G by A3,A4,NAT_1:13; set I1 = Int cell(G,i,width G), I2 = Int cell(G,i+1,width G); A7: I1 \/ I2 \/ { p2+|[0,1]| } = I1 \/ (I2 \/ ({ p2+|[0,1]| } \/ { p2+|[0,1 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { p2+|[0,1]| } \/ { p2+|[0,1]| }) by XBOOLE_1:4 .= I1 \/ { p2+|[0,1]| } \/ (I2 \/ { p2+|[0,1]| }) by XBOOLE_1:4; A8: LSeg(1/2*(p2+q3)+|[0,1]|,p2+|[0,1]|) c= I2 \/ { p2+|[0,1]| } by A3,A4,Th54, NAT_1:11; i < i+1 by XREAL_1:29; then p1`1 < p2`1 by A1,A2,A3,GOBOARD5:3; then A9: p2`1-p1`1 > 0 by XREAL_1:50; A10: i+1 >= 1 by NAT_1:11; then A11: p2`2 = G*(1,width G)`2 by A1,A3,GOBOARD5:1 .= q3`2 by A1,A6,A5,GOBOARD5:1; i <= i+1 by NAT_1:11; then A12: i < len G by A3,XXREAL_0:2; then p1`2 = G*(1,width G)`2 by A1,A2,GOBOARD5:1 .= p2`2 by A1,A3,A10,GOBOARD5:1; then A13: 1*p2`2 = (1-r)*p1`2+r*q3`2 by A11 .= ((1-r)*p1)`2+r*q3`2 by Lm3 .= ((1-r)*p1)`2+(r*q3)`2 by Lm3 .= ((1-r)*p1+r*q3)`2 by Lm1; i+1 < i+2 by XREAL_1:6; then p2`1 < q3`1 by A1,A10,A6,GOBOARD5:3; then A14: p2`1-p1`1 < q3`1-p1`1 by XREAL_1:9; then r*(q3`1-p1`1) = p2`1-p1`1 by A9,XCMPLX_1:87; then p2`1 = (1-r)*p1`1 +r*q3`1; then 1*p2`1 = ((1-r)*p1)`1+r*q3`1 by Lm3 .= ((1-r)*p1)`1+(r*q3)`1 by Lm3 .= ((1-r)*p1+(r*q3))`1 by Lm1; then A15: (1-r)*p1+r*q3 = |[p2`1,p2`2]| by A13,EUCLID:53 .= p2 by EUCLID:53; p2 = 1*p2 by RLVECT_1:def 8 .= 1/2*p2+1/2*p2 by Lm7,RLVECT_1:def 6 .= 1/2*(((1-r)+r)*p2) + 1/2*((1-r)*p1+r*q3) by A15,RLVECT_1:def 8 .= 1/2*((1-r)*p2+r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 6 .= 1/2*((1-r)*p2)+1/2*(r*p2) + 1/2*((1-r)*p1+r*q3) by RLVECT_1:def 5 .= 1/2*((1-r)*p2)+1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3)) by RLVECT_1:def 5 .= 1/2*((1-r)*p2)+(1/2*(r*p2) + (1/2*((1-r)*p1)+1/2*(r*q3))) by RLVECT_1:def 3 .= 1/2*((1-r)*p2)+(1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3))) by RLVECT_1:def 3 .= 1/2*((1-r)*p2)+1/2*((1-r)*p1)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3 .= (1/2*((1-r)*p2)+1/2*((1-r)*p1))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 3 .= 1/2*((1-r)*p2+(1-r)*p1)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5 .= 1/2*((1-r)*(p1+p2))+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 5 .= 1/2*(1-r)*(p1+p2)+1/2*(r*p2)+1/2*(r*q3) by RLVECT_1:def 7; then A16: p2 = 1/2*(1-r)*(p1+p2)+(1/2*(r*p2)+1/2*(r*q3)) by RLVECT_1:def 3 .= 1/2*(1-r)*(p1+p2)+1/2*(r*p2+r*q3) by RLVECT_1:def 5 .= 1/2*(1-r)*(p1+p2)+1/2*(r*(p2+q3)) by RLVECT_1:def 5 .= (1-r)*(1/2*(p1+p2))+1/2*(r*(p2+q3)) by RLVECT_1:def 7 .= (1-r)*(1/2*(p1+p2))+1/2*r*(p2+q3) by RLVECT_1:def 7 .= (1-r)*((1/2)*(p1+p2))+r*((1/2)*(p2+q3)) by RLVECT_1:def 7; A17: (1-r)*(1/2*(p1+p2)+|[0,1]|)+r*(1/2*(p2+q3)+|[0,1]|) = (1-r)*(1/2*(p1+p2 ))+(1-r)*|[0,1]|+r*(1/2*(p2+q3)+|[0,1]|) by RLVECT_1:def 5 .= (1-r)*(1/2*(p1+p2))+(1-r)*|[0,1]|+(r*(1/2*(p2+q3))+r*|[0,1]|) by RLVECT_1:def 5 .= r*(1/2*(p2+q3))+((1-r)*(1/2*(p1+p2))+(1-r)*|[0,1]|)+r*|[0,1]| by RLVECT_1:def 3 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+(1-r)*|[0,1]|+r*|[0,1]| by RLVECT_1:def 3 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+((1-r)*|[0,1]|+r*|[0,1]|) by RLVECT_1:def 3 .= r*(1/2*(p2+q3))+(1-r)*(1/2*(p1+p2))+((1-r)+r)*|[0,1]| by RLVECT_1:def 6 .= p2+|[0,1]| by A16,RLVECT_1:def 8; r < 1 by A14,A9,XREAL_1:189; then p2+|[0,1]| in LSeg(1/2*(p1+p2)+|[0,1]|,1/2*(p2+q3)+|[0,1]|) by A14,A9 ,A17; then A18: LSeg(1/2*(p1+p2)+|[0,1]|,1/2*(p2+q3)+|[0,1]|) = LSeg(1/2*(p1+p2)+|[0,1 ]|,p2+|[0,1]|) \/ LSeg(p2+|[0,1]|,1/2*(p2+q3)+|[0,1]|) by TOPREAL1:5; LSeg(1/2*(p1+p2)+|[0,1]|,p2+|[0,1]|) c= I1 \/ { p2+|[0,1]| } by A2,A12,Th55; hence thesis by A18,A8,A7,XBOOLE_1:13; end; theorem 1 < len G & 1 < width G implies LSeg(G*(1,1)-|[1,1]|,1/2*(G*(1,1)+G*(1 ,2))-|[1,0]|) c= Int cell(G,0,0) \/ Int cell(G,0,1) \/ { G*(1,1)-|[1,0]| } proof assume that A1: 1 < len G and A2: 1 < width G; set q2 = G*(1,1), q3 = G*(1,2), r = 1/(1/2*(q3`2-q2`2)+1); A3: 0+(1+1) <= width G by A2,NAT_1:13; then A4: q2`1 = q3`1 by A1,GOBOARD5:2; q2`2 < q3`2 by A1,A3,GOBOARD5:4; then A5: q3`2-q2`2 > 0 by XREAL_1:50; then 1 < 1/2*(q3`2-q2`2)+1 by XREAL_1:29,129; then A6: r < 1 by XREAL_1:212; A7: ((1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[1,0]|))`1 = ((1-r)*(q2-|[1,1]|))`1 +(r*(1/2*(q2+q3)-|[1,0]|))`1 by Lm1 .