### The Mizar article:

### Cages - the External Approximation of Jordan's Curve

**by****Czeslaw Bylinski, and****Mariusz Zynel**

- Received June 22, 1999
Copyright (c) 1999 Association of Mizar Users

- MML identifier: JORDAN9
- [ MML identifier index ]

environ vocabulary FINSEQ_1, BOOLE, PRE_TOPC, RELAT_2, CONNSP_1, RELAT_1, FINSEQ_5, ARYTM_1, GOBOARD1, EUCLID, MATRIX_1, ABSVALUE, GOBRD13, FUNCT_1, TOPREAL1, RFINSEQ, GOBOARD5, TOPS_1, TREES_1, SPPOL_1, MCART_1, TARSKI, SUBSET_1, SEQM_3, GOBOARD9, COMPTS_1, JORDAN8, PSCOMP_1, GROUP_1, ARYTM_3, SPRECT_2, CARD_1, PCOMPS_1, METRIC_1, JORDAN9, FINSEQ_4, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, GOBOARD5, STRUCT_0, ORDINAL1, NUMBERS, XREAL_0, REAL_1, NAT_1, BINARITH, ABSVALUE, RELAT_1, FUNCT_1, FUNCT_2, CARD_1, CARD_4, FINSEQ_1, FINSEQ_2, FINSEQ_4, FINSEQ_5, RFINSEQ, MATRIX_1, METRIC_1, PRE_TOPC, TOPS_1, COMPTS_1, CONNSP_1, PCOMPS_1, EUCLID, TOPREAL1, GOBOARD1, SPPOL_1, PSCOMP_1, SPRECT_2, GOBOARD9, JORDAN8, GOBRD13; constructors REAL_1, FINSEQ_4, CARD_4, RFINSEQ, BINARITH, TOPS_1, CONNSP_1, COMPTS_1, SPPOL_1, PSCOMP_1, REALSET1, SPRECT_2, GOBOARD9, JORDAN8, GOBRD13, GROUP_1, MEMBERED; clusters PSCOMP_1, RELSET_1, FINSEQ_1, SPPOL_2, REVROT_1, SPRECT_1, SPRECT_2, XREAL_0, GOBOARD9, JORDAN8, GOBRD13, EUCLID, TOPREAL1, NAT_1, MEMBERED, ZFMISC_1, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; definitions TARSKI, GOBOARD1, GOBOARD5, GOBRD13, XBOOLE_0; theorems NAT_1, FINSEQ_1, GOBOARD1, FINSEQ_4, EUCLID, FINSEQ_3, AXIOMS, REAL_1, REAL_2, HEINE, TOPREAL4, SPPOL_2, TARSKI, JORDAN3, SQUARE_1, PSCOMP_1, FINSEQ_5, FINSEQ_6, CQC_THE1, GOBOARD7, TOPREAL1, BINARITH, AMI_5, JORDAN5B, RLVECT_1, GOBOARD5, SPRECT_2, SPPOL_1, ABSVALUE, FUNCT_1, FUNCT_2, GROUP_5, GOBOARD9, RELAT_1, FINSEQ_2, UNIFORM1, SUBSET_1, GOBRD11, JORDAN4, GOBOARD2, SPRECT_3, CARD_1, RFINSEQ, GOBOARD6, TOPREAL3, TOPMETR, TOPS_1, JORDAN8, GOBRD13, SPRECT_4, CONNSP_1, PARTFUN2, RELSET_1, SCMFSA_7, SPRECT_1, XBOOLE_0, XBOOLE_1, XREAL_0, XCMPLX_0, XCMPLX_1; schemes NAT_1, LATTICE5; begin :: Generalities reserve i,j,k,n for Nat, D for non empty set, f, g for FinSequence of D; Lm1: for n st 1 <= n holds n-'1+2 = n+1 proof let n; assume 1 <= n; hence n-'1+2 = n+2-'1 by JORDAN4:3 .= n+(1+1) - 1 by JORDAN4:2 .= n+1+1 - 1 by XCMPLX_1:1 .= n+1 by XCMPLX_1:26; end; canceled 2; theorem Th3: for T being non empty TopSpace for B,C1,C2,D being Subset of T st B is connected & C1 is_a_component_of D & C2 is_a_component_of D & B meets C1 & B meets C2 & B c= D holds C1 = C2 proof let T be non empty TopSpace; let B,C1,C2,D be Subset of T; assume that A1: B is connected and A2: C1 is_a_component_of D and A3: C2 is_a_component_of D and A4: B meets C1 and A5: B meets C2 and A6: B c= D; A7: B c= C1 by A1,A2,A4,A6,GOBOARD9:6; A8: B c= C2 by A1,A3,A5,A6,GOBOARD9:6; B <> {} by A4,XBOOLE_1:65; then C1 meets C2 by A7,A8,XBOOLE_1:68; hence C1 = C2 by A2,A3,GOBOARD9:3; end; theorem Th4: (for n holds f|n = g|n) implies f = g proof assume A1: for n holds f|n = g|n; A2: now assume A3: len f <> len g; per cases by A3,REAL_1:def 5; suppose A4: len f < len g; then f|len g = f & g|len g = g by TOPREAL1:2; hence contradiction by A1,A4; suppose A5: len g < len f; then f|len f = f & g|len f = g by TOPREAL1:2; hence contradiction by A1,A5; end; f|len f = f & g|len g = g by TOPREAL1:2; hence thesis by A1,A2; end; theorem Th5: n in dom f implies ex k st k in dom Rev f & n+k = len f+1 & f/.n = (Rev f)/.k proof assume A1: n in dom f; then A2: 1 <= n & n <= len f by FINSEQ_3:27; take k = len f+1-'n; n <= len f+1 by A2,SPPOL_1:5; then A3: k+n = len f+1 by AMI_5:4; then A4: k+n-'1 = len f by BINARITH:39; n+1 <= len f+1 by A2,AXIOMS:24; then A5: 1 <= k by SPRECT_3:8; k+1 <= k+n by A2,AXIOMS:24; then k+1-'1 <= k+n -'1 by JORDAN3:5; then k <= len f by A4,BINARITH:39; then k in dom f by A5,FINSEQ_3:27; hence thesis by A1,A3,FINSEQ_5:60,69; end; theorem Th6: n in dom Rev f implies ex k st k in dom f & n+k = len f+1 & (Rev f)/.n = f/.k proof assume n in dom Rev f; then n in dom f by FINSEQ_5:60; then consider k such that A1: k in dom Rev f and A2: n+k = len f+1 and f/.n = (Rev f)/.k by Th5; A3: len f = len Rev f & dom f = dom Rev f by FINSEQ_5:60,def 3; then (Rev f)/.n = f/.k by A1,A2,FINSEQ_5:69; hence thesis by A1,A2,A3; end; begin :: Go-board preliminaries reserve G for Go-board, f, g for FinSequence of TOP-REAL 2, p for Point of TOP-REAL 2, r, s for Real, x for set; theorem Th7: for D being non empty set for G being Matrix of D for f being FinSequence of D holds f is_sequence_on G iff Rev f is_sequence_on G proof let D be non empty set; let G be Matrix of D; let f be FinSequence of D; hereby assume A1: f is_sequence_on G; A2:for n st n in dom Rev f ex i,j st [i,j] in Indices G & (Rev f)/.n = G*(i,j) proof let n; assume n in dom Rev f; then consider k such that A3: k in dom f and n+k = len f+1 and A4: (Rev f)/.n = f/.k by Th6; consider i,j such that A5: [i,j] in Indices G and A6: f/.k = G*(i,j) by A1,A3,GOBOARD1:def 11; take i,j; thus thesis by A4,A5,A6; end; for n st n in dom Rev f & n+1 in dom Rev f holds for m,k,i,j being Nat st [m,k] in Indices G & [i,j] in Indices G & (Rev f)/.n = G*(m,k) & (Rev f)/.(n+1) = G*(i,j) holds abs(m-i)+abs(k-j) = 1 proof let n such that A7: n in dom Rev f and A8: n+1 in dom Rev f; let m,k,i,j be Nat such that A9: [m,k] in Indices G and A10: [i,j] in Indices G and A11: (Rev f)/.n = G*(m,k) and A12: (Rev f)/.(n+1) = G*(i,j); consider l being Nat such that A13: l in dom f and A14: n+l = len f+1 and A15: (Rev f)/.n = f/.l by A7,Th6; consider l' being Nat such that A16: l' in dom f and A17: n+1+l' = len f+1 and A18: (Rev f)/.(n+1) = f/.l' by A8,Th6; n+(1+l') = n+l by A14,A17,XCMPLX_1:1; then A19: l'+1 = l by XCMPLX_1:2; abs(i-m) = abs(m-i) & abs(j-k) = abs(k-j) by UNIFORM1:13; hence abs(m-i)+abs(k-j) = 1 by A1,A9,A10,A11,A12,A13,A15,A16,A18,A19,GOBOARD1:def 11; end; hence Rev f is_sequence_on G by A2,GOBOARD1:def 11; end; assume A20: Rev f is_sequence_on G; A21:for n st n in dom f ex i,j st [i,j] in Indices G & f/.n = G*(i,j) proof let n; assume n in dom f; then consider k such that A22: k in dom Rev f and n+k = len f+1 and A23: f/.n = (Rev f)/.k by Th5; consider i,j such that A24: [i,j] in Indices G and A25: (Rev f)/.k = G*(i,j) by A20,A22,GOBOARD1:def 11; take i,j; thus thesis by A23,A24,A25; end; for n st n in dom f & n+1 in dom f holds for m,k,i,j being Nat st [m,k] in Indices G & [i,j] in Indices G & f/.n = G*(m,k) & f/.(n+1) = G*(i,j) holds abs(m-i)+abs(k-j) = 1 proof let n such that A26: n in dom f and A27: n+1 in dom f; let m,k,i,j be Nat such that A28: [m,k] in Indices G and A29: [i,j] in Indices G and A30: f/.n = G*(m,k) and A31: f/.(n+1) = G*(i,j); consider l being Nat such that A32: l in dom Rev f and A33: n+l = len f+1 and A34: f/.n = (Rev f)/.l by A26,Th5; consider l' being Nat such that A35: l' in dom Rev f and A36: n+1+l' = len f+1 and A37: f/.(n+1) = (Rev f)/.l' by A27,Th5; n+(1+l') = n+l by A33,A36,XCMPLX_1:1; then A38: l'+1 = l by XCMPLX_1:2; abs(i-m) = abs(m-i) & abs(j-k) = abs(k-j) by UNIFORM1:13; hence abs(m-i)+abs(k-j) = 1 by A20,A28,A29,A30,A31,A32,A34,A35,A37,A38,GOBOARD1:def 11; end; hence f is_sequence_on G by A21,GOBOARD1:def 11; end; theorem Th8: for G being Matrix of TOP-REAL 2 for f being FinSequence of TOP-REAL 2 holds f is_sequence_on G & 1 <= k & k <= len f implies f/.k in Values G proof let G be Matrix of TOP-REAL 2; let f be FinSequence of TOP-REAL 2; assume that A1: f is_sequence_on G and A2: 1 <= k and A3: k <= len f; A4: k in dom f by A2,A3,FINSEQ_3:27; then f/.k = f.k by FINSEQ_4:def 4; then A5: f/.k in rng f by A4,FUNCT_1:def 5; rng f c= Values G by A1,GOBRD13:9; hence f/.k in Values G by A5; end; Lm2: f is_sequence_on G & 1 <= k & k <= len f implies ex i,j being Nat st [i,j] in Indices G & f/.k = G*(i,j) proof assume that A1: f is_sequence_on G and A2: 1 <= k & k <= len f; k in dom f by A2,FINSEQ_3:27; then consider i,j such that A3: [i,j] in Indices G & f/.k = G*(i,j) by A1,GOBOARD1:def 11; take i,j; thus thesis by A3; end; theorem Th9: n <= len f & x in L~(f/^n) implies ex i being Nat st n+1 <= i & i+1 <= len f & x in LSeg(f,i) proof assume that A1: n <= len f and A2: x in L~(f/^n); consider j such that A3: 1 <= j and A4: j+1 <= len(f/^n) and A5: x in LSeg(f/^n,j) by A2,SPPOL_2:13; A6: j+1 <= len f - n by A1,A4,RFINSEQ:def 2; take n+j; j+1 <= len f - n by A1,A4,RFINSEQ:def 2; then n+(j+1) <= len f by REAL_1:84; hence thesis by A3,A5,A6,AXIOMS:24,SPPOL_2:5,XCMPLX_1:1; end; theorem Th10: f is_sequence_on G & 1 <= k & k+1 <= len f implies f/.k in left_cell(f,k,G) & f/.k in right_cell(f,k,G) proof assume that A1: f is_sequence_on G and A2: 1 <= k & k+1 <= len f; set p = f/.k; LSeg(f,k) = LSeg(f/.k, f/.(k+1)) by A2,TOPREAL1:def 5; then p in LSeg(f,k) by TOPREAL1:6; then p in left_cell(f,k,G) /\ right_cell(f,k,G) by A1,A2,GOBRD13:30; hence thesis by XBOOLE_0:def 3; end; theorem Th11: f is_sequence_on G & 1 <= k & k+1 <= len f implies Int left_cell(f,k,G) <> {} & Int right_cell(f,k,G) <> {} proof assume that A1: f is_sequence_on G and A2: 1 <= k and A3: k+1 <= len f; consider i1,j1,i2,j2 being Nat such that A4: [i1,j1] in Indices G and A5: f/.k = G*(i1,j1) and A6: [i2,j2] in Indices G and A7: f/.(k+1) = G*(i2,j2) and A8: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A1,A2,A3,JORDAN8:6; A9: i1+1 > i1 & j1+1 > j1 & i2+1 > i2 & j2+1 > j2 by NAT_1:38; A10: i1 <= len G & j1 <= width G by A4,GOBOARD5:1; A11: i2 <= len G & j2 <= width G by A6,GOBOARD5:1; A12: i1-'1 <= len G & j1-'1 <= width G by A10,JORDAN3:7; A13: i2-'1 <= len G & j2-'1 <= width G by A11,JORDAN3:7; per cases by A8; suppose A14: i1 = i2 & j1+1 = j2; then A15: right_cell(f,k,G) = cell(G,i1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1-'1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,A14,GOBRD13:def 3; hence thesis by A10,A12,A15,GOBOARD9:17; suppose A16: i1+1 = i2 & j1 = j2; then A17: right_cell(f,k,G) = cell(G,i1,j1-'1) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,A16,GOBRD13:def 3; hence thesis by A10,A12,A17,GOBOARD9:17; suppose A18: i1 = i2+1 & j1 = j2; then A19: right_cell(f,k,G) = cell(G,i2,j2) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i2,j2-'1) by A1,A2,A3,A4,A5,A6,A7,A9,A18,GOBRD13:def 3; hence thesis by A11,A13,A19,GOBOARD9:17; suppose A20: i1 = i2 & j1 = j2+1; then A21: right_cell(f,k,G) = cell(G,i1-'1,j2) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1,j2) by A1,A2,A3,A4,A5,A6,A7,A9,A20,GOBRD13:def 3; hence thesis by A10,A11,A12,A21,GOBOARD9:17; end; theorem Th12: f is_sequence_on G & 1 <= k & k+1 <= len f implies Int left_cell(f,k,G) is connected & Int right_cell(f,k,G) is connected proof assume that A1: f is_sequence_on G and A2: 1 <= k and A3: k+1 <= len f; consider i1,j1,i2,j2 being Nat such that A4: [i1,j1] in Indices G and A5: f/.k = G*(i1,j1) and A6: [i2,j2] in Indices G and A7: f/.(k+1) = G*(i2,j2) and A8: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A1,A2,A3,JORDAN8:6; A9: i1+1 > i1 & j1+1 > j1 & i2+1 > i2 & j2+1 > j2 by NAT_1:38; A10: i1 <= len G & j1 <= width G by A4,GOBOARD5:1; A11: i2 <= len G & j2 <= width G by A6,GOBOARD5:1; A12: i1-'1 <= len G & j1-'1 <= width G by A10,JORDAN3:7; A13: i2-'1 <= len G & j2-'1 <= width G by A11,JORDAN3:7; per cases by A8; suppose A14: i1 = i2 & j1+1 = j2; then A15: right_cell(f,k,G) = cell(G,i1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1-'1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,A14,GOBRD13:def 3; hence thesis by A10,A12,A15,GOBOARD9:21; suppose A16: i1+1 = i2 & j1 = j2; then A17: right_cell(f,k,G) = cell(G,i1,j1-'1) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,A16,GOBRD13:def 3; hence thesis by A10,A12,A17,GOBOARD9:21; suppose A18: i1 = i2+1 & j1 = j2; then A19: right_cell(f,k,G) = cell(G,i2,j2) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i2,j2-'1) by A1,A2,A3,A4,A5,A6,A7,A9,A18,GOBRD13:def 3; hence thesis by A11,A13,A19,GOBOARD9:21; suppose A20: i1 = i2 & j1 = j2+1; then A21: right_cell(f,k,G) = cell(G,i1-'1,j2) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1,j2) by A1,A2,A3,A4,A5,A6,A7,A9,A20,GOBRD13:def 3; hence thesis by A10,A11,A12,A21,GOBOARD9:21; end; theorem Th13: f is_sequence_on G & 1 <= k & k+1 <= len f implies Cl Int left_cell(f,k,G) = left_cell(f,k,G) & Cl Int right_cell(f,k,G) = right_cell(f,k,G) proof assume that A1: f is_sequence_on G and A2: 1 <= k and A3: k+1 <= len f; consider i1,j1,i2,j2 being Nat such that A4: [i1,j1] in Indices G and A5: f/.k = G*(i1,j1) and A6: [i2,j2] in Indices G and A7: f/.