= (1-r)*(q2-|[1,1]|)`1+(r*(1/2*(q2+q3)-|[1,0]|))`1 by Lm3 .= (1-r)*(q2-|[1,1]|)`1+r*(1/2*(q2+q3)-|[1,0]|)`1 by Lm3 .= (1-r)*(q2`1-|[1,1]|`1)+r*(1/2*(q2+q3)-|[1,0]|)`1 by Lm2 .= (1-r)*(q2`1-|[1,1]|`1)+r*((1/2*(q2+q3))`1-|[1,0]|`1) by Lm2 .= (1-r)*(q2`1-1)+r*((1/2*(q2+q3))`1-|[1,0]|`1) by EUCLID:52 .= (1-r)*q2`1-(1-r)*1+r*((1/2*(q2+q3))`1-1) by EUCLID:52 .= (1-r)*q2`1+r*(1/2*(q2+q3))`1-((1-r)+r) .= (1-r)*q2`1+r*(1/2*(q2+q3)`1)-1 by Lm3 .= (1-r)*q2`1+r*(1/2*(q2`1+q2`1))-1 by A4,Lm1 .= q2`1-|[1,0]|`1 by EUCLID:52 .= (q2-|[1,0]|)`1 by Lm2; A8: r*((1/2)*q3`2)-r*((1/2)*q2`2)+r = r*((1/2)*(q3`2-q2`2)+1) .= 1 by A5,XCMPLX_1:106; ((1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[1,0]|))`2 = ((1-r)*(q2-|[1,1]|))`2 +(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm1 .= ((1-r)*q2-(1-r)*|[1,1]|)`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by RLVECT_1:34 .= ((1-r)*q2)`2-((1-r)*|[1,1]|)`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm2 .= ((1-r)*q2)`2-(1-r)*|[1,1]|`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm3 .= ((1-r)*q2)`2-(1-r)*1+(r*(1/2*(q2+q3)-|[1,0]|))`2 by EUCLID:52 .= (1-r)*q2`2-(1-r)*1+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm3 .= (1-r)*q2`2-(1-r)+r*(1/2*(q2+q3)-|[1,0]|)`2 by Lm3 .= (1-r)*q2`2-(1-r)+r*((1/2*(q2+q3))`2-|[1,0]|`2) by Lm2 .= (1-r)*q2`2-(1-r)+r*((1/2*(q2+q3))`2-0) by EUCLID:52 .= (1-r)*q2`2-(1-r)+r*(1/2*(q2+q3)`2) by Lm3 .= (1-r)*q2`2-(1-r)+r*(1/2*(q2`2+q3`2)) by Lm1 .= q2`2-0 by A8 .= q2`2-|[1,0]|`2 by EUCLID:52 .= (q2-|[1,0]|)`2 by Lm2; then (1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[1,0]|) = |[(q2-|[1,0]|)`1,(q2-|[1,0 ]|)`2]| by A7,EUCLID:53 .= q2-|[1,0]| by EUCLID:53; then q2-|[1,0]| in LSeg(q2-|[1,1]|,1/2*(q2+q3)-|[1,0]|) by A5,A6; then A9: LSeg(q2-|[1,1]|,1/2*(q2+q3)-|[1,0]|) = LSeg(q2-|[1,1]|,q2-|[1,0]|) \/ LSeg(q2-|[1,0]|,1/2*(q2+q3)-|[1,0]|) by TOPREAL1:5; set I1 = Int cell(G,0,0), I2 = Int cell(G,0,1); 0+1+1 = 0+(1+1); then A10: LSeg(q2-|[1,0]|,1/2*(q2+q3)-|[1,0]|) c= I2 \/ { q2-|[1,0]| } by A2,Th48; A11: I1 \/ I2 \/ { q2-|[1,0]| } = I1 \/ (I2 \/ ({ q2-|[1,0]| } \/ { q2-|[1,0 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { q2-|[1,0]| } \/ { q2-|[1,0]| }) by XBOOLE_1:4 .= I1 \/ { q2-|[1,0]| } \/ (I2 \/ { q2-|[1,0]| }) by XBOOLE_1:4; LSeg(q2-|[1,1]|,q2-|[1,0]|) c= I1 \/ { q2-|[1,0]| } by Th56; hence thesis by A9,A10,A11,XBOOLE_1:13; end; theorem 1 < len G & 1 < width G implies LSeg(G*(len G,1)+|[1,-1]|,1/2*(G*(len G,1)+G*(len G,2))+|[1,0]|) c= Int cell(G,len G,0) \/ Int cell(G,len G,1) \/ { G *(len G,1)+|[1,0]| } proof assume that A1: 1 < len G and A2: 1 < width G; set q2 = G*(len G,1), q3 = G*(len G,2), r = 1/(1/2*(q3`2-q2`2)+1); A3: 0+(1+1) <= width G by A2,NAT_1:13; then A4: q2`1 = q3`1 by A1,GOBOARD5:2; q2`2 < q3`2 by A1,A3,GOBOARD5:4; then A5: q3`2-q2`2 > 0 by XREAL_1:50; then 1 < 1/2*(q3`2-q2`2)+1 by XREAL_1:29,129; then A6: r < 1 by XREAL_1:212; A7: ((1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)+|[1,0]|))`1 = ((1-r)*(q2+|[1,-1]|)) `1+(r*(1/2*(q2+q3)+|[1,0]|))`1 by Lm1 .= (1-r)*(q2+|[1,-1]|)`1+(r*(1/2*(q2+q3)+|[1,0]|))`1 by Lm3 .= (1-r)*(q2+|[1,-1]|)`1+r*(1/2*(q2+q3)+|[1,0]|)`1 by Lm3 .= (1-r)*(q2`1+|[1,-1]|`1)+r*(1/2*(q2+q3)+|[1,0]|)`1 by Lm1 .= (1-r)*(q2`1+|[1,-1]|`1)+r*((1/2*(q2+q3))`1+|[1,0]|`1) by Lm1 .= (1-r)*(q2`1+1)+r*((1/2*(q2+q3))`1+|[1,0]|`1) by EUCLID:52 .= (1-r)*q2`1+(1-r)*1+r*((1/2*(q2+q3))`1+1) by EUCLID:52 .= (1-r)*q2`1+r*(1/2*(q2+q3))`1+((1-r)+r) .= (1-r)*q2`1+r*(1/2*(q2+q3)`1)+1 by Lm3 .= (1-r)*q2`1+r*(1/2*(q2`1+q2`1))+1 by A4,Lm1 .= q2`1+|[1,0]|`1 by EUCLID:52 .= (q2+|[1,0]|)`1 by Lm1; A8: r*((1/2)*q3`2)-r*((1/2)*q2`2)+r = r*((1/2)*(q3`2-q2`2)+1) .= 1 by A5,XCMPLX_1:106; ((1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)+|[1,0]|))`2 = ((1-r)*(q2+|[1,-1]|)) `2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm1 .= ((1-r)*q2+(1-r)*|[1,-1]|)`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by RLVECT_1:def 5 .= ((1-r)*q2)`2+((1-r)*|[1,-1]|)`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm1 .= ((1-r)*q2)`2+(1-r)*|[1,-1]|`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm3 .= ((1-r)*q2)`2+(1-r)*(-1)+(r*(1/2*(q2+q3)+|[1,0]|))`2 by EUCLID:52 .= (1-r)*q2`2+-(1-r)+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm3 .= (1-r)*q2`2-(1-r)+r*(1/2*(q2+q3)+|[1,0]|)`2 by Lm3 .= (1-r)*q2`2-(1-r)+r*((1/2*(q2+q3))`2+|[1,0]|`2) by Lm1 .= (1-r)*q2`2-(1-r)+r*((1/2*(q2+q3))`2+0) by EUCLID:52 .= (1-r)*q2`2-(1-r)+r*(1/2*(q2+q3)`2) by Lm3 .= (1-r)*q2`2-(1-r)+r*(1/2*(q2`2+q3`2)) by Lm1 .= q2`2+0 by A8 .= q2`2+|[1,0]|`2 by EUCLID:52 .= (q2+|[1,0]|)`2 by Lm1; then (1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)+|[1,0]|) = |[(q2+|[1,0]|)`1,(q2+|[1, 0]|)`2]| by A7,EUCLID:53 .= q2+|[1,0]| by EUCLID:53; then q2+|[1,0]| in LSeg(q2+|[1,-1]|,1/2*(q2+q3)+|[1,0]|) by A5,A6; then A9: LSeg(q2+|[1,-1]|,1/2*(q2+q3)+|[1,0]|) = LSeg(q2+|[1,-1]|,q2+|[1,0]|) \/ LSeg(q2+|[1,0]|,1/2*(q2+q3)+|[1,0]|) by TOPREAL1:5; set I1 = Int cell(G,len G,0), I2 = Int cell(G,len G,1); 0+1+1 = 0+(1+1); then A10: LSeg(q2+|[1,0]|,1/2*(q2+q3)+|[1,0]|) c= I2 \/ { q2+|[1,0]| } by A2,Th50; A11: I1 \/ I2 \/ { q2+|[1,0]| } = I1 \/ (I2 \/ ({ q2+|[1,0]| } \/ { q2+|[1,0 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { q2+|[1,0]| } \/ { q2+|[1,0]| }) by XBOOLE_1:4 .= I1 \/ { q2+|[1,0]| } \/ (I2 \/ { q2+|[1,0]| }) by XBOOLE_1:4; LSeg(q2+|[1,-1]|,q2+|[1,0]|) c= I1 \/ { q2+|[1,0]| } by Th57; hence thesis by A9,A10,A11,XBOOLE_1:13; end; theorem 1 < len G & 1 < width G implies LSeg(G*(1,width G)+|[-1,1]|,1/2*(G*(1, width G)+G* (1,width G -' 1))-|[1,0]|) c= Int cell(G,0,width G) \/ Int cell(G,0 ,width G -' 1) \/ { G*(1,width G)-|[1,0]| } proof assume that A1: 1 < len G and A2: 1 < width G; set q2 = G*(1,width G), q3 = G*(1,width G -' 1), r = 1/(1/2*(q2`2-q3`2)+1); A3: width G -' 1 + 1 = width G by A2,XREAL_1:235; then A4: width G -' 1 >= 1 by A2,NAT_1:13; A5: width G -'1 < width G by A3,NAT_1:13; then q3`2 < q2`2 by A1,A4,GOBOARD5:4; then A6: q2`2-q3`2 > 0 by XREAL_1:50; then 1 < 1/2*(q2`2-q3`2)+1 by XREAL_1:29,129; then A7: r < 1 by XREAL_1:212; A8: q2`1 = G*(1,1)`1 by A1,A2,GOBOARD5:2 .= q3`1 by A1,A4,A5,GOBOARD5:2; A9: ((1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)-|[1,0]|))`1 = ((1-r)*(q2+|[-1,1]|)) `1+(r*(1/2*(q2+q3)-|[1,0]|))`1 by Lm1 .= (1-r)*(q2+|[-1,1]|)`1+(r*(1/2*(q2+q3)-|[1,0]|))`1 by Lm3 .= (1-r)*(q2+|[-1,1]|)`1+r*(1/2*(q2+q3)-|[1,0]|)`1 by Lm3 .= (1-r)*(q2`1+|[-1,1]|`1)+r*(1/2*(q2+q3)-|[1,0]|)`1 by Lm1 .= (1-r)*(q2`1+|[-1,1]|`1)+r*((1/2*(q2+q3))`1-|[1,0]|`1) by Lm2 .= (1-r)*(q2`1+-1)+r*((1/2*(q2+q3))`1-|[1,0]|`1) by EUCLID:52 .= (1-r)*(q2`1-1)+r*((1/2*(q2+q3))`1-1) by EUCLID:52 .= (1-r)*q2`1+r*(1/2*(q2+q3))`1-1 .= (1-r)*q2`1+r*(1/2*(q2+q3)`1)-1 by Lm3 .= (1-r)*q2`1+r*(1/2*(q2`1+q2`1))-1 by A8,Lm1 .= q2`1-|[1,0]|`1 by EUCLID:52 .= (q2-|[1,0]|)`1 by Lm2; A10: r*((1/2)*q2`2)-r*((1/2)*q3`2)+r = r*((1/2)*(q2`2-q3`2)+1) .= 1 by A6,XCMPLX_1:106; ((1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)-|[1,0]|))`2 = ((1-r)*(q2+|[-1,1]|)) `2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm1 .= ((1-r)*q2+(1-r)*|[-1,1]|)`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by RLVECT_1:def 5 .= ((1-r)*q2)`2+((1-r)*|[-1,1]|)`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm1 .= ((1-r)*q2)`2+(1-r)*|[-1,1]|`2+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm3 .= ((1-r)*q2)`2+(1-r)*1+(r*(1/2*(q2+q3)-|[1,0]|))`2 by EUCLID:52 .= (1-r)*q2`2+(1-r)*1+(r*(1/2*(q2+q3)-|[1,0]|))`2 by Lm3 .= (1-r)*q2`2+(1-r)+r*(1/2*(q2+q3)-|[1,0]|)`2 by Lm3 .= (1-r)*q2`2+(1-r)+r*((1/2*(q2+q3))`2-|[1,0]|`2) by Lm2 .= (1-r)*q2`2+(1-r)+r*((1/2*(q2+q3))`2-0) by EUCLID:52 .= (1-r)*q2`2+(1-r)+r*(1/2*(q2+q3)`2) by Lm3 .= (1-r)*q2`2+(1-r)+r*(1/2*(q2`2+q3`2)) by Lm1 .= q2`2-0 by A10 .= q2`2-|[1,0]|`2 by EUCLID:52 .= (q2-|[1,0]|)`2 by Lm2; then (1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)-|[1,0]|) = |[(q2-|[1,0]|)`1,(q2-|[1, 0]|)`2]| by A9,EUCLID:53 .= q2-|[1,0]| by EUCLID:53; then q2-|[1,0]| in LSeg(q2+|[-1,1]|,1/2*(q2+q3)-|[1,0]|) by A6,A7; then A11: LSeg(q2+|[-1,1]|,1/2*(q2+q3)-|[1,0]|) = LSeg(q2+|[-1,1]|,q2-|[1,0]|) \/ LSeg(q2-|[1,0]|,1/2*(q2+q3)-|[1,0]|) by TOPREAL1:5; set I1 = Int cell(G,0,width G), I2 = Int cell(G,0,width G -' 1); A12: I1 \/ I2 \/ { q2-|[1,0]| } = I1 \/ (I2 \/ ({ q2-|[1,0]| } \/ { q2-|[1,0 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { q2-|[1,0]| } \/ { q2-|[1,0]| }) by XBOOLE_1:4 .