(k+1) = G*(i2,j2) and A8: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A1,A2,A3,JORDAN8:6; A9: i1+1 > i1 & j1+1 > j1 & i2+1 > i2 & j2+1 > j2 by NAT_1:38; A10: i1 <= len G & j1 <= width G by A4,GOBOARD5:1; A11: i2 <= len G & j2 <= width G by A6,GOBOARD5:1; A12: i1-'1 <= len G & j1-'1 <= width G by A10,JORDAN3:7; A13: i2-'1 <= len G & j2-'1 <= width G by A11,JORDAN3:7; per cases by A8; suppose A14: i1 = i2 & j1+1 = j2; then A15: right_cell(f,k,G) = cell(G,i1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1-'1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,A14,GOBRD13:def 3; hence thesis by A10,A12,A15,GOBRD11:35; suppose A16: i1+1 = i2 & j1 = j2; then A17: right_cell(f,k,G) = cell(G,i1,j1-'1) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,A16,GOBRD13:def 3; hence thesis by A10,A12,A17,GOBRD11:35; suppose A18: i1 = i2+1 & j1 = j2; then A19: right_cell(f,k,G) = cell(G,i2,j2) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i2,j2-'1) by A1,A2,A3,A4,A5,A6,A7,A9,A18,GOBRD13:def 3; hence thesis by A11,A13,A19,GOBRD11:35; suppose A20: i1 = i2 & j1 = j2+1; then A21: right_cell(f,k,G) = cell(G,i1-'1,j2) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1,j2) by A1,A2,A3,A4,A5,A6,A7,A9,A20,GOBRD13:def 3; hence thesis by A10,A11,A12,A21,GOBRD11:35; end; theorem Th14: f is_sequence_on G & LSeg(f,k) is horizontal implies ex j st 1 <= j & j <= width G & for p st p in LSeg(f,k) holds p`2 = G*(1,j)`2 proof assume that A1: f is_sequence_on G and A2: LSeg(f,k) is horizontal; per cases; suppose A3: 1 <= k & k+1 <= len f; k <= k+1 by NAT_1:29; then k <= len f by A3,AXIOMS:22; then consider i,j such that A4: [i,j] in Indices G and A5: f/.k = G*(i,j) by A1,A3,Lm2; take j; thus A6: 1 <= j & j <= width G by A4,GOBOARD5:1; let p; A7: 1 <= i & i <= len G by A4,GOBOARD5:1; A8: f/.k in LSeg(f,k) by A3,TOPREAL1:27; assume p in LSeg(f,k); hence p`2 = (f/.k)`2 by A2,A8,SPPOL_1:def 2 .= G*(1,j)`2 by A5,A6,A7,GOBOARD5:2; suppose A9: not(1 <= k & k+1 <= len f); take 1; width G <> 0 by GOBOARD1:def 5; hence 1 <= 1 & 1 <= width G by RLVECT_1:99; thus thesis by A9,TOPREAL1:def 5; end; theorem Th15: f is_sequence_on G & LSeg(f,k) is vertical implies ex i st 1 <= i & i <= len G & for p st p in LSeg(f,k) holds p`1 = G*(i,1)`1 proof assume that A1: f is_sequence_on G and A2: LSeg(f,k) is vertical; per cases; suppose A3: 1 <= k & k+1 <= len f; k <= k+1 by NAT_1:29; then k <= len f by A3,AXIOMS:22; then consider i,j such that A4: [i,j] in Indices G and A5: f/.k = G*(i,j) by A1,A3,Lm2; take i; thus A6: 1 <= i & i <= len G by A4,GOBOARD5:1; let p; A7: 1 <= j & j <= width G by A4,GOBOARD5:1; A8: f/.k in LSeg(f,k) by A3,TOPREAL1:27; assume p in LSeg(f,k); hence p`1 = (f/.k)`1 by A2,A8,SPPOL_1:def 3 .= G*(i,1)`1 by A5,A6,A7,GOBOARD5:3; suppose A9: not(1 <= k & k+1 <= len f); take 1; 0 <> len G by GOBOARD1:def 5; hence 1 <= 1 & 1 <= len G by RLVECT_1:99; thus thesis by A9,TOPREAL1:def 5; end; theorem Th16: f is_sequence_on G & f is special & i <= len G & j <= width G implies Int cell(G,i,j) misses L~f proof assume that A1: f is_sequence_on G and A2: f is special and A3: i <= len G and A4: j <= width G; assume Int cell(G,i,j) meets L~f; then consider x being set such that A5: x in Int cell(G,i,j) and A6: x in L~f by XBOOLE_0:3; L~f = union { LSeg(f,k) : 1 <= k & k+1 <= len f } by TOPREAL1:def 6; then consider X being set such that A7: x in X and A8: X in { LSeg(f,k) : 1 <= k & k+1 <= len f } by A6,TARSKI:def 4; consider k such that A9: X = LSeg(f,k) and 1 <= k & k+1 <= len f by A8; reconsider p = x as Point of TOP-REAL 2 by A7,A9; A10: Int cell(G,i,j) = Int(v_strip(G,i) /\ h_strip(G,j)) by GOBOARD5:def 3 .= Int v_strip(G,i) /\ Int h_strip(G,j) by TOPS_1:46; per cases by A2,SPPOL_1:41; suppose LSeg(f,k) is horizontal; then consider j0 being Nat such that A11: 1 <= j0 & j0 <= width G and A12: for p being Point of TOP-REAL 2 st p in LSeg(f,k) holds p`2 = G*(1,j0)`2 by A1,Th14; now assume A13: p in Int h_strip(G,j); A14: j0 > j implies j0 >= j+1 by NAT_1:38; per cases by A14,REAL_1:def 5; suppose A15: j0 < j; then j >= 1 by A11,AXIOMS:22; then A16: p`2 > G*(1,j)`2 by A4,A13,GOBOARD6:30; 0 <> len G by GOBOARD1:def 5; then 1 <= len G by RLVECT_1:99; then G*(1,j)`2 > G*(1,j0)`2 by A4,A11,A15,GOBOARD5:5; hence contradiction by A7,A9,A12,A16; suppose j0 = j; then p`2 > G*(1,j0)`2 by A11,A13,GOBOARD6:30; hence contradiction by A7,A9,A12; suppose A17: j0 > j+1; then j+1 <= width G by A11,AXIOMS:22; then j < width G by NAT_1:38; then A18: p`2 < G*(1,j+1)`2 by A13,GOBOARD6:31; 0 <> len G by GOBOARD1:def 5; then 1 <= len G & j+1 >= 1 by RLVECT_1:99; then G*(1,j+1)`2 < G*(1,j0)`2 by A11,A17,GOBOARD5:5; hence contradiction by A7,A9,A12,A18; suppose A19: j0 = j+1; then j < width G by A11,NAT_1:38; then p`2 < G*(1,j0)`2 by A13,A19,GOBOARD6:31; hence contradiction by A7,A9,A12; end; hence contradiction by A5,A10,XBOOLE_0:def 3; suppose LSeg(f,k) is vertical; then consider i0 being Nat such that A20: 1 <= i0 & i0 <= len G and A21: for p being Point of TOP-REAL 2 st p in LSeg(f,k) holds p`1 = G*(i0,1)`1 by A1,Th15; now assume A22: p in Int v_strip(G,i); A23: i0 > i implies i0 >= i+1 by NAT_1:38; per cases by A23,REAL_1:def 5; suppose A24: i0 < i; then i >= 1 by A20,AXIOMS:22; then A25: p`1 > G*(i,1)`1 by A3,A22,GOBOARD6:32; 0 <> width G by GOBOARD1:def 5; then 1 <= width G by RLVECT_1:99; then G*(i,1)`1 > G*(i0,1)`1 by A3,A20,A24,GOBOARD5:4; hence contradiction by A7,A9,A21,A25; suppose i0 = i; then p`1 > G*(i0,1)`1 by A20,A22,GOBOARD6:32; hence contradiction by A7,A9,A21; suppose A26: i0 > i+1; then i+1 <= len G by A20,AXIOMS:22; then i < len G by NAT_1:38; then A27: p`1 < G*(i+1,1)`1 by A22,GOBOARD6:33; 0 <> width G by GOBOARD1:def 5; then 1 <= width G & i+1 >= 1 by RLVECT_1:99; then G*(i+1,1)`1 < G*(i0,1)`1 by A20,A26,GOBOARD5:4; hence contradiction by A7,A9,A21,A27; suppose A28: i0 = i+1; then i < len G by A20,NAT_1:38; then p`1 < G*(i0,1)`1 by A22,A28,GOBOARD6:33; hence contradiction by A7,A9,A21; end; hence contradiction by A5,A10,XBOOLE_0:def 3; end; theorem Th17: f is_sequence_on G & f is special & 1 <= k & k+1 <= len f implies Int left_cell(f,k,G) misses L~f & Int right_cell(f,k,G) misses L~f proof assume that A1: f is_sequence_on G & f is special and A2: 1 <= k and A3: k+1 <= len f; consider i1,j1,i2,j2 being Nat such that A4: [i1,j1] in Indices G and A5: f/.k = G*(i1,j1) and A6: [i2,j2] in Indices G and A7: f/.(k+1) = G*(i2,j2) and A8: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A1,A2,A3,JORDAN8:6; A9: i1+1 > i1 & j1+1 > j1 & i2+1 > i2 & j2+1 > j2 by NAT_1:38; A10: i1 <= len G & j1 <= width G by A4,GOBOARD5:1; A11: i2 <= len G & j2 <= width G by A6,GOBOARD5:1; A12: i1-'1 <= len G & j1-'1 <= width G by A10,JORDAN3:7; A13: i2-'1 <= len G & j2-'1 <= width G by A11,JORDAN3:7; per cases by A8; suppose A14: i1 = i2 & j1+1 = j2; then A15: right_cell(f,k,G) = cell(G,i1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1-'1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,A14,GOBRD13:def 3; hence thesis by A1,A10,A12,A15,Th16; suppose A16: i1+1 = i2 & j1 = j2; then A17: right_cell(f,k,G) = cell(G,i1,j1-'1) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1,j1) by A1,A2,A3,A4,A5,A6,A7,A9,A16,GOBRD13:def 3; hence thesis by A1,A10,A12,A17,Th16; suppose A18: i1 = i2+1 & j1 = j2; then A19: right_cell(f,k,G) = cell(G,i2,j2) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i2,j2-'1) by A1,A2,A3,A4,A5,A6,A7,A9,A18,GOBRD13:def 3; hence thesis by A1,A11,A13,A19,Th16; suppose A20: i1 = i2 & j1 = j2+1; then A21: right_cell(f,k,G) = cell(G,i1-'1,j2) by A1,A2,A3,A4,A5,A6,A7,A9,GOBRD13:def 2; left_cell(f,k,G) = cell(G,i1,j2) by A1,A2,A3,A4,A5,A6,A7,A9,A20,GOBRD13:def 3; hence thesis by A1,A10,A11,A12,A21,Th16; end; theorem Th18: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies G*(i,j)`1 = G*(i,j+1)`1 & G*(i,j)`2 = G*(i+1,j)`2 & G*(i+1,j+1)`1 = G*(i+1,j)`1 & G*(i+1,j+1)`2 = G*(i,j+1)`2 proof assume that A1: 1 <= i and A2: i+1 <= len G and A3: 1 <= j and A4: j+1 <= width G; A5: i < len G by A2,NAT_1:38; A6: j < width G by A4,NAT_1:38; A7: 1 <= i+1 by NAT_1:29; A8: 1 <= j+1 by NAT_1:29; thus G*(i,j)`1 = G*(i,1)`1 by A1,A3,A5,A6,GOBOARD5:3 .= G*(i,j+1)`1 by A1,A4,A5,A8,GOBOARD5:3; thus G*(i,j)`2 = G*(1,j)`2 by A1,A3,A5,A6,GOBOARD5:2 .= G*(i+1,j)`2 by A2,A3,A6,A7,GOBOARD5:2; thus G*(i+1,j+1)`1 = G*(i+1,1)`1 by A2,A4,A7,A8,GOBOARD5:3 .= G*(i+1,j)`1 by A2,A3,A6,A7,GOBOARD5:3; thus G*(i+1,j+1)`2 = G*(1,j+1)`2 by A2,A4,A7,A8,GOBOARD5:2 .= G*(i,j+1)`2 by A1,A4,A5,A8,GOBOARD5:2; end; theorem Th19: for i,j being Nat st 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G holds p in cell(G,i,j) iff G*(i,j)`1 <= p`1 & p`1 <= G*(i+1,j)`1 & G*(i,j)`2 <= p`2 & p`2 <= G*(i,j+1)`2 proof let i,j be Nat such that A1: 1 <= i & i+1 <= len G and A2: 1 <= j & j+1 <= width G; A3: i < len G by A1,NAT_1:38; A4: j < width G by A2,NAT_1:38; then A5: v_strip(G,i) = { |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 } by A1, A2,A3,GOBOARD5:9; A6: h_strip(G,j) = { |[r,s]| : G*(i,j)`2 <= s & s <= G*(i,j+1)`2 } by A1,A2,A3, A4,GOBOARD5:6; hereby assume p in cell(G,i,j); then p in v_strip(G,i) /\ h_strip(G,j) by GOBOARD5:def 3; then A7: p in v_strip(G,i) & p in h_strip(G,j) by XBOOLE_0:def 3; then ex r,s st |[r,s]| = p & G*(i,j)`1 <= r & r <= G*(i+1,j)`1 by A5; hence G*(i,j)`1 <= p`1 & p`1 <= G*(i+1,j)`1 by EUCLID:56; ex r,s st |[r,s]| = p & G*(i,j)`2 <= s & s <= G*(i,j+1)`2 by A6,A7; hence G*(i,j)`2 <= p`2 & p`2 <= G*(i,j+1)`2 by EUCLID:56; end; assume that A8: G*(i,j)`1 <= p`1 & p`1 <= G*(i+1,j)`1 and A9: G*(i,j)`2 <= p`2 & p`2 <= G*(i,j+1)`2; A10: p = |[p`1,p`2]| by EUCLID:57; then A11: p in v_strip(G,i) by A5,A8; p in h_strip(G,j) by A6,A9,A10; then p in v_strip(G,i) /\ h_strip(G,j) by A11,XBOOLE_0:def 3; hence thesis by GOBOARD5:def 3; end; theorem 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies cell(G,i,j) = { |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 & G*(i,j)`2 <= s & s <= G*(i,j+1)`2 } proof assume that A1: 1 <= i & i+1 <= len G and A2: 1 <= j & j+1 <= width G; set A = { |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 & G*(i,j)`2 <= s & s <= G*(i,j+1)`2 }; now let p be set; assume A3: p in cell(G,i,j); then reconsider q=p as Point of TOP-REAL 2; A4: p = |[q`1,q`2]| by EUCLID:57; G*(i,j)`1 <= q`1 & q`1 <= G*(i+1,j)`1 & G*(i,j)`2 <= q`2 & q`2 <= G*(i,j+1)`2 by A1,A2,A3,Th19; hence p in A by A4; end; hence cell(G,i,j) c= A by TARSKI:def 3; now let p be set; assume p in A; then consider r,s such that A5: |[r,s]| = p and A6: G*(i,j)`1 <= r & r <= G*(i+1,j)`1 and A7: G*(i,j)`2 <= s & s <= G*(i,j+1)`2; reconsider q=p as Point of TOP-REAL 2 by A5; r = q`1 & s = q`2 by A5,EUCLID:56; hence p in cell(G,i,j) by A1,A2,A6,A7,Th19; end; hence A c= cell(G,i,j) by TARSKI:def 3; end; theorem Th21: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G & p in Values G & p in cell(G,i,j) implies p = G*(i,j) or p = G*(i,j+1) or p = G*(i+1,j+1) or p = G*(i+1,j) proof assume that A1: 1 <= i and A2: i+1 <= len G and A3: 1 <= j and A4: j+1 <= width G and A5: p in Values G and A6: p in cell(G,i,j); A7: Values G = { G*(k,l) where k,l is Nat: [k,l] in Indices G } by GOBRD13:7; A8: i < len G by A2,NAT_1:38; A9: j < width G by A4,NAT_1:38; A10: 1 <= i+1 by NAT_1:29; A11: 1 <= j+1 by NAT_1:29; consider k,l being Nat such that A12: p = G*(k,l) and A13: [k,l] in Indices G by A5,A7; A14: 1 <= k & k <= len G & 1 <= l & l <= width G by A13,GOBOARD5:1; A15: now assume A16: k <> i & k <> i+1; per cases by A16,NAT_1:27; suppose k < i; then G*(k,l)`1 < G*(i,l)`1 by A8,A14,GOBOARD5:4; then G*(k,l)`1 < G*(i,1)`1 by A1,A8,A14,GOBOARD5:3; then G*(k,l)`1 < G*(i,j)`1 by A1,A3,A8,A9,GOBOARD5:3; hence contradiction by A1,A2,A3,A4,A6,A12,Th19; suppose i+1 < k; then G*(i+1,l)`1 < G*(k,l)`1 by A10,A14,GOBOARD5:4; then G*(i+1,1)`1 < G*(k,l)`1 by A2,A10,A14,GOBOARD5:3; then G*(i+1,j)`1 < G*(k,l)`1 by A2,A3,A9,A10,GOBOARD5:3; hence contradiction by A1,A2,A3,A4,A6,A12,Th19; end; now assume A17: l <> j & l <> j+1; per cases by A17,NAT_1:27; suppose l < j; then G*(k,l)`2 < G*(k,j)`2 by A9,A14,GOBOARD5:5; then G*(k,l)`2 < G*(1,j)`2 by A3,A9,A14,GOBOARD5:2; then G*(k,l)`2 < G*(i,j)`2 by A1,A3,A8,A9,GOBOARD5:2; hence contradiction by A1,A2,A3,A4,A6,A12,Th19; suppose j+1 < l; then G*(k,j+1)`2 < G*(k,l)`2 by A11,A14,GOBOARD5:5; then G*(1,j+1)`2 < G*(k,l)`2 by A4,A11,A14,GOBOARD5:2; then G*(i,j+1)`2 < G*(k,l)`2 by A1,A4,A8,A11,GOBOARD5:2; hence contradiction by A1,A2,A3,A4,A6,A12,Th19; end; hence thesis by A12,A15; end; theorem Th22: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies G*(i,j) in cell(G,i,j) & G*(i,j+1) in cell(G,i,j) & G*(i+1,j+1) in cell(G,i,j) & G*(i+1,j) in 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: i < i+1 & j < j+1 by NAT_1:38; A6: i < len G by A2,NAT_1:38; A7: j < width G by A4,NAT_1:38; then A8: G*(i,j)`1 <= G*(i+1,j)`1 by A1,A2,A3,A5,GOBOARD5:4; G*(i,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,A5,A6,GOBOARD5:5; hence G*(i,j) in cell(G,i,j) by A1,A2,A3,A4,A8,Th19; A9: G*(i,j)`1 = G*(i,j+1)`1 by A1,A2,A3,A4,Th18; then A10: G*(i,j+1)`1 <= G*(i+1,j)`1 by A1,A2,A3,A5,A7,GOBOARD5:4; G*(i,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,A5,A6,GOBOARD5:5; hence