= I1 \/ { q2-|[1,0]| } \/ (I2 \/ { q2-|[1,0]| }) by XBOOLE_1:4; A13: LSeg(q2+|[-1,1]|,q2-|[1,0]|) c= I1 \/ { q2-|[1,0]| } by Th58; LSeg(q2-|[1,0]|,1/2*(q2+q3)-|[1,0]|) c= I2 \/ { q2-|[1,0]| } by A3,A4,A5,Th49 ; hence thesis by A11,A13,A12,XBOOLE_1:13; end; theorem 1 < len G & 1 < width G implies LSeg(G*(len G,width G)+|[1,1]|, 1/2*(G *(len G,width G)+G*(len G,width G -' 1))+|[1,0]|) c= Int cell(G,len G,width G) \/ Int cell(G,len G,width G -' 1) \/ { G*(len G,width G)+|[1,0]| } proof assume that A1: 1 < len G and A2: 1 < width G; set q2 = G*(len G,width G), q3 = G*(len G,width G -' 1), r = 1/(1/2*(q2`2-q3 `2)+1); A3: width G -' 1 + 1 = width G by A2,XREAL_1:235; then A4: width G -' 1 >= 1 by A2,NAT_1:13; A5: width G -' 1 < width G by A3,NAT_1:13; then q3`2 < q2`2 by A1,A4,GOBOARD5:4; then A6: q2`2-q3`2 > 0 by XREAL_1:50; then 1 < 1/2*(q2`2-q3`2)+1 by XREAL_1:29,129; then A7: r < 1 by XREAL_1:212; A8: q2`1 = G*(len G,1)`1 by A1,A2,GOBOARD5:2 .= q3`1 by A1,A4,A5,GOBOARD5:2; A9: ((1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[1,0]|))`1 = ((1-r)*(q2+|[1,1]|))`1 +(r*(1/2*(q2+q3)+|[1,0]|))`1 by Lm1 .= (1-r)*(q2+|[1,1]|)`1+(r*(1/2*(q2+q3)+|[1,0]|))`1 by Lm3 .= (1-r)*(q2+|[1,1]|)`1+r*(1/2*(q2+q3)+|[1,0]|)`1 by Lm3 .= (1-r)*(q2`1+|[1,1]|`1)+r*(1/2*(q2+q3)+|[1,0]|)`1 by Lm1 .= (1-r)*(q2`1+|[1,1]|`1)+r*((1/2*(q2+q3))`1+|[1,0]|`1) by Lm1 .= (1-r)*(q2`1+1)+r*((1/2*(q2+q3))`1+|[1,0]|`1) by EUCLID:52 .= (1-r)*q2`1+(1-r)*1+r*((1/2*(q2+q3))`1+1) by EUCLID:52 .= (1-r)*q2`1+r*(1/2*(q2+q3))`1+((1-r)+r) .= (1-r)*q2`1+r*(1/2*(q2+q3)`1)+1 by Lm3 .= (1-r)*q2`1+r*(1/2*(q2`1+q2`1))+1 by A8,Lm1 .= q2`1+|[1,0]|`1 by EUCLID:52 .= (q2+|[1,0]|)`1 by Lm1; A10: r*((1/2)*q2`2)-r*((1/2)*q3`2)+r = r*((1/2)*(q2`2-q3`2)+1) .= 1 by A6,XCMPLX_1:106; ((1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[1,0]|))`2 = ((1-r)*(q2+|[1,1]|))`2 +(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm1 .= ((1-r)*q2+(1-r)*|[1,1]|)`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by RLVECT_1:def 5 .= ((1-r)*q2)`2+((1-r)*|[1,1]|)`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm1 .= ((1-r)*q2)`2+(1-r)*|[1,1]|`2+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm3 .= ((1-r)*q2)`2+(1-r)*1+(r*(1/2*(q2+q3)+|[1,0]|))`2 by EUCLID:52 .= (1-r)*q2`2+(1-r)+(r*(1/2*(q2+q3)+|[1,0]|))`2 by Lm3 .= (1-r)*q2`2+(1-r)+r*(1/2*(q2+q3)+|[1,0]|)`2 by Lm3 .= (1-r)*q2`2+(1-r)+r*((1/2*(q2+q3))`2+|[1,0]|`2) by Lm1 .= (1-r)*q2`2+(1-r)+r*((1/2*(q2+q3))`2+0) by EUCLID:52 .= (1-r)*q2`2+(1-r)+r*(1/2*(q2+q3)`2) by Lm3 .= (1-r)*q2`2+(1-r)+r*(1/2*(q2`2+q3`2)) by Lm1 .= q2`2+0 by A10 .= q2`2+|[1,0]|`2 by EUCLID:52 .= (q2+|[1,0]|)`2 by Lm1; then (1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[1,0]|) = |[(q2+|[1,0]|)`1,(q2+|[1,0 ]|)`2]| by A9,EUCLID:53 .= q2+|[1,0]| by EUCLID:53; then q2+|[1,0]| in LSeg(q2+|[1,1]|,1/2*(q2+q3)+|[1,0]|) by A6,A7; then A11: LSeg(q2+|[1,1]|,1/2*(q2+q3)+|[1,0]|) = LSeg(q2+|[1,1]|,q2+|[1,0]|) \/ LSeg(q2+|[1,0]|,1/2*(q2+q3)+|[1,0]|) by TOPREAL1:5; set I1 = Int cell(G,len G,width G), I2 = Int cell(G,len G,width G -' 1); A12: I1 \/ I2 \/ { q2+|[1,0]| } = I1 \/ (I2 \/ ({ q2+|[1,0]| } \/ { q2+|[1,0 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { q2+|[1,0]| } \/ { q2+|[1,0]| }) by XBOOLE_1:4 .= I1 \/ { q2+|[1,0]| } \/ (I2 \/ { q2+|[1,0]| }) by XBOOLE_1:4; A13: LSeg(q2+|[1,1]|,q2+|[1,0]|) c= I1 \/ { q2+|[1,0]| } by Th59; LSeg(q2+|[1,0]|,1/2*(q2+q3)+|[1,0]|) c= I2 \/ { q2+|[1,0]| } by A3,A4,A5,Th51 ; hence thesis by A11,A13,A12,XBOOLE_1:13; end; theorem 1 < width G & 1 < len G implies LSeg(G*(1,1)-|[1,1]|,1/2*(G*(1,1)+G*(2 ,1))-|[0,1]|) c= Int cell(G,0,0) \/ Int cell(G,1,0) \/ { G*(1,1)-|[0,1]| } proof assume that A1: 1 < width G and A2: 1 < len G; set q2 = G*(1,1), q3 = G*(2,1), r = 1/(1/2*(q3`1-q2`1)+1); A3: 0+(1+1) <= len G by A2,NAT_1:13; then A4: q2`2 = q3`2 by A1,GOBOARD5:1; q2`1 < q3`1 by A1,A3,GOBOARD5:3; then A5: q3`1-q2`1 > 0 by XREAL_1:50; then 1 < 1/2*(q3`1-q2`1)+1 by XREAL_1:29,129; then A6: r < 1 by XREAL_1:212; A7: ((1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[0,1]|))`2 = ((1-r)*(q2-|[1,1]|))`2 +(r*(1/2*(q2+q3)-|[0,1]|))`2 by Lm1 .= (1-r)*(q2-|[1,1]|)`2+(r*(1/2*(q2+q3)-|[0,1]|))`2 by Lm3 .= (1-r)*(q2-|[1,1]|)`2+r*(1/2*(q2+q3)-|[0,1]|)`2 by Lm3 .= (1-r)*(q2`2-|[1,1]|`2)+r*(1/2*(q2+q3)-|[0,1]|)`2 by Lm2 .= (1-r)*(q2`2-|[1,1]|`2)+r*((1/2*(q2+q3))`2-|[0,1]|`2) by Lm2 .= (1-r)*(q2`2-1)+r*((1/2*(q2+q3))`2-|[0,1]|`2) by EUCLID:52 .= (1-r)*q2`2-(1-r)*1+r*((1/2*(q2+q3))`2-1) by EUCLID:52 .= (1-r)*q2`2+r*(1/2*(q2+q3))`2-((1-r)+r) .= (1-r)*q2`2+r*(1/2*(q2+q3)`2)-1 by Lm3 .= (1-r)*q2`2+r*(1/2*(q2`2+q2`2))-1 by A4,Lm1 .= q2`2-|[0,1]|`2 by EUCLID:52 .= (q2-|[0,1]|)`2 by Lm2; A8: r*((1/2)*q3`1)-r*((1/2)*q2`1)+r = r*((1/2)*(q3`1-q2`1)+1) .= 1 by A5,XCMPLX_1:106; ((1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[0,1]|))`1 = ((1-r)*(q2-|[1,1]|))`1 +(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm1 .= ((1-r)*q2-(1-r)*|[1,1]|)`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by RLVECT_1:34 .= ((1-r)*q2)`1-((1-r)*|[1,1]|)`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm2 .= ((1-r)*q2)`1-(1-r)*|[1,1]|`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm3 .= ((1-r)*q2)`1-(1-r)*1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by EUCLID:52 .= (1-r)*q2`1-(1-r)*1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm3 .= (1-r)*q2`1-(1-r)+r*(1/2*(q2+q3)-|[0,1]|)`1 by Lm3 .= (1-r)*q2`1-(1-r)+r*((1/2*(q2+q3))`1-|[0,1]|`1) by Lm2 .= (1-r)*q2`1-(1-r)+r*((1/2*(q2+q3))`1-0) by EUCLID:52 .= (1-r)*q2`1-(1-r)+r*(1/2*(q2+q3)`1) by Lm3 .= (1-r)*q2`1-(1-r)+r*(1/2*(q2`1+q3`1)) by Lm1 .= q2`1-0 by A8 .= q2`1-|[0,1]|`1 by EUCLID:52 .= (q2-|[0,1]|)`1 by Lm2; then (1-r)*(q2-|[1,1]|)+r*(1/2*(q2+q3)-|[0,1]|) = |[(q2-|[0,1]|)`1,(q2-|[0,1 ]|)`2]| by A7,EUCLID:53 .= q2-|[0,1]| by EUCLID:53; then q2-|[0,1]| in LSeg(q2-|[1,1]|,1/2*(q2+q3)-|[0,1]|) by A5,A6; then A9: LSeg(q2-|[1,1]|,1/2*(q2+q3)-|[0,1]|) = LSeg(q2-|[1,1]|,q2-|[0,1]|) \/ LSeg(q2-|[0,1]|,1/2*(q2+q3)-|[0,1]|) by TOPREAL1:5; set I1 = Int cell(G,0,0), I2 = Int cell(G,1,0); 0+1+1 = 0+(1+1); then A10: LSeg(q2-|[0,1]|,1/2*(q2+q3)-|[0,1]|) c= I2 \/ { q2-|[0,1]| } by A2,Th52; A11: I1 \/ I2 \/ { q2-|[0,1]| } = I1 \/ (I2 \/ ({ q2-|[0,1]| } \/ { q2-|[0,1 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { q2-|[0,1]| } \/ { q2-|[0,1]| }) by XBOOLE_1:4 .= I1 \/ { q2-|[0,1]| } \/ (I2 \/ { q2-|[0,1]| }) by XBOOLE_1:4; LSeg(q2-|[1,1]|,q2-|[0,1]|) c= I1 \/ { q2-|[0,1]| } by Th60; hence thesis by A9,A10,A11,XBOOLE_1:13; end; theorem 1 < width G & 1 < len G implies LSeg(G*(1,width G)+|[-1,1]|,1/2*(G*(1, width G)+G*(2,width G))+|[0,1]|) c= Int cell(G,0,width G) \/ Int cell(G,1,width G) \/ { G*(1,width G)+|[0,1]| } proof assume that A1: 1 < width G and A2: 1 < len G; set q2 = G*(1,width G), q3 = G*(2,width G), r = 1/(1/2*(q3`1-q2`1)+1); A3: 0+(1+1) <= len G by A2,NAT_1:13; then A4: q2`2 = q3`2 by A1,GOBOARD5:1; q2`1 < q3`1 by A1,A3,GOBOARD5:3; then A5: q3`1-q2`1 > 0 by XREAL_1:50; then 1 < 1/2*(q3`1-q2`1)+1 by XREAL_1:29,129; then A6: r < 1 by XREAL_1:212; A7: ((1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)+|[0,1]|))`2 = ((1-r)*(q2+|[-1,1]|)) `2+(r*(1/2*(q2+q3)+|[0,1]|))`2 by Lm1 .= (1-r)*(q2+|[-1,1]|)`2+(r*(1/2*(q2+q3)+|[0,1]|))`2 by Lm3 .= (1-r)*(q2+|[-1,1]|)`2+r*(1/2*(q2+q3)+|[0,1]|)`2 by Lm3 .= (1-r)*(q2`2+|[-1,1]|`2)+r*(1/2*(q2+q3)+|[0,1]|)`2 by Lm1 .= (1-r)*(q2`2+|[-1,1]|`2)+r*((1/2*(q2+q3))`2+|[0,1]|`2) by Lm1 .= (1-r)*(q2`2+1)+r*((1/2*(q2+q3))`2+|[0,1]|`2) by EUCLID:52 .= (1-r)*q2`2+(1-r)*1+r*((1/2*(q2+q3))`2+1) by EUCLID:52 .= (1-r)*q2`2+r*(1/2*(q2+q3))`2+((1-r)+r) .= (1-r)*q2`2+r*(1/2*(q2+q3)`2)+1 by Lm3 .= (1-r)*q2`2+r*(1/2*(q2`2+q2`2))+1 by A4,Lm1 .= q2`2+|[0,1]|`2 by EUCLID:52 .= (q2+|[0,1]|)`2 by Lm1; A8: r*((1/2)*q3`1)-r*((1/2)*q2`1)+r = r*((1/2)*(q3`1-q2`1)+1) .= 1 by A5,XCMPLX_1:106; ((1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)+|[0,1]|))`1 = ((1-r)*(q2+|[-1,1]|)) `1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm1 .= ((1-r)*q2+(1-r)*|[-1,1]|)`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by RLVECT_1:def 5 .= ((1-r)*q2)`1+((1-r)*|[-1,1]|)`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm1 .= ((1-r)*q2)`1+(1-r)*|[-1,1]|`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm3 .= ((1-r)*q2)`1+(1-r)*(-1)+(r*(1/2*(q2+q3)+|[0,1]|))`1 by EUCLID:52 .= ((1-r)*q2)`1-(1-r)+(r*(1/2*(q2+q3)+|[0,1]|))`1 .= (1-r)*q2`1-(1-r)+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm3 .= (1-r)*q2`1-(1-r)+r*(1/2*(q2+q3)+|[0,1]|)`1 by Lm3 .= (1-r)*q2`1-(1-r)+r*((1/2*(q2+q3))`1+|[0,1]|`1) by Lm1 .= (1-r)*q2`1-(1-r)+r*((1/2*(q2+q3))`1+0) by EUCLID:52 .