G*(i,j+1) in cell(G,i,j) by A1,A2,A3,A4,A9,A10,Th19; A11: G*(i+1,j+1)`1 = G*(i+1,j)`1 by A1,A2,A3,A4,Th18; then A12: G*(i,j)`1 <= G*(i+1,j+1)`1 by A1,A2,A3,A5,A7,GOBOARD5:4; A13: G*(i+1,j+1)`2 = G*(i,j+1)`2 by A1,A2,A3,A4,Th18; then G*(i,j)`2 <= G*(i+1,j+1)`2 by A1,A3,A4,A5,A6,GOBOARD5:5; hence G*(i+1,j+1) in cell(G,i,j) by A1,A2,A3,A4,A11,A12,A13,Th19; A14: G*(i,j)`1 <= G*(i+1,j)`1 by A1,A2,A3,A5,A7,GOBOARD5:4; A15: G*(i,j)`2 = G*(i+1,j)`2 by A1,A2,A3,A4,Th18; then G*(i+1,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,A5,A6,GOBOARD5:5; hence G*(i+1,j) in cell(G,i,j) by A1,A2,A3,A4,A14,A15,Th19; end; theorem Th23: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G & p in Values G & p in cell(G,i,j) implies p is_extremal_in 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 and A5: p in Values G and A6: p in cell(G,i,j); for a,b being Point of TOP-REAL 2 st p in LSeg(a,b) & LSeg(a,b) c= cell(G,i,j) holds p = a or p = b proof let a,b be Point of TOP-REAL 2 such that A7: p in LSeg(a,b) and A8: LSeg(a,b) c= cell(G,i,j); assume A9: a <> p & b <> p; A10: a in LSeg(a,b) & b in LSeg(a,b) by TOPREAL1:6; per cases by A1,A2,A3,A4,A5,A6,Th21; suppose A11: p = G*(i,j); then A12: p`1 <= a`1 & p`2 <= a`2 by A1,A2,A3,A4,A8,A10,Th19; A13: p`1 <= b`1 & p`2 <= b`2 by A1,A2,A3,A4,A8,A10,A11,Th19; now per cases; suppose a`1 <= b`1 & a`2 <= b`2; then a`1 <= p`1 & a`2 <= p`2 by A7,TOPREAL1:9,10; then a`1 = p`1 & a`2 = p`2 by A12,REAL_1:def 5; hence contradiction by A9,TOPREAL3:11; suppose a`1 <= b`1 & b`2 < a`2; then a`1 <= p`1 & b`2 <= p`2 by A7,TOPREAL1:9,10; then A14: a`1 = p`1 & b`2 = p`2 by A12,A13,REAL_1:def 5; then a`2 <> p`2 by A9,TOPREAL3:11; then LSeg(a,b) is vertical by A7,A10,A14,SPPOL_1:39; then a`1 = b`1 by SPPOL_1:37; hence contradiction by A9,A14,TOPREAL3:11; suppose b`1 < a`1 & a`2 <= b`2; then b`1 <= p`1 & a`2 <= p`2 by A7,TOPREAL1:9,10; then A15: b`1 = p`1 & a`2 = p`2 by A12,A13,REAL_1:def 5; then b`2 <> p`2 by A9,TOPREAL3:11; then LSeg(a,b) is vertical by A7,A10,A15,SPPOL_1:39; then a`1 = b`1 by SPPOL_1:37; hence contradiction by A9,A15,TOPREAL3:11; suppose b`1 < a`1 & b`2 < a`2; then b`1 <= p`1 & b`2 <= p`2 by A7,TOPREAL1:9,10; then b`1 = p`1 & b`2 = p`2 by A13,REAL_1:def 5; hence contradiction by A9,TOPREAL3:11; end; hence contradiction; suppose A16: p = G*(i,j+1); then A17: p`1 = G*(i,j)`1 by A1,A2,A3,A4,Th18; then A18: p`1 <= a`1 & a`2 <= p`2 by A1,A2,A3,A4,A8,A10,A16,Th19; A19: p`1 <= b`1 & b`2 <= p`2 by A1,A2,A3,A4,A8,A10,A16,A17,Th19; now per cases; suppose a`1 <= b`1 & a`2 <= b`2; then a`1 <= p`1 & p`2 <= b`2 by A7,TOPREAL1:9,10; then A20: a`1 = p`1 & b`2 = p`2 by A18,A19,REAL_1:def 5; then a`2 <> p`2 by A9,TOPREAL3:11; then LSeg(a,b) is vertical by A7,A10,A20,SPPOL_1:39; then a`1 = b`1 by SPPOL_1:37; hence contradiction by A9,A20,TOPREAL3:11; suppose a`1 <= b`1 & b`2 < a`2; then a`1 <= p`1 & p`2 <= a`2 by A7,TOPREAL1:9,10; then a`1 = p`1 & a`2 = p`2 by A18,REAL_1:def 5; hence contradiction by A9,TOPREAL3:11; suppose b`1 < a`1 & a`2 <= b`2; then b`1 <= p`1 & p`2 <= b`2 by A7,TOPREAL1:9,10; then b`1 = p`1 & b`2 = p`2 by A19,REAL_1:def 5; hence contradiction by A9,TOPREAL3:11; suppose b`1 < a`1 & b`2 < a`2; then b`1 <= p`1 & p`2 <= a`2 by A7,TOPREAL1:9,10; then A21: b`1 = p`1 & a`2 = p`2 by A18,A19,REAL_1:def 5; then b`2 <> p`2 by A9,TOPREAL3:11; then LSeg(a,b) is vertical by A7,A10,A21,SPPOL_1:39; then a`1 = b`1 by SPPOL_1:37; hence contradiction by A9,A21,TOPREAL3:11; end; hence contradiction; suppose p = G*(i+1,j+1); then A22: p`1 = G*(i+1,j)`1 & p`2 = G*(i,j+1)`2 by A1,A2,A3,A4,Th18; then A23: a`1 <= p`1 & a`2 <= p`2 by A1,A2,A3,A4,A8,A10,Th19; A24: b`1 <= p`1 & b`2 <= p`2 by A1,A2,A3,A4,A8,A10,A22,Th19; now per cases; suppose a`1 <= b`1 & a`2 <= b`2; then p`1 <= b`1 & p`2 <= b`2 by A7,TOPREAL1:9,10; then b`1 = p`1 & b`2 = p`2 by A24,REAL_1:def 5; hence contradiction by A9,TOPREAL3:11; suppose a`1 <= b`1 & b`2 < a`2; then p`1 <= b`1 & p`2 <= a`2 by A7,TOPREAL1:9,10; then A25: b`1 = p`1 & a`2 = p`2 by A23,A24,REAL_1:def 5; then b`2 <> p`2 by A9,TOPREAL3:11; then LSeg(a,b) is vertical by A7,A10,A25,SPPOL_1:39; then a`1 = b`1 by SPPOL_1:37; hence contradiction by A9,A25,TOPREAL3:11; suppose b`1 < a`1 & a`2 <= b`2; then p`1 <= a`1 & p`2 <= b`2 by A7,TOPREAL1:9,10; then A26: a`1 = p`1 & b`2 = p`2 by A23,A24,REAL_1:def 5; then a`2 <> p`2 by A9,TOPREAL3:11; then LSeg(a,b) is vertical by A7,A10,A26,SPPOL_1:39; then a`1 = b`1 by SPPOL_1:37; hence contradiction by A9,A26,TOPREAL3:11; suppose b`1 < a`1 & b`2 < a`2; then p`1 <= a`1 & p`2 <= a`2 by A7,TOPREAL1:9,10; then a`1 = p`1 & a`2 = p`2 by A23,REAL_1:def 5; hence contradiction by A9,TOPREAL3:11; end; hence contradiction; suppose A27: p = G*(i+1,j); then A28: p`2 = G*(i,j)`2 by A1,A2,A3,A4,Th18; then A29: a`1 <= p`1 & p`2 <= a`2 by A1,A2,A3,A4,A8,A10,A27,Th19; A30: b`1 <= p`1 & p`2 <= b`2 by A1,A2,A3,A4,A8,A10,A27,A28,Th19; now per cases; suppose a`1 <= b`1 & a`2 <= b`2; then p`1 <= b`1 & a`2 <= p`2 by A7,TOPREAL1:9,10; then A31: b`1 = p`1 & a`2 = p`2 by A29,A30,REAL_1:def 5; then b`2 <> p`2 by A9,TOPREAL3:11; then LSeg(a,b) is vertical by A7,A10,A31,SPPOL_1:39; then a`1 = b`1 by SPPOL_1:37; hence contradiction by A9,A31,TOPREAL3:11; suppose a`1 <= b`1 & b`2 < a`2; then p`1 <= b`1 & b`2 <= p`2 by A7,TOPREAL1:9,10; then b`1 = p`1 & b`2 = p`2 by A30,REAL_1:def 5; hence contradiction by A9,TOPREAL3:11; suppose b`1 < a`1 & a`2 <= b`2; then p`1 <= a`1 & a`2 <= p`2 by A7,TOPREAL1:9,10; then a`1 = p`1 & a`2 = p`2 by A29,REAL_1:def 5; hence contradiction by A9,TOPREAL3:11; suppose b`1 < a`1 & b`2 < a`2; then p`1 <= a`1 & b`2 <= p`2 by A7,TOPREAL1:9,10; then A32: a`1 = p`1 & b`2 = p`2 by A29,A30,REAL_1:def 5; then a`2 <> p`2 by A9,TOPREAL3:11; then LSeg(a,b) is vertical by A7,A10,A32,SPPOL_1:39; then a`1 = b`1 by SPPOL_1:37; hence contradiction by A9,A32,TOPREAL3:11; end; hence contradiction; end; hence thesis by A6,SPPOL_1:def 1; end; theorem Th24: 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k+1 <= len f implies ex i,j st 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G & LSeg(f,k) c= cell(G,i,j) proof assume that A1: 2 <= len G and A2: 2 <= width G and A3: f is_sequence_on G and A4: 1 <= k and A5: k+1 <= len f; A6: LSeg(f,k) = LSeg(f/.k, f/.(k+1)) by A4,A5,TOPREAL1:def 5; consider i1,j1,i2,j2 being Nat such that A7: [i1,j1] in Indices G and A8: f/.k = G*(i1,j1) and A9: [i2,j2] in Indices G and A10: f/.(k+1) = G*(i2,j2) and A11: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A3,A4,A5,JORDAN8:6; A12: 1 <= i1 & i1 <= len G & 1 <= j1 & j1 <= width G by A7,GOBOARD5:1; A13: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A9,GOBOARD5:1; per cases by A11; suppose A14: i1 = i2 & j1+1 = j2; then A15: j1 < width G by A13,SPPOL_1:5; now per cases by A12,REAL_1:def 5; suppose i1 < len G; then A16: i1+1 <= len G by NAT_1:38; take i1,j1; LSeg(f,k) c= cell(G,i1,j1) by A6,A8,A10,A12,A14,A15,GOBOARD5:20; hence thesis by A12,A13,A14,A16; suppose A17: i1 = len G; take i1'=i1-'1,j1; 2-1 <= 2-'1 & 2-'1 <= i1' by A1,A17,JORDAN3:3,5; then A18: 1 <= i1' by AXIOMS:22; A19: i1'+1 = i1 by A12,AMI_5:4; then i1' < len G by A12,NAT_1:38; then LSeg(f,k) c= cell(G,i1',j1) by A6,A8,A10,A12,A14,A15,A19,GOBOARD5: 19; hence thesis by A12,A13,A14,A18,A19; end; hence thesis; suppose A20: i1+1 = i2 & j1 = j2; then A21: i1 < len G by A13,SPPOL_1:5; now per cases by A12,REAL_1:def 5; suppose j1 < width G; then A22: j1+1 <= width G by NAT_1:38; take i1,j1; LSeg(f,k) c= cell(G,i1,j1) by A6,A8,A10,A12,A20,A21,GOBOARD5:23; hence thesis by A12,A13,A20,A22; suppose A23: j1 = width G; take i1,j1'=j1-'1; 2-1 <= 2-'1 & 2-'1 <= j1' by A2,A23,JORDAN3:3,5; then A24: 1 <= j1' by AXIOMS:22; A25: j1'+1=j1 by A12,AMI_5:4; then j1' < width G by A23,NAT_1:38; then LSeg(f,k) c= cell(G,i1,j1') by A6,A8,A10,A12,A20,A21,A25,GOBOARD5: 22; hence thesis by A12,A13,A20,A24,A25; end; hence thesis; suppose A26: i1 = i2+1 & j1 = j2; then A27: i2 < len G by A12,SPPOL_1:5; now per cases by A12,REAL_1:def 5; suppose j1 < width G; then A28: j1+1 <= width G by NAT_1:38; take i2,j1; LSeg(f,k) c= cell(G,i2,j1) by A6,A8,A10,A13,A26,A27,GOBOARD5:23; hence thesis by A12,A13,A26,A28; suppose A29: j1 = width G; take i2,j1'=j1-'1; 2-1 <= 2-'1 & 2-'1 <= j1' by A2,A29,JORDAN3:3,5; then A30: 1 <= j1' by AXIOMS:22; A31: j1'+1=j1 by A12,AMI_5:4; then j1' < width G by A29,NAT_1:38; then LSeg(f,k) c= cell(G,i2,j1') by A6,A8,A10,A13,A26,A27,A31,GOBOARD5: 22; hence thesis by A12,A13,A26,A30,A31; end; hence thesis; suppose A32: i1 = i2 & j1 = j2+1; then A33: j2 < width G by A12,SPPOL_1:5; now per cases by A12,REAL_1:def 5; suppose i1 < len G; then A34: i1+1 <= len G by NAT_1:38; take i1,j2; LSeg(f,k) c= cell(G,i1,j2) by A6,A8,A10,A13,A32,A33,GOBOARD5:20; hence thesis by A12,A13,A32,A34; suppose A35: i1 = len G; take i1'=i1-'1,j2; 2-1 <= 2-'1 & 2-'1 <= i1' by A1,A35,JORDAN3:3,5; then A36: 1 <= i1' by AXIOMS:22; A37: i1'+1 = i1 by A12,AMI_5:4; then i1' < len G by A12,NAT_1:38; then LSeg(f,k) c= cell(G,i1',j2) by A6,A8,A10,A13,A32,A33,A37,GOBOARD5: 19; hence thesis by A12,A13,A32,A36,A37; end; hence thesis; end; theorem Th25: 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k+1 <= len f & p in Values G & p in LSeg(f,k) implies p = f/.k or p = f/.(k+1) proof assume that A1: 2 <= len G and A2: 2 <= width G and A3: f is_sequence_on G and A4: 1 <= k and A5: k+1 <= len f and A6: p in Values G and A7: p in LSeg(f,k); consider i,j such that A8: 1 <= i & i+1 <= len G and A9: 1 <= j & j+1 <= width G and A10: LSeg(f,k) c= cell(G,i,j) by A1,A2,A3,A4,A5,Th24; A11: LSeg(f,k) = LSeg(f/.k, f/.(k+1)) by A4,A5,TOPREAL1:def 5; p is_extremal_in cell(G,i,j) by A6,A7,A8,A9,A10,Th23; hence thesis by A7,A10,A11,SPPOL_1:def 1; end; theorem [i,j] in Indices G & 1 <= k & k <= width G implies G*(i,j)`1 <= G* (len G,k)`1 proof assume that A1: [i,j] in Indices G and A2: 1 <= k & k <= width G; A3: 1 <= i & i <= len G & 1 <= j & j <= width G by A1,GOBOARD5:1; then A4: G*(i,j)`1 = G*(i,1)`1 by GOBOARD5:3 .= G*(i,k)`1 by A2,A3,GOBOARD5:3; i < len G or i = len G by A3,REAL_1:def 5; hence thesis by A2,A3,A4,GOBOARD5:4; end; theorem [i,j] in Indices G & 1 <= k & k <= len G implies G*(i,j)`2 <= G* (k,width G)`2 proof assume that A1: [i,j] in Indices G and A2: 1 <= k & k <= len G; A3: 1 <= i & i <= len G & 1 <= j & j <= width G by A1,GOBOARD5:1; then A4: G*(i,j)`2 = G*(1,j)`2 by GOBOARD5:2 .= G*(k,j)`2 by A2,A3,GOBOARD5:2; j < width G or j = width G by A3,REAL_1:def 5; hence thesis by A2,A3,A4,GOBOARD5:5; end; theorem Th28: f is_sequence_on G & f is special & L~g c= L~f & 1 <= k & k+1 <= len f implies for A being Subset of TOP-REAL 2 st A = right_cell(f,k,G)\L~g or A = left_cell(f,k,G)\L~g holds A is connected proof assume that A1: f is_sequence_on G & f is special & L~g c= L~f and A2: 1 <= k and A3: k+1 <= len f; let A be Subset of TOP-REAL 2 such that A4: A = right_cell(f,k,G)\L~g or A = left_cell(f,k,G)\L~g; per cases by A4; suppose A5: A = right_cell(f,k,G)\L~g; then A6: A = right_cell(f,k,G) /\ (L~g)` by SUBSET_1:32; A7: A c= right_cell(f,k,G) by A5,XBOOLE_1:36; A8: Int right_cell(f,k,G) is connected by A1,A2,A3,Th12; Int right_cell(f,k,G) misses L~f by A1,A2,A3,Th17; then Int right_cell(f,k,G) misses L~g by A1,XBOOLE_1:63; then A9: Int right_cell(f,k,G) c= (L~g)` by SUBSET_1:43; Int right_cell(f,k,G) c= right_cell(f,k,G) by TOPS_1:44; then A10: Int right_cell(f,k,G) c= A by A6,A9,XBOOLE_1:19; A c= Cl Int right_cell(f,k,G) by A1,A2,A3,A7,Th13; hence A is connected by A8,A10,CONNSP_1:19; suppose A11: A = left_cell(f,k,G)\L~g; then A12: A = left_cell(f,k,G) /\ (L~g)` by SUBSET_1:32; A13: A c= left_cell(f,k,G) by A11,XBOOLE_1:36; A14: Int left_cell(f,k,G) is connected by A1,A2,A3,Th12; Int left_cell(f,k,G) misses L~f by A1,A2,A3,Th17; then Int left_cell(f,k,G) misses L~g by A1,XBOOLE_1:63; then A15: Int left_cell(f,k,G) c= (L~g)` by SUBSET_1:43; Int left_cell(f,k,G) c= left_cell(f,k,G) by TOPS_1:44; then A16: Int left_cell(f,k,G) c= A by A12,A15,XBOOLE_1:19; A c= Cl Int left_cell(f,k,G) by A1,A2,A3,A13,Th13; hence A is connected by A14,A16,CONNSP_1:19; end; theorem Th29: for f being non constant standard special_circular_sequence st f is_sequence_on G for k st 1 <= k & k+1 <= len f holds right_cell(f,k,G)\L~f c= RightComp f & left_cell(f,k,G)\L~f c= LeftComp f proof let f be non constant standard special_circular_sequence such that A1: f is_sequence_on G; let k such that A2: 1 <= k & k+1 <= len f; set rc = right_cell(f,k,G)\L~f; A3: rc = right_cell(f,k,G) /\ (L~f)` by SUBSET_1:32; A4: rc c= right_cell(f,k,G) by XBOOLE_1:36; right_cell(f,k,G) c= right_cell(f,k) by A1,A2,GOBRD13:34; then rc c= right_cell(f,k) by A4,XBOOLE_1:1; then A5: Int rc c= Int right_cell(f,k) by TOPS_1:48; Int right_cell(f,k) c= RightComp f by A2,GOBOARD9:28; then A6: Int rc c= RightComp f by A5,XBOOLE_1:1; A7: Int right_cell(f,k,G) <> {} by A1,A2,Th11; A8: Int