= (1-r)*q2`1-(1-r)+r*(1/2*(q2+q3)`1) by Lm3 .= (1-r)*q2`1-(1-r)+r*(1/2*(q2`1+q3`1)) by Lm1 .= q2`1+0 by A8 .= q2`1+|[0,1]|`1 by EUCLID:52 .= (q2+|[0,1]|)`1 by Lm1; then (1-r)*(q2+|[-1,1]|)+r*(1/2*(q2+q3)+|[0,1]|) = |[(q2+|[0,1]|)`1,(q2+|[0, 1]|)`2]| by A7,EUCLID:53 .= q2+|[0,1]| by EUCLID:53; then q2+|[0,1]| in LSeg(q2+|[-1,1]|,1/2*(q2+q3)+|[0,1]|) by A5,A6; then A9: LSeg(q2+|[-1,1]|,1/2*(q2+q3)+|[0,1]|) = LSeg(q2+|[-1,1]|,q2+|[0,1]|) \/ LSeg(q2+|[0,1]|,1/2*(q2+q3)+|[0,1]|) by TOPREAL1:5; set I1 = Int cell(G,0,width G), I2 = Int cell(G,1,width G); 0+1+1 = 0+(1+1); then A10: LSeg(q2+|[0,1]|,1/2*(q2+q3)+|[0,1]|) c= I2 \/ { q2+|[0,1]| } by A2,Th54; A11: I1 \/ I2 \/ { q2+|[0,1]| } = I1 \/ (I2 \/ ({ q2+|[0,1]| } \/ { q2+|[0,1 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { q2+|[0,1]| } \/ { q2+|[0,1]| }) by XBOOLE_1:4 .= I1 \/ { q2+|[0,1]| } \/ (I2 \/ { q2+|[0,1]| }) by XBOOLE_1:4; LSeg(q2+|[-1,1]|,q2+|[0,1]|) c= I1 \/ { q2+|[0,1]| } by Th62; hence thesis by A9,A10,A11,XBOOLE_1:13; end; theorem 1 < width G & 1 < len G implies LSeg(G*(len G,1)+|[1,-1]|,1/2*(G*(len G,1)+G*(len G -' 1,1))-|[0,1]|) c= Int cell(G,len G,0) \/ Int cell(G,len G -' 1 ,0) \/ { G*(len G,1)-|[0,1]| } proof assume that A1: 1 < width G and A2: 1 < len G; set q2 = G*(len G,1), q3 = G*(len G -' 1,1), r = 1/(1/2*(q2`1-q3`1)+1); A3: len G -' 1 + 1 = len G by A2,XREAL_1:235; then A4: len G -' 1 >= 1 by A2,NAT_1:13; A5: len G -'1 < len G by A3,NAT_1:13; then q3`1 < q2`1 by A1,A4,GOBOARD5:3; then A6: q2`1-q3`1 > 0 by XREAL_1:50; then 1 < 1/2*(q2`1-q3`1)+1 by XREAL_1:29,129; then A7: r < 1 by XREAL_1:212; A8: q2`2 = G*(1,1)`2 by A1,A2,GOBOARD5:1 .= q3`2 by A1,A4,A5,GOBOARD5:1; A9: ((1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)-|[0,1]|))`2 = ((1-r)*(q2+|[1,-1]|)) `2+(r*(1/2*(q2+q3)-|[0,1]|))`2 by Lm1 .= (1-r)*(q2+|[1,-1]|)`2+(r*(1/2*(q2+q3)-|[0,1]|))`2 by Lm3 .= (1-r)*(q2+|[1,-1]|)`2+r*(1/2*(q2+q3)-|[0,1]|)`2 by Lm3 .= (1-r)*(q2`2+|[1,-1]|`2)+r*(1/2*(q2+q3)-|[0,1]|)`2 by Lm1 .= (1-r)*(q2`2+|[1,-1]|`2)+r*((1/2*(q2+q3))`2-|[0,1]|`2) by Lm2 .= (1-r)*(q2`2+-1)+r*((1/2*(q2+q3))`2-|[0,1]|`2) by EUCLID:52 .= (1-r)*(q2`2-1)+r*((1/2*(q2+q3))`2-1) by EUCLID:52 .= (1-r)*q2`2+r*(1/2*(q2+q3))`2-1 .= (1-r)*q2`2+r*(1/2*(q2+q3)`2)-1 by Lm3 .= (1-r)*q2`2+r*(1/2*(q2`2+q2`2))-1 by A8,Lm1 .= q2`2-|[0,1]|`2 by EUCLID:52 .= (q2-|[0,1]|)`2 by Lm2; A10: r*((1/2)*q2`1)-r*((1/2)*q3`1)+r = r*((1/2)*(q2`1-q3`1)+1) .= 1 by A6,XCMPLX_1:106; ((1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)-|[0,1]|))`1 = ((1-r)*(q2+|[1,-1]|)) `1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm1 .= ((1-r)*q2+(1-r)*|[1,-1]|)`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by RLVECT_1:def 5 .= ((1-r)*q2)`1+((1-r)*|[1,-1]|)`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm1 .= ((1-r)*q2)`1+(1-r)*|[1,-1]|`1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm3 .= ((1-r)*q2)`1+(1-r)*1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by EUCLID:52 .= (1-r)*q2`1+(1-r)*1+(r*(1/2*(q2+q3)-|[0,1]|))`1 by Lm3 .= (1-r)*q2`1+(1-r)+r*(1/2*(q2+q3)-|[0,1]|)`1 by Lm3 .= (1-r)*q2`1+(1-r)+r*((1/2*(q2+q3))`1-|[0,1]|`1) by Lm2 .= (1-r)*q2`1+(1-r)+r*((1/2*(q2+q3))`1-0) by EUCLID:52 .= (1-r)*q2`1+(1-r)+r*(1/2*(q2+q3)`1) by Lm3 .= (1-r)*q2`1+(1-r)+r*(1/2*(q2`1+q3`1)) by Lm1 .= q2`1-0 by A10 .= q2`1-|[0,1]|`1 by EUCLID:52 .= (q2-|[0,1]|)`1 by Lm2; then (1-r)*(q2+|[1,-1]|)+r*(1/2*(q2+q3)-|[0,1]|) = |[(q2-|[0,1]|)`1,(q2-|[0, 1]|)`2]| by A9,EUCLID:53 .= q2-|[0,1]| by EUCLID:53; then q2-|[0,1]| in LSeg(q2+|[1,-1]|,1/2*(q2+q3)-|[0,1]|) by A6,A7; then A11: LSeg(q2+|[1,-1]|,1/2*(q2+q3)-|[0,1]|) = LSeg(q2+|[1,-1]|,q2-|[0,1]|) \/ LSeg(q2-|[0,1]|,1/2*(q2+q3)-|[0,1]|) by TOPREAL1:5; set I1 = Int cell(G,len G,0), I2 = Int cell(G,len G -' 1,0); A12: I1 \/ I2 \/ { q2-|[0,1]| } = I1 \/ (I2 \/ ({ q2-|[0,1]| } \/ { q2-|[0,1 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { q2-|[0,1]| } \/ { q2-|[0,1]| }) by XBOOLE_1:4 .= I1 \/ { q2-|[0,1]| } \/ (I2 \/ { q2-|[0,1]| }) by XBOOLE_1:4; A13: LSeg(q2+|[1,-1]|,q2-|[0,1]|) c= I1 \/ { q2-|[0,1]| } by Th61; LSeg(q2-|[0,1]|,1/2*(q2+q3)-|[0,1]|) c= I2 \/ { q2-|[0,1]| } by A3,A4,A5,Th53 ; hence thesis by A11,A13,A12,XBOOLE_1:13; end; theorem 1 < width G & 1 < len G implies LSeg(G*(len G,width G)+|[1,1]|, 1/2*(G *(len G,width G)+G*(len G -' 1,width G))+|[0,1]|) c= Int cell(G,len G,width G) \/ Int cell(G,len G -' 1,width G) \/ { G*(len G,width G)+|[0,1]| } proof assume that A1: 1 < width G and A2: 1 < len G; set q2 = G*(len G,width G), q3 = G*(len G -' 1,width G), r = 1/(1/2*(q2`1-q3 `1)+1); A3: len G -' 1 + 1 = len G by A2,XREAL_1:235; then A4: len G -' 1 >= 1 by A2,NAT_1:13; A5: len G -' 1 < len G by A3,NAT_1:13; then q3`1 < q2`1 by A1,A4,GOBOARD5:3; then A6: q2`1-q3`1 > 0 by XREAL_1:50; then 1 < 1/2*(q2`1-q3`1)+1 by XREAL_1:29,129; then A7: r < 1 by XREAL_1:212; A8: q2`2 = G*(1,width G)`2 by A1,A2,GOBOARD5:1 .= q3`2 by A1,A4,A5,GOBOARD5:1; A9: ((1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[0,1]|))`2 = ((1-r)*(q2+|[1,1]|))`2 +(r*(1/2*(q2+q3)+|[0,1]|))`2 by Lm1 .= (1-r)*(q2+|[1,1]|)`2+(r*(1/2*(q2+q3)+|[0,1]|))`2 by Lm3 .= (1-r)*(q2+|[1,1]|)`2+r*(1/2*(q2+q3)+|[0,1]|)`2 by Lm3 .= (1-r)*(q2`2+|[1,1]|`2)+r*(1/2*(q2+q3)+|[0,1]|)`2 by Lm1 .= (1-r)*(q2`2+|[1,1]|`2)+r*((1/2*(q2+q3))`2+|[0,1]|`2) by Lm1 .= (1-r)*(q2`2+1)+r*((1/2*(q2+q3))`2+|[0,1]|`2) by EUCLID:52 .= (1-r)*q2`2+(1-r)*1+r*((1/2*(q2+q3))`2+1) by EUCLID:52 .= (1-r)*q2`2+r*(1/2*(q2+q3))`2+((1-r)+r) .= (1-r)*q2`2+r*(1/2*(q2+q3)`2)+1 by Lm3 .= (1-r)*q2`2+r*(1/2*(q2`2+q2`2))+1 by A8,Lm1 .= q2`2+|[0,1]|`2 by EUCLID:52 .= (q2+|[0,1]|)`2 by Lm1; A10: r*((1/2)*q2`1)-r*((1/2)*q3`1)+r = r*((1/2)*(q2`1-q3`1)+1) .= 1 by A6,XCMPLX_1:106; ((1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[0,1]|))`1 = ((1-r)*(q2+|[1,1]|))`1 +(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm1 .= ((1-r)*q2+(1-r)*|[1,1]|)`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by RLVECT_1:def 5 .= ((1-r)*q2)`1+((1-r)*|[1,1]|)`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm1 .= ((1-r)*q2)`1+(1-r)*|[1,1]|`1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm3 .= ((1-r)*q2)`1+(1-r)*1+(r*(1/2*(q2+q3)+|[0,1]|))`1 by EUCLID:52 .= (1-r)*q2`1+(1-r)+(r*(1/2*(q2+q3)+|[0,1]|))`1 by Lm3 .= (1-r)*q2`1+(1-r)+r*(1/2*(q2+q3)+|[0,1]|)`1 by Lm3 .= (1-r)*q2`1+(1-r)+r*((1/2*(q2+q3))`1+|[0,1]|`1) by Lm1 .= (1-r)*q2`1+(1-r)+r*((1/2*(q2+q3))`1+0) by EUCLID:52 .= (1-r)*q2`1+(1-r)+r*(1/2*(q2+q3)`1) by Lm3 .= (1-r)*q2`1+(1-r)+r*(1/2*(q2`1+q3`1)) by Lm1 .= q2`1+0 by A10 .= q2`1+|[0,1]|`1 by EUCLID:52 .= (q2+|[0,1]|)`1 by Lm1; then (1-r)*(q2+|[1,1]|)+r*(1/2*(q2+q3)+|[0,1]|) = |[(q2+|[0,1]|)`1,(q2+|[0,1 ]|)`2]| by A9,EUCLID:53 .= q2+|[0,1]| by EUCLID:53; then q2+|[0,1]| in LSeg(q2+|[1,1]|,1/2*(q2+q3)+|[0,1]|) by A6,A7; then A11: LSeg(q2+|[1,1]|,1/2*(q2+q3)+|[0,1]|) = LSeg(q2+|[1,1]|,q2+|[0,1]|) \/ LSeg(q2+|[0,1]|,1/2*(q2+q3)+|[0,1]|) by TOPREAL1:5; set I1 = Int cell(G,len G,width G), I2 = Int cell(G,len G -' 1,width G); A12: I1 \/ I2 \/ { q2+|[0,1]| } = I1 \/ (I2 \/ ({ q2+|[0,1]| } \/ { q2+|[0,1 ]| })) by XBOOLE_1:4 .= I1 \/ (I2 \/ { q2+|[0,1]| } \/ { q2+|[0,1]| }) by XBOOLE_1:4 .= I1 \/ { q2+|[0,1]| } \/ (I2 \/ { q2+|[0,1]| }) by XBOOLE_1:4; A13: LSeg(q2+|[1,1]|,q2+|[0,1]|) c= I1 \/ { q2+|[0,1]| } by Th63; LSeg(q2+|[0,1]|,1/2*(q2+q3)+|[0,1]|) c= I2 \/ { q2+|[0,1]| } by A3,A4,A5,Th55 ; hence thesis by A11,A13,A12,XBOOLE_1:13; end; theorem 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies LSeg(1/2*(G*(i ,j)+G*(i+1,j+1)),p) meets Int cell(G,i,j) proof assume A1: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G; now take a = 1/2*(G*(i,j)+G*(i+1,j+1)); thus a in LSeg(1/2*(G*(i,j)+G*(i+1,j+1)),p) by RLTOPSP1:68; thus a in Int cell(G,i,j) by A1,Th31; end; hence thesis by XBOOLE_0:3; end; theorem 1 <= i & i+1 <= len G implies LSeg(p,1/2*(G*(i,width G)+G*(i+1,width G ))+|[0,1]|) meets Int cell(G,i,width G) proof assume A1: 1 <= i & i+1 <= len G; now take a = 1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|; thus a in LSeg(p,1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]|) by RLTOPSP1:68; thus a in Int cell(G,i,width G) by A1,Th32; end; hence thesis by XBOOLE_0:3; end; theorem 1 <= i & i+1 <= len G implies LSeg(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|,p) meets Int cell(G,i,0) proof assume A1: 1 <= i & i+1 <= len G; now take a = 1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|; thus a in LSeg(1/2*(G*(i,1)+G*(i+1,1))-|[0,1]|,p) by RLTOPSP1:68; thus a in Int cell(G,i,0) by A1,Th33; end; hence thesis