right_cell(f,k,G) misses L~f by A1,A2,Th17; A9: Int right_cell(f,k,G) c= right_cell(f,k,G) by TOPS_1:44; rc \/ L~f = right_cell(f,k,G) \/ L~f by XBOOLE_1:39; then right_cell(f,k,G) c= rc \/ L~f by XBOOLE_1:7; then Int right_cell(f,k,G) c= rc \/ L~f by A9,XBOOLE_1:1; then A10: Int right_cell(f,k,G) c= rc by A8,XBOOLE_1:73; then Int Int right_cell(f,k,G) c= Int rc by TOPS_1:48; then Int right_cell(f,k,G) c= Int rc by TOPS_1:45; then A11: rc meets Int rc by A7,A10,XBOOLE_1:68; A12: rc is connected by A1,A2,Th28; A13: RightComp f is_a_component_of (L~f)` by GOBOARD9:def 2; rc c= (L~f)` by A3,XBOOLE_1:17; hence right_cell(f,k,G)\L~f c= RightComp f by A6,A11,A12,A13,GOBOARD9:6; set lc = left_cell(f,k,G)\L~f; A14: lc = left_cell(f,k,G) /\ (L~f)` by SUBSET_1:32; A15: lc c= left_cell(f,k,G) by XBOOLE_1:36; left_cell(f,k,G) c= left_cell(f,k) by A1,A2,GOBRD13:34; then lc c= left_cell(f,k) by A15,XBOOLE_1:1; then A16: Int lc c= Int left_cell(f,k) by TOPS_1:48; Int left_cell(f,k) c= LeftComp f by A2,GOBOARD9:24; then A17: Int lc c= LeftComp f by A16,XBOOLE_1:1; A18: Int left_cell(f,k,G) <> {} by A1,A2,Th11; A19: Int left_cell(f,k,G) misses L~f by A1,A2,Th17; A20: Int left_cell(f,k,G) c= left_cell(f,k,G) by TOPS_1:44; lc \/ L~f = left_cell(f,k,G) \/ L~f by XBOOLE_1:39; then left_cell(f,k,G) c= lc \/ L~f by XBOOLE_1:7; then Int left_cell(f,k,G) c= lc \/ L~f by A20,XBOOLE_1:1; then A21: Int left_cell(f,k,G) c= lc by A19,XBOOLE_1:73; then Int Int left_cell(f,k,G) c= Int lc by TOPS_1:48; then Int left_cell(f,k,G) c= Int lc by TOPS_1:45; then A22: lc meets Int lc by A18,A21,XBOOLE_1:68; A23: lc is connected by A1,A2,Th28; A24: LeftComp f is_a_component_of (L~f)` by GOBOARD9:def 1; lc c= (L~f)` by A14,XBOOLE_1:17; hence left_cell(f,k,G)\L~f c= LeftComp f by A17,A22,A23,A24,GOBOARD9:6; end; begin :: Cages reserve C for compact non vertical non horizontal non empty Subset of TOP-REAL 2, l, m, i1, i2, j1, j2 for Nat; theorem Th30: ex i st 1 <= i & i+1 <= len Gauge(C,n) & N-min C in cell(Gauge(C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i,width Gauge(C,n)-'1) proof set G = Gauge(C,n); A1: len G = width G by JORDAN8:def 1; defpred P[Nat] means 1 <= $1 & $1 < len G & G*($1,(width G)-'1)`1 < (N-min C)`1; A2: for k st P[k] holds k <= len G; A3: len G = 2|^n+3 & len G = width G by JORDAN8:def 1; A4: len G >= 4 by JORDAN8:13; then A5: 1 < len G by AXIOMS:22; then A6: 1 <= (len G)-'1 by JORDAN3:12; A7: (len G)-'1 <= len G by GOBOARD9:2; (NW-corner C)`1 <= (N-min C)`1 by PSCOMP_1:97; then A8: W-bound C <= (N-min C)`1 by PSCOMP_1:74; A9: 2 <= len G by A4,AXIOMS:22; G*(2,(width G)-'1)`1 = W-bound C by A1,A6,A7,JORDAN8:14; then G*(1,(width G)-'1)`1 < W-bound C by A3,A6,A7,A9,GOBOARD5:4; then G*(1,(width G)-'1)`1 < (N-min C)`1 by A8,AXIOMS:22; then A10: ex k st P[k] by A5; ex i st P[i] & for n st P[n] holds n <= i from Max(A2,A10); then consider i such that A11: 1 <= i & i < len G and A12: G*(i,(width G)-'1)`1 < (N-min C)`1 and A13: for n st P[n] holds n <= i; take i; thus 1 <= i & i+1 <= len G by A11,NAT_1:38; A14: LSeg(G*(i,(width G)-'1),G*(i+1,(width G)-'1)) c= cell(G,i,(width G)-'1) by A3,A6,A7,A11,GOBOARD5:23; A15: i+1 <= len G by A11,NAT_1:38; A16: 1 <= i+1 by NAT_1:37; (N-min C)`2 = N-bound C by PSCOMP_1:94; then A17: G*(i,(width G)-'1)`2 = (N-min C)`2 & (N-min C)`2 = G* (i+1,(width G)-'1)`2 by A1,A11,A15,A16,JORDAN8:17; now assume i+1 = len G; then len G-'1 = i by BINARITH:39; then A18: G*(i,(width G)-'1)`1 = E-bound C by A1,A6,A7,JORDAN8:15; (NE-corner C)`1 >= (N-min C)`1 by PSCOMP_1:97; hence contradiction by A12,A18,PSCOMP_1:76; end; then i+1 < len G & i < i+1 by A15,NAT_1:38,REAL_1:def 5; then (N-min C)`1 <= G*(i+1,(width G)-'1)`1 by A13,A16; then N-min C in LSeg(G*(i,(width G)-'1),G*(i+1,(width G)-'1)) by A12,A17,GOBOARD7:9; hence N-min C in cell(G,i,(width G)-'1) by A14; thus N-min C <> G*(i,(width G)-'1) by A12; end; theorem Th31: 1 <= i1 & i1+1 <= len Gauge(C,n) & N-min C in cell(Gauge(C,n),i1,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i1,width Gauge(C,n)-'1) & 1 <= i2 & i2+1 <= len Gauge(C,n) & N-min C in cell(Gauge(C,n),i2,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i2,width Gauge(C,n)-'1) implies i1 = i2 proof set G = Gauge(C,n), j = width G-'1; assume that A1: 1 <= i1 & i1+1 <= len G and A2: N-min C in cell(G,i1,j) and A3: N-min C <> G*(i1,j) and A4: 1 <= i2 & i2+1 <= len G and A5: N-min C in cell(G,i2,j) and A6: N-min C <> G*(i2,j) and A7: i1 <> i2; A8: i1 < len G & i2 < len G & len G = width G by A1,A4,JORDAN8:def 1,NAT_1:38; A9: cell(G,i1,j) meets cell(G,i2,j) by A2,A5,XBOOLE_0:3; A10: len G = 2|^n+3 & len G = width G by JORDAN8:def 1; A11: n+3 > 0 by NAT_1:19; A12: 2|^n >= n+1 by HEINE:7; then len G >= n+1+3 by A10,AXIOMS:24; then len G >= 1+ (n+3) & 1+ (n+3) > 1+0 by A11,REAL_1:53,XCMPLX_1:1; then A13: len G > 1 by AXIOMS:22; then len G >= 1+1 by NAT_1:38; then A14: 1 <= j by A8,JORDAN5B:2; A15: j+1 = len G by A8,A13,AMI_5:4; then A16: j < len G by NAT_1:38; A17: (N-min C)`2 = N-bound C by PSCOMP_1:94; per cases by A7,REAL_1:def 5; suppose A18: i1 < i2; then A19: i2-'i1+i1 = i2 by AMI_5:4; then i2-'i1 <= 1 by A8,A9,A14,A16,JORDAN8:10; then i2-'i1 < 1 or i2-'i1 = 1 by REAL_1:def 5; then i2-'i1 = 0 or i2-'i1 = 1 by RLVECT_1:98; then cell(G,i1,j) /\ cell(G,i2,j) = LSeg(G*(i2,j),G*(i2,j+1)) by A8,A14,A16,A18,A19,GOBOARD5:26; then A20: N-min C in LSeg(G*(i2,j),G*(i2,j+1)) by A2,A5,XBOOLE_0:def 3; A21: [i2,j] in Indices G by A4,A8,A14,A16,GOBOARD7:10; 1 <= j+1 by NAT_1:37; then A22: [i2,j+1] in Indices G by A4,A8,A15,GOBOARD7:10; set x = (W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*(i2-2); set y1 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-2); set y2 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-1); A23: j+1-(1+1) = j+1-1-1 by XCMPLX_1:36 .= j-1 by XCMPLX_1:26; A24: G*(i2,j) = |[x,y1]| by A21,JORDAN8:def 1; G*(i2,j+1) = |[x,y2]| by A22,A23,JORDAN8:def 1; then A25: G*(i2,j)`1 = x & G*(i2,j+1)`1 = x by A24,EUCLID:56; then LSeg(G*(i2,j),G*(i2,j+1)) is vertical by SPPOL_1:37; then A26: (N-min C)`1 = G*(i2,j)`1 by A20,SPRECT_3:20; A27: 2|^n > 0 by A12,NAT_1:37; j = (2|^n+(2+1))-'1 by A10 .= (2|^n+2+1)-'1 by XCMPLX_1:1 .= (2|^n+2) by BINARITH:39; then j-2 = 2|^n by XCMPLX_1:26; then (((N-bound C)-(S-bound C))/(2|^n))*(j-2) = (N-bound C)-(S-bound C) by A27,XCMPLX_1:88; then y1 = N-bound C by XCMPLX_1:27; hence contradiction by A6,A17,A24,A25,A26,EUCLID:57; suppose A28: i2 < i1; then A29: i1-'i2+i2 = i1 by AMI_5:4; then i1-'i2 <= 1 by A8,A9,A14,A16,JORDAN8:10; then i1-'i2 < 1 or i1-'i2 = 1 by REAL_1:def 5; then i1-'i2 = 0 or i1-'i2 = 1 by RLVECT_1:98; then cell(G,i2,j) /\ cell(G,i1,j) = LSeg(G*(i1,j),G*(i1,j+1)) by A8,A14,A16,A28,A29,GOBOARD5:26; then A30: N-min C in LSeg(G*(i1,j),G*(i1,j+1)) by A2,A5,XBOOLE_0:def 3; A31: [i1,j] in Indices G by A1,A8,A14,A16,GOBOARD7:10; 1 <= j+1 by NAT_1:37; then A32: [i1,j+1] in Indices G by A1,A8,A15,GOBOARD7:10; set x = (W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*(i1-2); set y1 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-2); set y2 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-1); A33: j+1-(1+1) = j+1-1-1 by XCMPLX_1:36 .= j-1 by XCMPLX_1:26; A34: G*(i1,j) = |[x,y1]| by A31,JORDAN8:def 1; G*(i1,j+1) = |[x,y2]| by A32,A33,JORDAN8:def 1; then A35: G*(i1,j)`1 = x & G*(i1,j+1)`1 = x by A34,EUCLID:56; then LSeg(G*(i1,j),G*(i1,j+1)) is vertical by SPPOL_1:37; then A36: (N-min C)`1 = G*(i1,j)`1 by A30,SPRECT_3:20; A37: 2|^n > 0 by A12,NAT_1:37; j = (2|^n+(2+1))-'1 by A10 .= (2|^n+2+1)-'1 by XCMPLX_1:1 .= (2|^n+2) by BINARITH:39; then j-2 = 2|^n by XCMPLX_1:26; then (((N-bound C)-(S-bound C))/(2|^n))*(j-2) = (N-bound C)-(S-bound C) by A37,XCMPLX_1:88; then y1 = N-bound C by XCMPLX_1:27; hence contradiction by A3,A17,A34,A35,A36,EUCLID:57; end; theorem Th32: for f being standard non constant special_circular_sequence st f is_sequence_on Gauge(C,n) & (for k st 1 <= k & k+1 <= len f holds left_cell(f,k,Gauge(C,n)) misses C & right_cell(f,k,Gauge(C,n)) meets C) & (ex i st 1 <= i & i+1 <= len Gauge(C,n) & f/.1 = Gauge(C,n)*(i,width Gauge(C,n)) & f/.2 = Gauge(C,n)*(i+1,width Gauge(C,n)) & N-min C in cell(Gauge(C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i,width Gauge(C,n)-'1)) holds N-min L~f = f/.1 proof set G = Gauge(C,n); let f be standard non constant special_circular_sequence such that A1: f is_sequence_on G and A2: for k st 1 <= k & k+1 <= len f holds left_cell(f,k,G) misses C & right_cell(f,k,G) meets C; given i' being Nat such that A3: 1 <= i' & i'+1 <= len G and A4: f/.1 = G*(i',width G) and A5: f/.2 = G*(i'+1,width G) and A6: N-min C in cell(G,i',width G-'1) and A7: N-min C <> G*(i',width G-'1); A8: i' < len G & len G = width G by A3,JORDAN8:def 1,NAT_1:38; len f > 4 by GOBOARD7:36; then A9: len f >= 2 by AXIOMS:22; A10: len G = 2|^n+3 & len G = width G by JORDAN8:def 1; then len G >= 3 by NAT_1:37; then A11: 1 < len G by AXIOMS:22; set W = W-bound C, S = S-bound C, E = E-bound C, N = N-bound C; N-min L~f in rng f by SPRECT_2:43; then consider m such that A12: m in dom f and A13: f.m = N-min L~f by FINSEQ_2:11; A14: f/.m = f.m by A12,FINSEQ_4:def 4; A15: (N-min L~f)`2 = N-bound L~f by PSCOMP_1:94; A16: 2|^n > 0 by HEINE:5; N > S by JORDAN8:12; then N-S > 0 by SQUARE_1:11; then A17: (N-S)/(2|^n) > 0 by A16,REAL_2:127; A18: 1 <= m & m <= len f by A12,FINSEQ_3:27; consider i,j such that A19: [i,j] in Indices G and A20: f/.m = G*(i,j) by A1,A12,GOBOARD1:def 11; A21: 1 <= i & i <= len G & 1 <= j & j <= width G by A19,GOBOARD5:1; G*(i,j) = |[W+((E-W)/(2|^n))*(i-2), S+((N-S)/(2|^n))*(j-2)]| by A19,JORDAN8:def 1; then A22: S+((N-S)/(2|^n))*(j-2) = N-bound L~f by A13,A14,A15,A20,EUCLID:56 ; 1 in dom f by FINSEQ_5:6; then A23: f/.1 in L~f by A9,GOBOARD1:16; then A24: N-bound L~f >= (f/.1)`2 by PSCOMP_1:71; G*(i',j)`2 = G*(1,j)`2 & G*(i,j)`2 = G* (1,j)`2 by A3,A8,A21,GOBOARD5:2; then G*(i,j)`2 <= G*(i',len G)`2 by A3,A8,A21,SPRECT_3:24; then A25: N-bound L~f = (f/.1)`2 by A4,A8,A13,A14,A15,A20,A24,AXIOMS:21; [i',len G] in Indices G by A3,A8,A11,GOBOARD7:10; then G*(i',len G)=|[W+((E-W)/(2|^n))*(i'-2),S+((N-S)/(2|^n))*(len G-2)]| by JORDAN8:def 1; then S+((N-S)/(2|^n))*(len G-2) = N-bound L~f by A4,A8,A25,EUCLID:56; then ((N-S)/(2|^n))*(j-2) = ((N-S)/(2|^n))*(len G-2) by A22,XCMPLX_1:2; then len G-2 = j-2 by A17,XCMPLX_1:5; then A26: len G = j by XCMPLX_1:19; A27: (NW-corner C)`1 = W & (NE-corner C)`1 = E & (NW-corner C)`2 = N & (NE-corner C)`2 = N by PSCOMP_1:74,75,76,77; A28: G*(i',len G)`1 = G*(i',1)`1 by A3,A8,A21,A26,GOBOARD5:3; A29: G*(i,len G)`1 = G*(i,1)`1 by A21,A26,GOBOARD5:3; A30: (NW-corner L~f)`1 = W-bound L~f & (NE-corner L~f)`1 = E-bound L~f & (NW-corner L~f)`2 = N-bound L~f & (NE-corner L~f)`2 = N-bound L~f by PSCOMP_1:74,75,76,77; W-bound L~f <= (f/.1)`1 & (f/.1)`1 <= E-bound L~f by A23,PSCOMP_1:71; then f/.1 in LSeg(NW-corner L~f, NE-corner L~f) by A25,A30,GOBOARD7:9; then f/.1 in LSeg(NW-corner L~f, NE-corner L~f) /\ L~f by A23,XBOOLE_0:def 3; then f/.1 in N-most L~f by PSCOMP_1:def 39; then A31: (N-min L~f)`1 <= (f/.1)`1 by PSCOMP_1:98; A32: 1 <= len G-'1 by A11,JORDAN3:12; then A33: len G-'1 < len G by JORDAN3:14; A34: len G-'1+1 = len G by A11,AMI_5:4; then N-min C in { |[r',s']| where r',s' is Real: G*(i',1)`1 <= r' & r' <= G*(i'+1,1)`1 & G*(1,len G-'1)`2 <= s' & s' <= G*(1,len G)`2 } by A3,A6,A8,A32,A33,GOBRD11:32; then consider r',s' being Real such that A35: N-min C = |[r',s']| and A36: G*(i',1)`1 <= r' & r' <= G*(i'+1,1)`1 and G*(1,len G-'1)`2 <= s' & s' <= G*(1,len G)`2; A37: (f/.1)`1 <= (N-min C)`1 by A4,A8,A28,A35,A36,EUCLID:56; then A38: (N-min L~f)`1 <= (N-min C)`1 by A31,AXIOMS:22; A39: G*(i',len G-'1)`1 = G*(i',1)`1 by A3,A8,A32,A33,GOBOARD5:3; A40: N-min C = |[(N-min C)`1,(N-min C)`2]| by EUCLID:57; A41: G*(i',len G-'1) = |[G*(i',len G-'1)`1,G*(i',len G-'1)`2]| by EUCLID:57; A42: (N-min C)`2 = N by PSCOMP_1:94; G*(i',len G-'1)`2 = N by A3,A8,JORDAN8:17; then A43: G*(i',len G-'1)`1 < (N-min C)`1 by A4,A7,A8,A28,A37,A39,A40,A41,A42,REAL_1:def 5; A44: i <= i' by A3,A4,A8,A13,A14,A20,A21,A26,A31,GOBOARD5:4; then A45: i < len G by A8,AXIOMS:22; then A46: i+1 <= len G by NAT_1:38; per cases by A18,REAL_1:def 5; suppose m = len f; hence N-min L~f = f/.1 by A13,A14,FINSEQ_6:def 1; suppose m < len f; then A47: m+1 <= len f by NAT_1:38; then A48: right_cell(f,m,G) meets C by A2,A18; then consider p being set such that A49: p in right_cell(f,m,G) & p in C by XBOOLE_0:3; reconsider p as Point of TOP-REAL 2 by A49; consider i1,j1,i2,j2 being Nat such that A50: [i1,j1] in Indices G & f/.m = G*(i1,j1) and A51: [i2,j2] in Indices G & f/.(m+1) = G*(i2,j2) and A52: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A1,A18,A47,JORDAN8:6; A53: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= len G by A10,A51,GOBOARD5:1; A54: (N-min C)`2 = N by PSCOMP_1:94; then A55: p`2 <= (N-min C)`2 by A49,PSCOMP_1:71; A56: G*(1,len G-'1)`2 = N by A11,JORDAN8:17; A57: G*(1,len G-'1)`2 < G*(1,len G)`2 by A10,A11,A32,A33,GOBOARD5:5; A58: W <= p`1 & p`1 <= E by A49,PSCOMP_1:71; now per cases by A19,A20,A26,A50,A52,GOBOARD1:21; suppose i = i2 & len G+1 = j2; hence N-min L~f = f/.1 by A53,NAT_1:38; suppose i+1 = i2 & len G = j2; then A59: right_cell(f,m,G) = cell(G,i,len G-'1) by A1,A18,A19,A20,A26,A47,A51,GOBRD13:25; A60: cell(G,i,len G-'1) = {|[r,s]| : G*(i,1)`1 <= r & r <= G*(i+1,1)`1 & G*(1,len G-'1)`2 <= s & s <= G*(1,len G-'1+1)`2 } by A10,A21,A32,A33,A45,GOBRD11:32; then consider r,s such that A61: p = |[r,s]| and A62: G*(i,1)`1 <= r & r <= G*(i+1,1)`1 and A63: G*(1,len G-'1)`2 <= s & s <= G*(1,len G-'1+1)`2 by A49,A59; A64: p`1 = r & p`2 = s by A61,EUCLID:56; then p`2 = N by A54,A55,A56,A63,AXIOMS:21; then p in LSeg(NW-corner C, NE-corner C) by A27,A58,GOBOARD7:9; then p in LSeg(NW-corner C, NE-corner C) /\ C by A49,XBOOLE_0:def 3; then p in N-most C by PSCOMP_1:def 39; then (N-min C)`1 <= p`1 by PSCOMP_1:98; then (N-min C)`1 <= G*(i+1,1)`1 by A62,A64,AXIOMS:22; then A65: N-min C in cell(G,i,width G-'1) by A8,A13,A14,A20,A26,A29,A34,A38,A40,A54,A56,A57,A60; N-min C <> G*(i,len G-'1) by A8,A21,A32,A33,A43,A44,SPRECT_3:25; hence N-min L~f = f/.1 by A3,A4,A6,A7,A8,A13,A14,A20,A21,A26,A46,A65,Th31; suppose A66: i = i2+1 & len G = j2; then A67: right_cell(f,m,G) = cell(G,i2,len G) by A1,A18,A19,A20,A26,A47,A51,GOBRD13:27; i2 < len G by A21,A66,NAT_1:38; hence N-min L~f = f/.1 by A48,A67,JORDAN8:18; suppose A68: i = i2 & len G = j2+1; then A69: j2 = len G-'1 by BINARITH:39; then A70: right_cell(f,m,G) = cell(G,i-'1,len G-'1) by A1,A18,A19,A20,A26,A47,A51,A68,GOBRD13:29; now assume A71: m = 1; 1 <= i'+1 & i'+1<= len G by A3,NAT_1:38; then G*(i',len G)`2 = G*(1,len G)`2 & G*(i'+1,len G)`2 = G* (1,len G)`2 by A3,A8,A11,GOBOARD5:2; hence contradiction by A4,A5,A8,A20,A21,A26,A32,A33,A51,A68,A69,A71,GOBOARD5:5; end; then m > 1 by A18,REAL_1:def 5; then A72: m-'1 >= 1 by JORDAN3:12; A73: m-'1+1 = m by A18,AMI_5:4; m-'1 <= m by GOBOARD9:2; then A74: m-'1 <= len f by A18,AXIOMS:22; consider i1',j1',i2',j2' being Nat such that A75: [i1',j1'] in Indices G & f/.(m-'1) = G*(i1',j1') and A76: [i2',j2'] in Indices G & f/.m = G*(i2',j2') and A77: i1' = i2' & j1'+1 = j2' or i1'+1 = i2' & j1' = j2' or i1' = i2'+1 & j1' = j2' or i1' = i2' & j1' = j2'+1 by A1,A18,A72,A73,JORDAN8:6; A78: 1 <= i1' & i1' <= len G & 1 <= j1' & j1' <= len G by A10,A75,GOBOARD5:1; now per cases by A19,A20,A26,A76,A77,GOBOARD1:21; suppose A79: i1' = i & j1'+1 = len G; then j1' = len G-'1 by BINARITH:39; then left_cell(f,m-'1,G) = cell(G,i-'1,len G-'1) by A1,A18,A19,A20,A26,A72,A73,A75,A79,GOBRD13:22; hence contradiction by A2,A18,A48,A70,A72,A73; suppose A80: i1'+1 = i & j1' = len G; then i1' < i by NAT_1:38; then A81: (f/.(m-'1))`1 < (f/.m)`1 by A20,A21,A26,A75,A78,A80,GOBOARD5:4; A82: G*(i1',j)`2 = G*(1,j)`2 & G*(i,j)`2 = G*(1,j)`2 by A21,A78,GOBOARD5:2; m-'1 in dom f by A72,A74,FINSEQ_3:27; then A83: f/.(m-'1) in L~f by A9,GOBOARD1:16; then W-bound L~f <= (f/.(m-'1))`1 & (f/.(m-'1))`1 <= E-bound L~f by PSCOMP_1:71; then f/.(m-'1) in LSeg(NW-corner L~f, NE-corner L~f) by A13,A14,A15,A20,A26,A30,A75,A80,A82,GOBOARD7:9; then f/.(m-'1) in LSeg(NW-corner L~f, NE-corner L~f) /\ L~f by A83,XBOOLE_0:def 3; then f/.(m-'1) in N-most L~f by PSCOMP_1:def 39; hence contradiction by A13,A14,A81,PSCOMP_1:98; suppose i1' = i+1 & j1' = len G; then right_cell(f,m-'1,G) = cell(G,i,len G) by A1,A18,A19,A20,A26,A72,A73,A75,GOBRD13:27; then cell(G,i,len G) meets C by A2,A18,A72,A73; hence contradiction by A21,JORDAN8:18; suppose i1' = i & j1' = len G+1; then len G+1 <= len G+0 by A10,A75,GOBOARD5:1; hence contradiction by REAL_1:53; end; hence N-min L~f = f/.1; end; hence N-min L~f = f/.1; end; definition let C be compact non vertical non horizontal non empty Subset of TOP-REAL 2; let n be Nat; assume A1: C is connected; func Cage(C,n) -> clockwise_oriented (standard non constant special_circular_sequence) means :Def1: it is_sequence_on Gauge(C,n) & (ex i st 1 <= i & i+1 <= len Gauge(C,n) & it/.1 = Gauge(C,n)*(i,width Gauge(C,n)) & it/.2 = Gauge(C,n)*(i+1,width Gauge(C,n)) & N-min C in cell(Gauge(C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i,width Gauge(C,n)-'1)) & for k st 1 <= k & k+2 <= len it holds (front_left_cell(it,k,Gauge(C,n)) misses C & front_right_cell(it,k,Gauge(C,n)) misses C implies it turns_right k,Gauge(C,n)) & (front_left_cell(it,k,Gauge(C,n)) misses C & front_right_cell(it,k,Gauge(C,n)) meets C implies it goes_straight k,Gauge(C,n)) & (front_left_cell(it,k,Gauge(C,n)) meets C implies it turns_left k,Gauge(C,n)); existence proof set G = Gauge(C,n); set W = W-bound C, E = E-bound C, S = S-bound C, N = N-bound C; A2: len G = 2|^n+3 & len G = width G by JORDAN8:def 1; defpred P[Nat,set,set] means ($1 = 0 implies ex i st 1 <= i & i+1 <= len G & N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) & $3 = <*G*(i,width G)*>) & ($1 = 1 implies ex i st 1 <= i & i+1 <= len G & N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) & $3 = <*G*(i,width G),G*(i+1,width G)*>) & ($1 > 1 & $2 is FinSequence of TOP-REAL 2 implies for f being FinSequence of TOP-REAL 2 st $2 = f holds (len f = $1 implies (f is_sequence_on G & right_cell(f,len f-'1,G) meets C implies (front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) misses C implies ex i,j st f^<*G*(i,j)*> turns_right (len f)-'1,G & $3 = f^<*G*(i,j)*>) & (front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) meets C implies ex i,j st f^<*G*(i,j)*> goes_straight (len f)-'1,G & $3 = f^<*G*(i,j)*>) & (front_left_cell(f,(len f)-'1,G) meets C implies ex i,j st f^<*G*(i,j)*> turns_left (len f)-'1,G & $3 = f^<*G*(i,j)*>)) & (not f is_sequence_on G or right_cell(f,len f-'1,G) misses C implies $3 = f^<*G*(1,1)*>)) & (len f <> $1 implies $3 = {})) & ($1 > 1 & $2 is not FinSequence of TOP-REAL 2 implies $3 = {}); A3: for k being Nat, x being set ex y being set st P[k,x,y] proof let k be Nat, x be set; consider m being Nat such that A4: 1 <= m & m+1 <= len G and A5: N-min C in cell(G,m,width G-'1) & N-min C <> G*(m,width G-'1) by Th30; per cases by CQC_THE1:2; suppose A6: k=0; take <*G*(m,width G)*>; thus thesis by A4,A5,A6; suppose A7: k = 1; take <*G*(m,width G),G*(m+1,width G)*>; thus thesis by A4,A5,A7; suppose that A8: k > 1 and A9: x is FinSequence of TOP-REAL 2; reconsider f = x as FinSequence of TOP-REAL 2 by A9; thus thesis proof per cases; suppose A10: len f = k; thus thesis proof per cases; suppose A11: f is_sequence_on G & right_cell(f,len f-'1,G) meets C; A12: 1 <= (len f)-'1 by A8,A10,JORDAN3:12; A13: (len f) -'1 +1 = len f by A8,A10,AMI_5:4; then consider i1,j1,i2,j2 being Nat such that A14: [i1,j1] in Indices G & f/.((len f) -'1) = G*(i1,j1) and A15: [i2,j2] in Indices G & f/.((len f) -'1+1) = G*(i2,j2) and A16: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A11,A12,JORDAN8:6; A17: 1 <= i1 & i1 <= len G & 1 <= j1 & j1 <= width G by A14,GOBOARD5:1; A18: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A15,GOBOARD5:1; A19: i1-'1 <= len G & j1-'1 <= width G by A17,JORDAN3:7; A20: 1 <= i2+1 & 1 <= j2+1 by NAT_1:37; A21: i2-'1 <= len G & j2-'1 <= width G by A18,JORDAN3:7; (len f)-'1 <= len f by GOBOARD9:2; then A22: (len f)-'1 in dom f & (len f)-'1+1 in dom f by A8,A10,A12,A13,FINSEQ_3:27; A23: (len f)-'1+(1+1) = (len f)+1 by A13,XCMPLX_1:1; thus thesis proof per cases; suppose A24: front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) misses C; thus thesis proof per cases by A16; suppose A25: i1 = i2 & j1+1 = j2; take f1 = f^<*G*(i2+1,j2)*>; now take i=i2+1 ,j=j2; thus f1 turns_right (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A26: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A27: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A27,GROUP_5:95; then A28: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A26,GOBOARD1:21; per cases by A16,A28; case i1' = i2' & j1'+1 = j2'; now assume i2+1 > len G; then i2+1 <= (len G)+1 & (len G)+1 <= i2+1 by A18,AXIOMS:24,NAT_1:38; then i2+1 = (len G)+1 by AXIOMS:21; then i2 = len G by XCMPLX_1:2; then cell(G,len G,j1) meets C by A11,A12,A13,A14,A15,A25,GOBRD13:23; hence contradiction by A2,A17,JORDAN8:19; end; hence [i2'+1,j2'] in Indices G by A18,A20,A28,GOBOARD7:10; thus thesis by A23,A28,TOPREAL4:1; case i1'+1 = i2' & j1' = j2'; hence thesis by A25,A28,NAT_1:38; case i1' = i2'+1 & j1' = j2'; hence thesis by A25,A28,NAT_1:38; case A29: i1' = i2' & j1' = j2'+1; j1+1+1 = j1+(1+1) by XCMPLX_1:1; hence thesis by A25,A28,A29,REAL_1:69; end; end; hence thesis by A8,A10,A11,A24; suppose A30: i1+1 = i2 & j1 = j2; take f1 = f^<*G*(i2,j2-'1)*>; now take i=i2 ,j=j2-'1; thus f1 turns_right (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A31: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A32: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A32,GROUP_5:95; then A33: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A31,GOBOARD1:21; per cases by A16,A33; case i1' = i2' & j1'+1 = j2'; hence thesis by A30,A33,NAT_1:38; case i1'+1 = i2' & j1' = j2'; now assume j2-'1 < 1; then j2-'1 = 0 by RLVECT_1:98; then j2 <= 1 by JORDAN4:1; then j2 = 1 by A18,AXIOMS:21; then cell(G,i1,1-'1) meets C by A11,A12,A13,A14,A15,A30,GOBRD13:25; then cell(G,i1,0) meets C by GOBOARD9:1; hence contradiction by A17,JORDAN8:20; end; hence [i2',j2'-'1] in Indices G by A18,A21,A33,GOBOARD7:10; thus f1/.((len f)-'1+2) = G*(i2',j2'-'1) by A23,A33,TOPREAL4:1; case A34: i1' = i2'+1 & j1' = j2'; i1+1+1 = i1+(1+1) by XCMPLX_1:1; hence thesis by A30,A33,A34,REAL_1:69; case i1' = i2' & j1' = j2'+1; hence thesis by A30,A33,NAT_1:38; end; end; hence thesis by A8,A10,A11,A24; suppose A35: i1 = i2+1 & j1 = j2; take f1 = f^<*G*(i2,j2+1)*>; now take i=i2 ,j=j2+1; thus f1 turns_right (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A36: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A37: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A37,GROUP_5:95; then A38: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A36,GOBOARD1:21; per cases by A16,A38; case i1' = i2' & j1'+1 = j2'; hence thesis by A35,A38,NAT_1:38; case A39: i1'+1 = i2' & j1' = j2'; i1+1+1 = i1+(1+1) by XCMPLX_1:1; hence thesis by A35,A38,A39,REAL_1:69; case i1' = i2'+1 & j1' = j2'; now assume j2+1 > len G; then j2+1 <= (len G)+1 & (len G)+1 <= j2+1 by A2,A18,AXIOMS:24,NAT_1:38; then j2+1 = (len G)+1 by AXIOMS:21; then j2 = len G by XCMPLX_1:2; then cell(G,i2,len G) meets C by A11,A12,A13,A14,A15,A35,GOBRD13:27; hence contradiction by A18,JORDAN8:18; end; hence [i2',j2'+1] in Indices G by A2,A18,A20,A38,GOBOARD7:10 ; thus f1/.((len f)-'1+2) = G*(i2',j2'+1) by A23,A38,TOPREAL4:1 ; case i1' = i2' & j1' = j2'+1; hence thesis by A35,A38,NAT_1:38; end; end; hence thesis by A8,A10,A11,A24; suppose A40: i1 = i2 & j1 = j2+1; take f1 = f^<*G*(i2-'1,j2)*>; now take i=i2-'1 ,j=j2; thus f1 turns_right (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A41: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A42: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A42,GROUP_5:95; then A43: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A41,GOBOARD1:21; per cases by A16,A43; case A44: i1' = i2' & j1'+1 = j2'; j1+1+1 = j1+(1+1) by XCMPLX_1:1; hence thesis by A40,A43,A44,REAL_1:69; case i1'+1 = i2' & j1' = j2'; hence thesis by A40,A43,NAT_1:38; case i1' = i2'+1 & j1' = j2'; hence thesis by A40,A43,NAT_1:38; case i1' = i2' & j1' = j2'+1; now assume i2-'1 < 1; then i2-'1 = 0 by RLVECT_1:98; then i2 <= 1 by JORDAN4:1; then i2 = 1 by A18,AXIOMS:21; then cell(G,1-'1,j2) meets C by A11,A12,A13,A14,A15,A40,GOBRD13:29; then cell(G,0,j2) meets C by GOBOARD9:1; hence contradiction by A2,A18,JORDAN8:21; end; hence [i2'-'1,j2'] in Indices G by A18,A21,A43,GOBOARD7:10; thus f1/.((len f)-'1+2) = G*(i2'-'1,j2') by A23,A43,TOPREAL4:1; end; end; hence thesis by A8,A10,A11,A24; end; suppose A45: front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) meets C; thus thesis proof per cases by A16; suppose A46: i1 = i2 & j1+1 = j2; take f1 = f^<*G*(i2,j2+1)*>; now take i=i2 ,j=j2+1; thus f1 goes_straight (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A47: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A48: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A48,GROUP_5:95; then A49: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A47,GOBOARD1:21; per cases by A16,A49; case i1' = i2' & j1'+1 = j2'; now assume j2+1 > len G; then j2+1 <= (len G)+1 & (len G)+1 <= j2+1 by A2,A18,AXIOMS:24,NAT_1:38; then j2+1 = (len G)+1 by AXIOMS:21; then j2 = len G by XCMPLX_1:2; then cell(G,i1,len G) meets C by A11,A12,A13,A14,A15,A45,A46,GOBRD13:36; hence contradiction by A17,JORDAN8:18; end; hence [i2',j2'+1] in Indices G by A2,A18,A20,A49,GOBOARD7:10 ; thus f1/.