by XBOOLE_0:3; end; theorem 1 <= j & j+1 <= width G implies LSeg(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|,p ) meets Int cell(G,0,j) proof assume A1: 1 <= j & j+1 <= width G; now take a = 1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|; thus a in LSeg(1/2*(G*(1,j)+G*(1,j+1))-|[1,0]|,p) by RLTOPSP1:68; thus a in Int cell(G,0,j) by A1,Th35; end; hence thesis by XBOOLE_0:3; end; theorem 1 <= j & j+1 <= width G implies LSeg(p,1/2*(G*(len G,j)+G*(len G,j+1)) +|[1,0]|) meets Int cell(G,len G,j) proof assume A1: 1 <= j & j+1 <= width G; now take a = 1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|; thus a in LSeg(p,1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]|) by RLTOPSP1:68; thus a in Int cell(G,len G,j) by A1,Th34; end; hence thesis by XBOOLE_0:3; end; theorem LSeg(p,G*(1,1)-|[1,1]|) meets Int cell(G,0,0) proof now take a = G*(1,1)-|[1,1]|; thus a in LSeg(p,G*(1,1)-|[1,1]|) by RLTOPSP1:68; thus a in Int cell(G,0,0) by Th36; end; hence thesis by XBOOLE_0:3; end; theorem LSeg(p,G*(len G,width G)+|[1,1]|) meets Int cell(G,len G,width G) proof now take a = G*(len G,width G)+|[1,1]|; thus a in LSeg(p,G*(len G,width G)+|[1,1]|) by RLTOPSP1:68; thus a in Int cell(G,len G,width G) by Th37; end; hence thesis by XBOOLE_0:3; end; theorem LSeg(p,G*(1,width G)+|[-1,1]|) meets Int cell(G,0,width G) proof now take a = G*(1,width G)+|[-1,1]|; thus a in LSeg(p,G*(1,width G)+|[-1,1]|) by RLTOPSP1:68; thus a in Int cell(G,0,width G) by Th38; end; hence thesis by XBOOLE_0:3; end; theorem LSeg(p,G*(len G,1)+|[1,-1]|) meets Int cell(G,len G,0) proof now take a = G*(len G,1)+|[1,-1]|; thus a in LSeg(p,G*(len G,1)+|[1,-1]|) by RLTOPSP1:68; thus a in Int cell(G,len G,0) by Th39; end; hence thesis by XBOOLE_0:3; end; theorem Th91: for M being non empty MetrSpace, p being Point of M, q being Point of TopSpaceMetr M, r being Real st p = q & r > 0 holds Ball (p, r) is a_neighborhood of q proof let M be non empty MetrSpace, p be Point of M, q be Point of TopSpaceMetr M, r be Real; reconsider A = Ball (p, r) as Subset of TopSpaceMetr M by TOPMETR:12; assume p = q & r > 0; then q in A by Th1; hence thesis by CONNSP_2:3,TOPMETR:14; end; theorem for M being non empty MetrSpace, A being Subset of TopSpaceMetr M, p being Point of M holds p in Cl A iff for r being Real st r > 0 holds Ball (p, r) meets A proof let M be non empty MetrSpace, A be Subset of TopSpaceMetr M, p be Point of M; reconsider p9 = p as Point of TopSpaceMetr M by TOPMETR:12; hereby assume A1: p in Cl A; let r be Real; reconsider B = Ball (p, r) as Subset of TopSpaceMetr M by TOPMETR:12; assume r > 0; then B is a_neighborhood of p9 by Th91; hence Ball (p, r) meets A by A1,CONNSP_2:27; end; assume A2: for r being Real st r > 0 holds Ball (p, r) meets A; for G being a_neighborhood of p9 holds G meets A proof let G be a_neighborhood of p9; p in Int G by CONNSP_2:def 1; then ex r being Real st r > 0 & Ball (p, r) c= G by Th4; hence thesis by A2,XBOOLE_1:63; end; hence thesis by CONNSP_2:27; end; :: Moved from JORDAN19, AG 20.01.2006 theorem for A be Subset of TOP-REAL n for p be Point of TOP-REAL n for p9 be Point of Euclid n st p = p9 for s be Real st s > 0 holds p in Cl A iff for r be Real st 0 < r & r < s holds Ball (p9,r) meets A proof let A be Subset of TOP-REAL n; let p be Point of TOP-REAL n; let p9 be Point of Euclid n; assume A1: p = p9; let s be Real; assume A2: s > 0; hereby assume A3: p in Cl A; let r be Real; assume that A4: 0 < r and r < s; reconsider B = Ball (p9,r) as Subset of TOP-REAL n by TOPREAL3:8; B is a_neighborhood of p by A1,A4,Th2; hence Ball (p9, r) meets A by A3,CONNSP_2:27; end; assume A5: for r be Real st 0 < r & r < s holds Ball (p9,r) meets A; for G be a_neighborhood of p holds G meets A proof let G be a_neighborhood of p; p in Int G by CONNSP_2:def 1; then consider r9 be Real such that A6: r9 > 0 and A7: Ball (p9,r9) c= G by A1,Th5; set r = min(r9,s/2); Ball (p9,r) c= Ball (p9,r9) by PCOMPS_1:1,XXREAL_0:17; then A8: Ball (p9,r) c= G by A7; s/2 < s & r <= s/2 by A2,XREAL_1:216,XXREAL_0:17; then A9: r < s by XXREAL_0:2; s/2 > 0 by A2,XREAL_1:215; then r > 0 by A6,XXREAL_0:15; hence thesis by A5,A8,A9,XBOOLE_1:63; end; hence thesis by CONNSP_2:27; end;