((len f)-'1+2) = G*(i2',j2'+1) by A23,A49,TOPREAL4:1; case i1'+1 = i2' & j1' = j2'; hence thesis by A46,A49,NAT_1:38; case i1' = i2'+1 & j1' = j2'; hence thesis by A46,A49,NAT_1:38; case A50: i1' = i2' & j1' = j2'+1; j1+1+1 = j1+(1+1) by XCMPLX_1:1; hence thesis by A46,A49,A50,REAL_1:69; end; end; hence thesis by A8,A10,A11,A45; suppose A51: i1+1 = i2 & j1 = j2; take f1 = f^<*G*(i2+1,j2)*>; now take i=i2+1 ,j=j2; thus f1 goes_straight (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A52: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A53: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A53,GROUP_5:95; then A54: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A52,GOBOARD1:21; per cases by A16,A54; case i1' = i2' & j1'+1 = j2'; hence thesis by A51,A54,NAT_1:38; case i1'+1 = i2' & j1' = j2'; now assume i2+1 > len G; then i2+1 <= (len G)+1 & (len G)+1 <= i2+1 by A18,AXIOMS:24,NAT_1:38; then i2+1 = (len G)+1 by AXIOMS:21; then i2 = len G by XCMPLX_1:2; then cell(G,len G,j1-'1) meets C by A11,A12,A13,A14,A15,A45,A51,GOBRD13:38; hence contradiction by A2,A19,JORDAN8:19; end; hence [i2'+1,j2'] in Indices G by A18,A20,A54,GOBOARD7:10; thus f1/.((len f)-'1+2) = G*(i2'+1,j2') by A23,A54,TOPREAL4:1; case A55: i1' = i2'+1 & j1' = j2'; i1+1+1 = i1+(1+1) by XCMPLX_1:1; hence thesis by A51,A54,A55,REAL_1:69; case i1' = i2' & j1' = j2'+1; hence thesis by A51,A54,NAT_1:38; end; end; hence thesis by A8,A10,A11,A45; suppose A56: i1 = i2+1 & j1 = j2; take f1 = f^<*G*(i2-'1,j2)*>; now take i=i2-'1 ,j=j2; thus f1 goes_straight (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A57: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A58: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A58,GROUP_5:95; then A59: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A57,GOBOARD1:21; per cases by A16,A59; case i1' = i2' & j1'+1 = j2'; hence thesis by A56,A59,NAT_1:38; case A60: i1'+1 = i2' & j1' = j2'; i1+1+1 = i1+(1+1) by XCMPLX_1:1; hence thesis by A56,A59,A60,REAL_1:69; case i1' = i2'+1 & j1' = j2'; now assume i2-'1 < 1; then i2-'1 = 0 by RLVECT_1:98; then i2 <= 1 by JORDAN4:1; then i2 = 1 by A18,AXIOMS:21; then cell(G,1-'1,j1) meets C by A11,A12,A13,A14,A15,A45,A56,GOBRD13:40; then cell(G,0,j1) meets C by GOBOARD9:1; hence contradiction by A2,A17,JORDAN8:21; end; hence [i2'-'1,j2'] in Indices G by A18,A21,A59,GOBOARD7:10; thus f1/.((len f)-'1+2) = G*(i2'-'1,j2') by A23,A59,TOPREAL4:1; case i1' = i2' & j1' = j2'+1; hence thesis by A56,A59,NAT_1:38; end; end; hence thesis by A8,A10,A11,A45; suppose A61: i1 = i2 & j1 = j2+1; take f1 = f^<*G*(i2,j2-'1)*>; now take i=i2 ,j=j2-'1; thus f1 goes_straight (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A62: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A63: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A63,GROUP_5:95; then A64: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A62,GOBOARD1:21; per cases by A16,A64; case A65: i1' = i2' & j1'+1 = j2'; j1+1+1 = j1+(1+1) by XCMPLX_1:1; hence thesis by A61,A64,A65,REAL_1:69; case i1'+1 = i2' & j1' = j2'; hence thesis by A61,A64,NAT_1:38; case i1' = i2'+1 & j1' = j2'; hence thesis by A61,A64,NAT_1:38; case i1' = i2' & j1' = j2'+1; now assume j2-'1 < 1; then j2-'1 = 0 by RLVECT_1:98; then j2 <= 1 by JORDAN4:1; then j2 = 1 by A18,AXIOMS:21; then cell(G,i1-'1,1-'1) meets C by A11,A12,A13,A14,A15,A45,A61,GOBRD13:42; then cell(G,i1-'1,0) meets C by GOBOARD9:1; hence contradiction by A19,JORDAN8:20; end; hence [i2',j2'-'1] in Indices G by A18,A21,A64,GOBOARD7:10; thus f1/.((len f)-'1+2) = G*(i2',j2'-'1) by A23,A64,TOPREAL4:1; end; end; hence thesis by A8,A10,A11,A45; end; suppose A66: front_left_cell(f,(len f)-'1,G) meets C; thus thesis proof per cases by A16; suppose A67: i1 = i2 & j1+1 = j2; take f1 = f^<*G*(i2-'1,j2)*>; now take i=i2-'1 ,j=j2; thus f1 turns_left (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A68: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A69: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A69,GROUP_5:95; then A70: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A68,GOBOARD1:21; per cases by A16,A70; case i1' = i2' & j1'+1 = j2'; now assume i2-'1 < 1; then i2-'1 = 0 by RLVECT_1:98; then i2 <= 1 by JORDAN4:1; then i2 = 1 by A18,AXIOMS:21; then cell(G,1-'1,j2) meets C by A11,A12,A13,A14,A15,A66,A67,GOBRD13:35; then cell(G,0,j2) meets C by GOBOARD9:1; hence contradiction by A2,A18,JORDAN8:21; end; hence [i2'-'1,j2'] in Indices G by A18,A21,A70,GOBOARD7:10; thus f1/.((len f)-'1+2) = G*(i2'-'1,j2') by A23,A70,TOPREAL4:1; case i1'+1 = i2' & j1' = j2'; hence thesis by A67,A70,NAT_1:38; case i1' = i2'+1 & j1' = j2'; hence thesis by A67,A70,NAT_1:38; case A71: i1' = i2' & j1' = j2'+1; j1+1+1 = j1+(1+1) by XCMPLX_1:1; hence thesis by A67,A70,A71,REAL_1:69; end; end; hence thesis by A8,A10,A11,A66; suppose A72: i1+1 = i2 & j1 = j2; take f1 = f^<*G*(i2,j2+1)*>; now take i=i2 ,j=j2+1; thus f1 turns_left (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A73: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A74: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A74,GROUP_5:95; then A75: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A73,GOBOARD1:21; per cases by A16,A75; case i1' = i2' & j1'+1 = j2'; hence thesis by A72,A75,NAT_1:38; case i1'+1 = i2' & j1' = j2'; now assume j2+1 > len G; then j2+1 <= (len G)+1 & (len G)+1 <= j2+1 by A2,A18,AXIOMS:24,NAT_1:38; then j2+1 = (len G)+1 by AXIOMS:21; then j2 = len G by XCMPLX_1:2; then cell(G,i2,len G) meets C by A11,A12,A13,A14,A15,A66,A72,GOBRD13:37; hence contradiction by A18,JORDAN8:18; end; hence [i2',j2'+1] in Indices G by A2,A18,A20,A75,GOBOARD7:10 ; thus f1/.((len f)-'1+2) = G*(i2',j2'+1) by A23,A75,TOPREAL4:1; case A76: i1' = i2'+1 & j1' = j2'; i1+1+1 = i1+(1+1) by XCMPLX_1:1; hence thesis by A72,A75,A76,REAL_1:69; case i1' = i2' & j1' = j2'+1; hence thesis by A72,A75,NAT_1:38; end; end; hence thesis by A8,A10,A11,A66; suppose A77: i1 = i2+1 & j1 = j2; take f1 = f^<*G*(i2,j2-'1)*>; now take i=i2 ,j=j2-'1; thus f1 turns_left (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A78: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A79: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A79,GROUP_5:95; then A80: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A78,GOBOARD1:21; per cases by A16,A80; case i1' = i2' & j1'+1 = j2'; hence thesis by A77,A80,NAT_1:38; case A81: i1'+1 = i2' & j1' = j2'; i1+1+1 = i1+(1+1) by XCMPLX_1:1; hence thesis by A77,A80,A81,REAL_1:69; case i1' = i2'+1 & j1' = j2'; now assume j2-'1 < 1; then j2-'1 = 0 by RLVECT_1:98; then j2 <= 1 by JORDAN4:1; then j2 = 1 by A18,AXIOMS:21; then cell(G,i2-'1,1-'1) meets C by A11,A12,A13,A14,A15,A66,A77,GOBRD13:39; then cell(G,i2-'1,0) meets C by GOBOARD9:1; hence contradiction by A21,JORDAN8:20; end; hence [i2',j2'-'1] in Indices G by A18,A21,A80,GOBOARD7:10; thus f1/.((len f)-'1+2) = G*(i2',j2'-'1) by A23,A80,TOPREAL4:1; case i1' = i2' & j1' = j2'+1; hence thesis by A77,A80,NAT_1:38; end; end; hence thesis by A8,A10,A11,A66; suppose A82: i1 = i2 & j1 = j2+1; take f1 = f^<*G*(i2+1,j2)*>; now take i=i2+1 ,j=j2; thus f1 turns_left (len f)-'1,G proof let i1',j1',i2',j2' be Nat; assume that A83: [i1',j1'] in Indices G & [i2',j2'] in Indices G and A84: f1/.((len f)-'1) = G*(i1',j1') & f1/.((len f)-'1+1) = G*(i2',j2'); f/.((len f)-'1) = G*(i1',j1') & f/.((len f)-'1+1) = G*(i2',j2') by A22,A84,GROUP_5:95; then A85: i1 = i1' & j1 = j1' & i2 = i2' & j2 = j2' by A14,A15,A83,GOBOARD1:21; per cases by A16,A85; case A86: i1' = i2' & j1'+1 = j2'; j1+1+1 = j1+(1+1) by XCMPLX_1:1; hence thesis by A82,A85,A86,REAL_1:69; case i1'+1 = i2' & j1' = j2'; hence thesis by A82,A85,NAT_1:38; case i1' = i2'+1 & j1' = j2'; hence thesis by A82,A85,NAT_1:38; case i1' = i2' & j1' = j2'+1; now assume i2+1 > len G; then i2+1 <= (len G)+1 & (len G)+1 <= i2+1 by A18,AXIOMS:24,NAT_1:38; then i2+1 = (len G)+1 by AXIOMS:21; then i2 = len G by XCMPLX_1:2; then cell(G,len G,j2-'1) meets C by A11,A12,A13,A14,A15,A66,A82,GOBRD13:41; hence contradiction by A2,A21,JORDAN8:19; end; hence [i2'+1,j2'] in Indices G by A18,A20,A85,GOBOARD7:10; thus f1/.((len f)-'1+2) = G*(i2'+1,j2') by A23,A85,TOPREAL4:1; end; end; hence thesis by A8,A10,A11,A66; end; end; suppose A87: not f is_sequence_on G or right_cell(f,len f-'1,G) misses C; take f^<*G*(1,1)*>; thus thesis by A8,A10,A87; end; suppose A88: len f <> k; take {}; thus thesis by A8,A88; end; suppose A89: k > 1 & x is not FinSequence of TOP-REAL 2; take {}; thus thesis by A89; end; consider F being Function such that A90: dom F = NAT and A91: F.0 = {} and A92: for k being Element of NAT holds P[k,F.k,F.(k+1)] from RecChoice(A3); defpred P[Nat] means F.$1 is FinSequence of TOP-REAL 2; A93: {} = <*>(the carrier of TOP-REAL 2); then A94: P[0] by A91; A95: for k st P[k] holds P[k+1] proof let k such that A96: F.k is FinSequence of TOP-REAL 2; A97: P[k,F.k,F.(k+1)] by A92; per cases by CQC_THE1:2; suppose k = 0; hence F.(k+1) is FinSequence of TOP-REAL 2 by A97; suppose k = 1; hence F.(k+1) is FinSequence of TOP-REAL 2 by A97; suppose A98: k > 1; thus thesis proof reconsider f = F.k as FinSequence of TOP-REAL 2 by A96; per cases; suppose A99: len f = k; thus thesis proof per cases; suppose f is_sequence_on G & right_cell(f,len f-'1,G) meets C ; then (front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) misses C implies ex i,j st f^<*G*(i,j)*> turns_right (len f)-'1,G & F.(k+1) = f^<*G*(i,j)*>) & (front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) meets C implies ex i,j st f^<*G*(i,j)*> goes_straight (len f)-'1,G & F.(k+1) = f^<*G*(i,j)*>) & (front_left_cell(f,(len f)-'1,G) meets C implies ex i,j st f^<*G*(i,j)*> turns_left (len f)-'1,G & F.(k+1) = f^<*G*(i,j)*>) by A92,A98,A99; hence F.(k+1) is FinSequence of TOP-REAL 2; suppose A100: not f is_sequence_on G or right_cell(f,len f-'1,G) misses C; f^<*G*(1,1)*> is FinSequence of TOP-REAL 2; hence F.(k+1) is FinSequence of TOP-REAL 2 by A92,A98,A99,A100; end; suppose len f <> k; hence F.(k+1) is FinSequence of TOP-REAL 2 by A92,A93,A98; end; end; A101: for k holds P[k] from Ind(A94,A95); rng F c= (the carrier of TOP-REAL 2)* proof let y be set; assume y in rng F; then ex x being set st x in dom F & F.x = y by FUNCT_1:def 5; then y is FinSequence of TOP-REAL 2 by A90,A101; hence thesis by FINSEQ_1:def 11; end; then reconsider F as Function of NAT,(the carrier of TOP-REAL 2)* by A90,FUNCT_2:def 1,RELSET_1:11; defpred P[Nat] means len(F.$1) = $1; A102: P[0] by A91,FINSEQ_1:25; A103: for k st P[k] holds P[k+1] proof let k such that A104: len(F.k) = k; A105: P[k,F.k,F.(k+1)] by A92; per cases by CQC_THE1:2; suppose k = 0; hence thesis by A105,FINSEQ_1:56; suppose k = 1; hence thesis by A105,FINSEQ_1:61; suppose A106: k > 1; thus thesis proof per cases; suppose F.k is_sequence_on G & right_cell(F.k,len(F.k)-'1,G) meets C; then (front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C implies ex i,j st (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*>) & (front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C implies ex i,j st (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*>) & (front_left_cell(F.k,(len(F.k))-'1,G) meets C implies ex i,j st (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*>) by A92,A104,A106; hence thesis by A104,FINSEQ_2:19; suppose not F.k is_sequence_on G or right_cell(F.k,len(F.k)-'1,G) misses C; then F.(k+1) = (F.k)^<*G*(1,1)*> by A92,A104,A106; hence thesis by A104,FINSEQ_2:19; end; end; A107: for k holds P[k] from Ind(A102,A103); defpred P[Nat] means F.$1 is_sequence_on G & for m st 1 <= m & m+1 <= len(F.$1) holds left_cell(F.$1,m,G) misses C & right_cell(F.$1,m,G) meets C; A108: P[0] proof (for n st n in dom(F.0) ex i,j st [i,j] in Indices G & (F.0)/.n = G*(i,j)) & (for n st n in dom(F.0) & n+1 in dom(F.0) holds for m,k,i,j st [m,k] in Indices G & [i,j] in Indices G & (F.0)/.n = G*(m,k) & (F.0)/.(n+1) = G*(i,j) holds abs(m-i)+abs(k-j) = 1) by A91,RELAT_1:60; hence F.0 is_sequence_on G by GOBOARD1:def 11; let m; assume A109: 1 <= m & m+1 <= len(F.0); 1 <= m+1 by NAT_1:37; then 1 <= len(F.0) by A109,AXIOMS:22; then 1 <= 0 by A91,FINSEQ_1:25; hence left_cell(F.0,m,G) misses C & right_cell(F.0,m,G) meets C; end; A110: now let k such that A111: F.k is_sequence_on G and A112: for m st 1 <= m & m+1 <= len(F.k) holds left_cell(F.k,m,G) misses C & right_cell(F.k,m,G) meets C and A113: k > 1; let i1,j1,i2,j2 be Nat such that A114: [i1,j1] in Indices G & (F.k)/.((len(F.k)) -'1) = G*(i1,j1) and A115: [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2); A116: len(F.k) = k by A107; then A117: 1 <= (len(F.k))-'1 by A113,JORDAN3:12; A118: (len(F.k)) -'1 +1 = len(F.k) by A113,A116,AMI_5:4; then A119: left_cell(F.k,(len(F.k))-'1,G) misses C & right_cell(F.k,(len(F.k))-'1,G) meets C by A112,A117; A120: 1 <= i1 & i1 <= len G & 1 <= j1 & j1 <= width G by A114,GOBOARD5:1; A121: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A115,GOBOARD5:1; A122: i1-'1 <= len G & j1-'1 <= width G by A120,JORDAN3:7; A123: 1 <= i2+1 & 1 <= j2+1 by NAT_1:37; A124: i2-'1 <= len G & j2-'1 <= width G by A121,JORDAN3:7; hereby assume A125: i1 = i2 & j1+1 = j2; now assume i2+1 > len G; then i2+1 <= (len G)+1 & (len G)+1 <= i2+1 by A121,AXIOMS:24,NAT_1:38; then i2+1 = (len G)+1 by AXIOMS:21; then i2 = len G by XCMPLX_1:2; then cell(G,len G,j1) meets C by A111,A114,A115,A117,A118,A119,A125,GOBRD13:23; hence contradiction by A2,A120,JORDAN8:19; end; hence [i2+1,j2] in Indices G by A121,A123,GOBOARD7:10; end; hereby assume A126: i1+1 = i2 & j1 = j2; now assume j2-'1 < 1; then j2-'1 = 0 by RLVECT_1:98; then j2 <= 1 by JORDAN4:1; then j2 = 1 by A121,AXIOMS:21; then cell(G,i1,1-'1) meets C by A111,A114,A115,A117,A118,A119,A126, GOBRD13:25; then cell(G,i1,0) meets C by GOBOARD9:1; hence contradiction by A120,JORDAN8:20; end; hence [i2,j2-'1] in Indices G by A121,A124,GOBOARD7:10; end; hereby assume A127: i1 = i2+1 & j1 = j2; now assume j2+1 > len G; then j2+1 <= (len G)+1 & (len G)+1 <= j2+1 by A2,A121,AXIOMS:24,NAT_1:38; then j2+1 = (len G)+1 by AXIOMS:21; then j2 = len G by XCMPLX_1:2; then cell(G,i2,len G) meets C by A111,A114,A115,A117,A118,A119,A127, GOBRD13:27; hence contradiction by A121,JORDAN8:18; end; hence [i2,j2+1] in Indices G by A2,A121,A123,GOBOARD7:10; end; hereby assume A128: i1 = i2 & j1 = j2+1; now assume i2-'1 < 1; then i2-'1 = 0 by RLVECT_1:98; then i2 <= 1 by JORDAN4:1; then i2 = 1 by A121,AXIOMS:21; then cell(G,1-'1,j2) meets C by A111,A114,A115,A117,A118,A119,A128, GOBRD13:29; then cell(G,0,j2) meets C by GOBOARD9:1; hence contradiction by A2,A121,JORDAN8:21; end; hence [i2-'1,j2] in Indices G by A121,A124,GOBOARD7:10; end; hereby assume that A129: front_right_cell(F.k,(len(F.k))-'1,G) meets C and A130: i1 = i2 & j1+1 = j2; now assume j2+1 > len G; then j2+1 <= (len G)+1 & (len G)+1 <= j2+1 by A2,A121,AXIOMS:24,NAT_1:38; then j2+1 = (len G)+1 by AXIOMS:21; then j2 = len G by XCMPLX_1:2; then cell(G,i1,len G) meets C by A111,A114,A115,A117,A118,A129,A130, GOBRD13:36; hence contradiction by A120,JORDAN8:18; end; hence [i2,j2+1] in Indices G by A2,A121,A123,GOBOARD7:10; end; hereby assume that A131: front_right_cell(F.k,(len(F.k))-'1,G) meets C and A132: i1+1 = i2 & j1 = j2; now assume i2+1 > len G; then i2+1 <= (len G)+1 & (len G)+1 <= i2+1 by A121,AXIOMS:24,NAT_1:38; then i2+1 = (len G)+1 by AXIOMS:21; then i2 = len G by XCMPLX_1:2; then cell(G,len G,j1-'1) meets C by A111,A114,A115,A117,A118,A131,A132,GOBRD13:38; hence contradiction by A2,A122,JORDAN8:19; end; hence [i2+1,j2] in Indices G by A121,A123,GOBOARD7:10; end; hereby assume that A133: front_right_cell(F.k,(len(F.k))-'1,G) meets C and A134: i1 = i2+1 & j1 = j2; now assume i2-'1 < 1; then i2-'1 = 0 by RLVECT_1:98; then i2 <= 1 by JORDAN4:1; then i2 = 1 by A121,AXIOMS:21; then cell(G,1-'1,j1) meets C by A111,A114,A115,A117,A118,A133,A134, GOBRD13:40; then cell(G,0,j1) meets C by GOBOARD9:1; hence contradiction by A2,A120,JORDAN8:21; end; hence [i2-'1,j2] in Indices G by A121,A124,GOBOARD7:10; end; hereby assume that A135: front_right_cell(F.k,(len(F.k))-'1,G) meets C and A136: i1 = i2 & j1 = j2+1; now assume j2-'1 < 1; then j2-'1 = 0 by RLVECT_1:98; then j2 <= 1 by JORDAN4:1; then j2 = 1 by A121,AXIOMS:21; then cell(G,i1-'1,1-'1) meets C by A111,A114,A115,A117,A118,A135,A136,GOBRD13:42; then cell(G,i1-'1,0) meets C by GOBOARD9:1; hence contradiction by A122,JORDAN8:20; end; hence [i2,j2-'1] in Indices G by A121,A124,GOBOARD7:10; end; hereby assume that A137: front_left_cell(F.k,(len(F.k))-'1,G) meets C and A138: i1 = i2 & j1+1 = j2; now assume i2-'1 < 1; then i2-'1 = 0 by RLVECT_1:98; then i2 <= 1 by JORDAN4:1; then i2 = 1 by A121,AXIOMS:21; then cell(G,1-'1,j2) meets C by A111,A114,A115,A117,A118,A137,A138, GOBRD13:35; then cell(G,0,j2) meets C by GOBOARD9:1; hence contradiction by A2,A121,JORDAN8:21; end; hence [i2-'1,j2] in Indices G by A121,A124,GOBOARD7:10; end; hereby assume that A139: front_left_cell(F.k,(len(F.k))-'1,G) meets C and A140: i1+1 = i2 & j1 = j2; now assume j2+1 > len G; then j2+1 <= (len G)+1 & (len G)+1 <= j2+1 by A2,A121,AXIOMS:24,NAT_1:38; then j2+1 = (len G)+1 by AXIOMS:21; then j2 = len G by XCMPLX_1:2; then cell(G,i2,len G) meets C by A111,A114,A115,A117,A118,A139,A140, GOBRD13:37; hence contradiction by A121,JORDAN8:18; end; hence [i2,j2+1] in Indices G by A2,A121,A123,GOBOARD7:10; end; hereby assume that A141: front_left_cell(F.k,(len(F.k))-'1,G) meets C and A142: i1 = i2+1 & j1 = j2; now assume j2-'1 < 1; then j2-'1 = 0 by RLVECT_1:98; then j2 <= 1 by JORDAN4:1; then j2 = 1 by A121,AXIOMS:21; then cell(G,i2-'1,1-'1) meets C by A111,A114,A115,A117,A118,A141,A142,GOBRD13:39; then cell(G,i2-'1,0) meets C by GOBOARD9:1; hence contradiction by A124,JORDAN8:20; end; hence [i2,j2-'1] in Indices G by A121,A124,GOBOARD7:10; end; hereby assume that A143: front_left_cell(F.k,(len(F.k))-'1,G) meets C and A144: i1 = i2 & j1 = j2+1; now assume i2+1 > len G; then i2+1 <= (len G)+1 & (len G)+1 <= i2+1 by A121,AXIOMS:24,NAT_1:38; then i2+1 = (len G)+1 by AXIOMS:21; then i2 = len G by XCMPLX_1:2; then cell(G,len G,j2-'1) meets C by A111,A114,A115,A117,A118,A143,A144,GOBRD13:41; hence contradiction by A2,A124,JORDAN8:19; end; hence [i2+1,j2] in Indices G by A121,A123,GOBOARD7:10; end; end; A145: for k st P[k] holds P[k+1] proof let k such that A146: F.k is_sequence_on G and A147: for m st 1 <= m & m+1 <= len(F.k) holds left_cell(F.k,m,G) misses C & right_cell(F.k,m,G) meets C; A148: len(F.k) = k by A107; A149: len(F.(k+1)) = k+1 by A107; A150: 1 <= len G by A2,NAT_1:37; A151: 2|^n > 0 by HEINE:5; per cases by CQC_THE1:2; suppose A152: k = 0; then consider i such that A153: 1 <= i & i+1 <= len G and N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) and A154: F.(k+1) = <*G*(i,width G)*> by A92; i < len G by A153,NAT_1:38; then A155: [i,len G] in Indices G by A2,A150,A153,GOBOARD7:10; A156: now let l; assume l in dom(F.(k+1)); then 1 <= l & l <= 1 by A149,A152,FINSEQ_3:27; then l = 1 by AXIOMS:21; then (F.(k+1))/.l = G*(i,width G) by A154,FINSEQ_4:25; hence ex i,j st [i,j] in Indices G & (F.(k+1))/.l = G* (i,j) by A2,A155; end; now let l; assume l in dom(F.(k+1)) & l+1 in dom(F.(k+1)); then 1 <= l & l <= 1 & 1 <= l+1 & l+1 <= 1 by A149,A152,FINSEQ_3:27; then l = 1 & 1 = l+1 by AXIOMS:21; hence for i1,j1,i2,j2 st [i1,j1] in Indices G & [i2,j2] in Indices G & (F.(k+1))/.l = G*(i1,j1) & (F.(k+1))/.(l+1) = G*(i2,j2) holds abs(i1-i2)+abs(j1-j2) = 1; end; hence F.(k+1) is_sequence_on G by A156,GOBOARD1:def 11; let m; assume A157: 1 <= m & m+1 <= len(F.(k+1)); 1 <= m+1 by NAT_1:37; then m+1 = 0+1 by A149,A152,A157,AXIOMS:21; then m = 0 by XCMPLX_1:2; hence left_cell(F.(k+1),m,G) misses C & right_cell(F.(k+1),m,G) meets C by A157; suppose A158: k = 1; then consider i such that A159: 1 <= i & i+1 <= len G and A160: N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) and A161: F.(k+1) = <*G*(i,width G),G*(i+1,width G)*> by A92; A162: (F.(k+1))/.1 = G*(i,width G) & (F.(k+1))/.2 = G*(i+1,width G) by A161,FINSEQ_4:26; A163: i < len G & 1 <= i+1 & i+1 <= len G by A159,NAT_1:38; then A164: [i,len G] in Indices G & [i+1,len G] in Indices G by A2,A150,A159,GOBOARD7:10; A165: now let l; assume A166: l in dom(F.(k+1)); then 1 <= l & l <= 2 by A149,A158,FINSEQ_3:27; then l = 0 or l = 1 or l = 2 by CQC_THE1:3; hence ex i,j st [i,j] in Indices G & (F.(k+1))/.l = G*(i,j) by A2,A162,A164,A166,FINSEQ_3:27; end; now let l; assume A167: l in dom(F.(k+1)) & l+1 in dom(F.(k+1)); then 1 <= l & l <= 2 & 1 <= l+1 & l+1 <= 2 by A149,A158,FINSEQ_3:27; then A168: l = 0 or l = 1 or l = 2 by CQC_THE1:3; let i1,j1,i2,j2 such that A169: [i1,j1] in Indices G & [i2,j2] in Indices G & (F.(k+1))/.l = G*(i1,j1) & (F.(k+1))/.(l+1) = G*(i2,j2); i1 = i & j1 = len G & i2 = i+1 & j2 = len G by A2,A149,A158,A162,A164,A167,A168,A169,FINSEQ_3:27,GOBOARD1:21; then i2-i1 = 1 & j1-j2 = 0 by XCMPLX_1:14,26; then abs(i2-i1) = 1 & abs(j1-j2) = 0 by ABSVALUE:def 1; hence abs(i1-i2)+abs(j1-j2) = 1 by UNIFORM1:13; end; hence A170: F.(k+1) is_sequence_on G by A165,GOBOARD1:def 11; let m; assume A171: 1 <= m & m+1 <= len(F.(k+1)); then 1+1 <= m+1 by AXIOMS:24; then m+1 = 1+1 by A149,A158,A171,AXIOMS:21; then A172: m = 1 by XCMPLX_1:2; A173: i < i+1 & i+1 < (i+1)+1 by NAT_1:38; then A174: left_cell(F.(k+1),m,G) = cell(G,i,len G) by A2,A162,A164,A170,A171,A172,GOBRD13:def 3; now assume left_cell(F.(k+1),m,G) meets C; then consider p being set such that A175: p in cell(G,i,len G) & p in C by A174,XBOOLE_0:3; reconsider p as Point of TOP-REAL 2 by A175; reconsider p as Element of TOP-REAL 2; cell(G,i,len G) = { |[r,s]|: G*(i,1)`1 <= r & r <= G*(i+1,1)`1 & G*(1,len G)`2 <= s } by A2,A159,A163,GOBRD11:31; then consider r,s such that A176: p = |[r,s]| and G*(i,1)`1 <= r & r <= G*(i+1,1)`1 and A177: G*(1,len G)`2 <= s by A175; [1,len G] in Indices G by A2,A150,GOBOARD7:10; then A178: G* (1,len G) = |[W+((E-W)/(2|^n))*(1-2),S+((N-S)/(2|^n))*((len G)-2)]| by JORDAN8:def 1; 2|^n+(2+1) = 2|^n+1+2 by XCMPLX_1:1; then (len G) - 2 = 2|^n+1 by A2,XCMPLX_1:26; then ((N-S)/(2|^n))*((len G)-2) = ((N-S)/(2|^n))*(2|^n)+((N-S)/(2|^n))*1 by XCMPLX_1:8 .= (N-S)+(N-S)/(2|^n) by A151,XCMPLX_1:88; then A179: S+((N-S)/(2|^n))*((len G) -2) = S+(N-S)+(N-S)/(2|^n) by XCMPLX_1:1 .= N+(N-S)/(2|^n) by XCMPLX_1:27; A180: G*(1,len G)`2 = S+((N-S)/(2|^n))*((len G)-2) by A178,EUCLID:56; N > S by JORDAN8:12; then N-S > S-S by REAL_1:54; then N-S > 0 by XCMPLX_1:14; then (N-S)/(2|^n) > 0 by A151,REAL_2:127; then N+0 < N+(N-S)/(2|^n) by REAL_1:53; then A181: N < s by A177,A179,A180,AXIOMS:22; p`2 <= N by A175,PSCOMP_1:71; hence contradiction by A176,A181,EUCLID:56; end; hence left_cell(F.(k+1),m,G) misses C; A182: N-min C in C by SPRECT_1:13; N-min C in right_cell(F.(k+1),m,G) by A2,A160,A162,A164,A170,A171,A172,A173,GOBRD13:def 2; hence right_cell(F.(k+1),m,G) meets C by A182,XBOOLE_0:3; suppose A183: k > 1; then k > 0 by AXIOMS:22; then A184: F.k is non empty by A148,FINSEQ_1:25; A185: 1 <= (len(F.k))-'1 by A148,A183,JORDAN3:12; A186: (len(F.k)) -'1 +1 = len(F.k) by A148,A183,AMI_5:4; then consider i1,j1,i2,j2 being Nat such that A187: [i1,j1] in Indices G & (F.k)/.((len(F.k)) -'1) = G*(i1,j1) and A188: [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2) and i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A146,A185,JORDAN8:6; A189: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A188,GOBOARD5:1; (len(F.k))-'1 <= len(F.k) by GOBOARD9:2; then A190: (len(F.k))-'1 in dom(F.k) & len(F.k) in dom(F.k) by A148,A183,A185,FINSEQ_3:27; A191: (len(F.k))-'1+(1+1) = (len(F.k))+1 by A186,XCMPLX_1:1; A192: left_cell(F.k,(len(F.k))-'1,G) misses C & right_cell(F.k,(len(F.k))-'1,G) meets C by A147,A185,A186; A193: i1 = i2 & j1+1 = j2 implies [i2+1,j2] in Indices G by A110,A146,A147,A183 ,A187,A188; A194: i1+1 = i2 & j1 = j2 implies [i2,j2-'1] in Indices G by A110,A146,A147, A183,A187,A188; A195: i1 = i2+1 & j1 = j2 implies [i2,j2+1] in Indices G by A110,A146,A147,A183 ,A187,A188; A196: i1 = i2 & j1 = j2+1 implies [i2-'1,j2] in Indices G by A110,A146,A147, A183,A187,A188; A197: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1+1 = j2 implies [i2,j2+1] in Indices G by A110,A146,A147,A183, A187,A188; A198: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1+1 = i2 & j1 = j2 implies [i2+1,j2] in Indices G by A110,A146,A147,A183, A187,A188; A199: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2+1 & j1 = j2 implies [i2-'1,j2] in Indices G by A110,A146,A147,A183 ,A187,A188; A200: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 = j2+1 implies [i2,j2-'1] in Indices G by A110,A146,A147,A183 ,A187,A188; A201: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1+1 = j2 implies [i2-'1,j2] in Indices G by A110,A146,A147,A183 ,A187,A188; A202: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1+1 = i2 & j1 = j2 implies [i2,j2+1] in Indices G by A110,A146,A147,A183, A187,A188; A203: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2+1 & j1 = j2 implies [i2,j2-'1] in Indices G by A110,A146,A147,A183 ,A187,A188; A204: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 = j2+1 implies [i2+1,j2] in Indices G by A110,A146,A147,A183, A187,A188; thus A205: F.(k+1) is_sequence_on G proof per cases; suppose front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; then consider i,j such that A206: (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G and A207: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A146,A148,A183,A192; thus thesis proof set f = (F.k)^<*G*(i,j)*>; A208: f/.(len(F.k)-'1) = G*(i1,j1) & f/.len(F.k) = G*(i2,j2) by A187,A188,A190,GROUP_5:95; A209: f/.(len(F.k)+1) = G*(i,j) by TOPREAL4:1; per cases by A186,A187,A188,A191,A206,A208,GOBRD13:def 6; suppose that A210: i1 = i2 & j1+1 = j2 and A211: f/.(len(F.k)+1) = G*(i2+1,j2); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2+1,j2) = G*(i2',j2'); then i2 = i1' & j2 = j1' & i2+1 = i2' & j2 = j2' by A188,A193,A210,GOBOARD1:21; then i2'-i1' = 1 & j2'-j1' = 0 by XCMPLX_1:14,26; then abs(i2'-i1') = 1 & abs(j2'-j1') = 0 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1; end; hence F.(k+1) is_sequence_on G by A146,A184,A193,A207,A209,A210,A211 ,JORDAN8:9; suppose that A212: i1+1 = i2 & j1 = j2 and A213: f/.(len(F.k)+1) = G*(i2,j2-'1); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2,j2-'1) = G* (i2',j2'); then A214: i2 = i1' & j2 = j1' & i2 = i2' & j2-'1 = j2' by A188,A194,A212,GOBOARD1:21; then j1'-j2' = j2-(j2-1) by A189,SCMFSA_7:3; then i2'-i1' = 0 & j1'-j2' = 1 by A214,XCMPLX_1:14,18; then abs(i2'-i1') = 0 & abs(j1'-j2') = 1 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1 by UNIFORM1:13; end; hence F.(k+1) is_sequence_on G by A146,A184,A194,A207,A209,A212,A213 ,JORDAN8:9; suppose that A215: i1 = i2+1 & j1 = j2 and A216: f/.(len(F.k)+1) = G*(i2,j2+1); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2,j2+1) = G*(i2',j2'); then i2 = i1' & j2 = j1' & i2 = i2' & j2+1 = j2' by A188,A195,A215,GOBOARD1:21; then i2'-i1' = 0 & j2'-j1' = 1 by XCMPLX_1:14,26; then abs(i2'-i1') = 0 & abs(j2'-j1') = 1 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1; end; hence F.(k+1) is_sequence_on G by A146,A184,A195,A207,A209,A215,A216 ,JORDAN8:9; suppose that A217: i1 = i2 & j1 = j2+1 and A218: f/.(len(F.k)+1) = G*(i2-'1,j2); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2-'1,j2) = G* (i2',j2'); then A219: i2 = i1' & j2 = j1' & i2-'1 = i2' & j2 = j2' by A188,A196,A217,GOBOARD1:21; then i1'-i2' = i2-(i2-1) by A189,SCMFSA_7:3; then j2'-j1' = 0 & i1'-i2' = 1 by A219,XCMPLX_1:14,18; then abs(j2'-j1') = 0 & abs(i1'-i2') = 1 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1 by UNIFORM1:13; end; hence F.(k+1) is_sequence_on G by A146,A184,A196,A207,A209,A217,A218 ,JORDAN8:9; end; suppose A220: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that A221: (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G and A222: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A146,A148,A183,A192; thus thesis proof set f = (F.k)^<*G*(i,j)*>; A223: f/.(len(F.k)-'1) = G*(i1,j1) & f/.len(F.k) = G*(i2,j2) by A187,A188,A190,GROUP_5:95; A224: f/.(len(F.k)+1) = G*(i,j) by TOPREAL4:1; per cases by A186,A187,A188,A191,A221,A223,GOBRD13:def 8; suppose that A225: i1 = i2 & j1+1 = j2 and A226: f/.(len(F.k)+1) = G*(i2,j2+1); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2,j2+1) = G*(i2',j2'); then i2 = i1' & j2 = j1' & i2 = i2' & j2+1 = j2' by A188,A197,A220,A225,GOBOARD1:21; then i2'-i1' = 0 & j2'-j1' = 1 by XCMPLX_1:14,26; then abs(i2'-i1') = 0 & abs(j2'-j1') = 1 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1; end; hence F.(k+1) is_sequence_on G by A146,A184,A197,A220,A222,A224,A225, A226,JORDAN8:9; suppose that A227: i1+1 = i2 & j1 = j2 and A228: f/.(len(F.k)+1) = G*(i2+1,j2); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2+1,j2) = G* (i2',j2'); then i2 = i1' & j2 = j1' & i2+1 = i2' & j2 = j2' by A188,A198,A220,A227,GOBOARD1:21; then i2'-i1' = 1 & j2'-j1' = 0 by XCMPLX_1:14,26; then abs(i2'-i1') = 1 & abs(j2'-j1') = 0 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1; end; hence F.(k+1) is_sequence_on G by A146,A184,A198,A220,A222,A224,A227, A228,JORDAN8:9; suppose that A229: i1 = i2+1 & j1 = j2 and A230: f/.(len(F.k)+1) = G*(i2-'1,j2); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2-'1,j2) = G*(i2',j2'); then A231: i2 = i1' & j2 = j1' & i2-'1 = i2' & j2 = j2' by A188,A199,A220,A229,GOBOARD1:21; then i1'-i2' = i2-(i2-1) by A189,SCMFSA_7:3; then j2'-j1' = 0 & i1'-i2' = 1 by A231,XCMPLX_1:14,18; then abs(j2'-j1') = 0 & abs(i1'-i2') = 1 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1 by UNIFORM1:13; end; hence F.(k+1) is_sequence_on G by A146,A184,A199,A220,A222,A224,A229, A230,JORDAN8:9; suppose that A232: i1 = i2 & j1 = j2+1 and A233: f/.(len(F.k)+1) = G*(i2,j2-'1); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2,j2-'1) = G* (i2',j2'); then A234: i2 = i1' & j2 = j1' & i2 = i2' & j2-'1 = j2' by A188,A200,A220,A232,GOBOARD1:21; then j1'-j2' = j2-(j2-1) by A189,SCMFSA_7:3; then i2'-i1' = 0 & j1'-j2' = 1 by A234,XCMPLX_1:14,18; then abs(i2'-i1') = 0 & abs(j1'-j2') = 1 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1 by UNIFORM1:13; end; hence F.(k+1) is_sequence_on G by A146,A184,A200,A220,A222,A224,A232, A233,JORDAN8:9; end; suppose A235: front_left_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that A236: (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G and A237: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A146,A148,A183,A192; thus thesis proof set f = (F.k)^<*G*(i,j)*>; A238: f/.(len(F.k)-'1) = G*(i1,j1) & f/.len(F.k) = G*(i2,j2) by A187,A188,A190,GROUP_5:95; A239: f/.(len(F.k)+1) = G*(i,j) by TOPREAL4:1; per cases by A186,A187,A188,A191,A236,A238,GOBRD13:def 7; suppose that A240: i1 = i2 & j1+1 = j2 and A241: f/.(len(F.k)+1) = G*(i2-'1,j2); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2-'1,j2) = G*(i2',j2'); then A242: i2 = i1' & j2 = j1' & i2-'1 = i2' & j2 = j2' by A188,A201,A235,A240,GOBOARD1:21; then i1'-i2' = i2-(i2-1) by A189,SCMFSA_7:3; then j2'-j1' = 0 & i1'-i2' = 1 by A242,XCMPLX_1:14,18; then abs(j2'-j1') = 0 & abs(i1'-i2') = 1 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1 by UNIFORM1:13; end; hence F.(k+1) is_sequence_on G by A146,A184,A201,A235,A237,A239,A240, A241,JORDAN8:9; suppose that A243: i1+1 = i2 & j1 = j2 and A244: f/.(len(F.k)+1) = G*(i2,j2+1); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2,j2+1) = G* (i2',j2'); then i2 = i1' & j2 = j1' & i2 = i2' & j2+1 = j2' by A188,A202,A235,A243,GOBOARD1:21; then i2'-i1' = 0 & j2'-j1' = 1 by XCMPLX_1:14,26; then abs(i2'-i1') = 0 & abs(j2'-j1') = 1 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1; end; hence F.(k+1) is_sequence_on G by A146,A184,A202,A235,A237,A239,A243, A244,JORDAN8:9; suppose that A245: i1 = i2+1 & j1 = j2 and A246: f/.(len(F.k)+1) = G*(i2,j2-'1); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2,j2-'1) = G*(i2',j2'); then A247: i2 = i1' & j2 = j1' & i2 = i2' & j2-'1 = j2' by A188,A203,A235,A245,GOBOARD1:21; then j1'-j2' = j2-(j2-1) by A189,SCMFSA_7:3; then i2'-i1' = 0 & j1'-j2' = 1 by A247,XCMPLX_1:14,18; then abs(i2'-i1') = 0 & abs(j1'-j2') = 1 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1 by UNIFORM1:13; end; hence F.(k+1) is_sequence_on G by A146,A184,A203,A235,A237,A239,A245, A246,JORDAN8:9; suppose that A248: i1 = i2 & j1 = j2+1 and A249: f/.(len(F.k)+1) = G*(i2+1,j2); now let i1',j1',i2',j2' be Nat; assume [i1',j1'] in Indices G & [i2',j2'] in Indices G & (F.k)/.len(F.k) = G*(i1',j1') & G*(i2+1,j2) = G* (i2',j2'); then i2 = i1' & j2 = j1' & i2+1 = i2' & j2 = j2' by A188,A204,A235,A248,GOBOARD1:21; then i2'-i1' = 1 & j2'-j1' = 0 by XCMPLX_1:14,26; then abs(i2'-i1') = 1 & abs(j2'-j1') = 0 by ABSVALUE:def 1; hence abs(i2'-i1')+abs(j2'-j1') = 1; end; hence F.(k+1) is_sequence_on G by A146,A184,A204,A235,A237,A239,A248, A249,JORDAN8:9; end; end; let m such that A250: 1 <= m & m+1 <= len(F.(k+1)); now per cases; suppose m+1 = len(F.(k+1)); then A251: m = len(F.k) by A148,A149,XCMPLX_1:2; A252: (i2-'1)+1 = i2 by A189,AMI_5:4; A253: (j2-'1)+1 = j2 by A189,AMI_5:4; thus thesis proof per cases; suppose A254: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; then consider i,j such that A255: (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G and A256: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A146,A148,A183,A192; A257: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) & (F.(k+1))/.len(F.k) = G* (i2,j2) by A187,A188,A190,A256,GROUP_5:95; now per cases by A186,A187,A188,A191,A255,A256,A257,GOBRD13:def 6; suppose that A258: i1 = i2 & j1+1 = j2 and A259: (F.(k+1))/.((len(F.k))+1) = G*(i2+1,j2); front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i1,j2) by A146,A185,A186,A187,A188,A258,GOBRD13:36; hence left_cell(F.(k+1),m,G) misses C by A188,A193,A205,A250,A251,A254,A257,A258,A259,GOBRD13:24; j2-'1 = j1 & cell(G,i1,j1) meets C by A146,A185,A186,A187,A188,A192,A258,BINARITH:39,GOBRD13:23; hence right_cell(F.(k+1),m,G) meets C by A188,A193,A205,A250,A251,A257,A258,A259,GOBRD13:25; suppose that A260: i1+1 = i2 & j1 = j2 and A261: (F.(k+1))/.((len(F.k))+1) = G*(i2,j2-'1); front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2-'1) by A146,A185,A186,A187,A188,A260,GOBRD13:38; hence left_cell(F.(k+1),m,G) misses C by A188,A194,A205,A250,A251,A253,A254,A257,A260,A261,GOBRD13:28; i2-'1 = i1 & cell(G,i1,j1-'1) meets C by A146,A185,A186,A187,A188,A192,A260,BINARITH:39,GOBRD13:25; hence right_cell(F.(k+1),m,G) meets C by A188,A194,A205,A250,A251,A253,A257,A260,A261,GOBRD13:29; suppose that A262: i1 = i2+1 & j1 = j2 and A263: (F.(k+1))/.((len(F.k))+1) = G*(i2,j2+1); front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1,j2) by A146,A185,A186,A187,A188,A262,GOBRD13:40; hence left_cell(F.(k+1),m,G) misses C by A188,A195,A205,A250,A251,A254,A257,A262,A263,GOBRD13:22; cell(G,i2,j2) meets C by A146,A185,A186,A187,A188,A192,A262,GOBRD13:27; hence right_cell(F.(k+1),m,G) meets C by A188,A195,A205,A250,A251,A257,A262,A263,GOBRD13:23; suppose that A264: i1 = i2 & j1 = j2+1 and A265: (F.(k+1))/.((len(F.k))+1) = G*(i2-'1,j2); front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1,j2-'1) by A146,A185,A186,A187,A188,A264,GOBRD13:42; hence left_cell(F.(k+1),m,G) misses C by A188,A196,A205,A250,A251,A252,A254,A257,A264,A265,GOBRD13:26; cell(G,i2-'1,j2) meets C by A146,A185,A186,A187,A188,A192,A264,GOBRD13:29; hence right_cell(F.(k+1),m,G) meets C by A188,A196,A205,A250,A251,A252,A257,A264,A265,GOBRD13:27; end; hence thesis; suppose A266: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that A267: (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G and A268: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A146,A148,A183,A192; A269: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) & (F.(k+1))/.len(F.k) = G*(i2,j2) by A187,A188,A190,A268,GROUP_5:95; now per cases by A186,A187,A188,A191,A267,A268,A269,GOBRD13:def 8; suppose that A270: i1 = i2 & j1+1 = j2 and A271: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2+1); front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i1-'1,j2) by A146,A185,A186,A187,A188,A270,GOBRD13:35; hence left_cell(F.(k+1),m,G) misses C by A188,A197,A205,A250,A251,A266,A269,A270,A271,GOBRD13:22; front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i1,j2) by A146,A185,A186,A187,A188,A270,GOBRD13:36; hence right_cell(F.(k+1),m,G) meets C by A188,A197,A205,A250,A251,A266,A269,A270,A271,GOBRD13:23; suppose that A272: i1+1 = i2 & j1 = j2 and A273: (F.(k+1))/.(len(F.k)+1) = G*(i2+1,j2); front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2) by A146,A185,A186,A187,A188,A272,GOBRD13:37; hence left_cell(F.(k+1),m,G) misses C by A188,A198,A205,A250,A251,A266,A269,A272,A273,GOBRD13:24; front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2-'1) by A146,A185,A186,A187,A188,A272,GOBRD13:38; hence right_cell(F.(k+1),m,G) meets C by A188,A198,A205,A250,A251,A266,A269,A272,A273,GOBRD13:25; suppose that A274: i1 = i2+1 & j1 = j2 and A275: (F.(k+1))/.(len(F.k)+1) = G*(i2-'1,j2); front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1,j2-'1) by A146,A185,A186,A187,A188,A274,GOBRD13:39; hence left_cell(F.(k+1),m,G) misses C by A188,A199,A205,A250,A251,A252,A266,A269,A274,A275,GOBRD13:26; front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1,j2) by A146,A185,A186,A187,A188,A274,GOBRD13:40; hence right_cell(F.(k+1),m,G) meets C by A188,A199,A205,A250,A251,A252,A266,A269,A274,A275,GOBRD13:27; suppose that A276: i1 = i2 & j1 = j2+1 and A277: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2-'1); front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2-'1) by A146,A185,A186,A187,A188,A276,GOBRD13:41; hence left_cell(F.(k+1),m,G) misses C by A188,A200,A205,A250,A251,A253,A266,A269,A276,A277,GOBRD13:28; front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1,j2-'1) by A146,A185,A186,A187,A188,A276,GOBRD13:42; hence right_cell(F.(k+1),m,G) meets C by A188,A200,A205,A250,A251,A253,A266,A269,A276,A277,GOBRD13:29; end; hence thesis; suppose A278: front_left_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that A279: (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G and A280: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A146,A148,A183,A192; A281: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) & (F.(k+1))/.len(F.k) = G* (i2,j2) by A187,A188,A190,A280,GROUP_5:95; now per cases by A186,A187,A188,A191,A279,A280,A281,GOBRD13:def 7; suppose that A282: i1 = i2 & j1+1 = j2 and A283: (F.(k+1))/.(len(F.k)+1) = G*(i2-'1,j2); j2-'1 = j1 & cell(G,i1-'1,j1) misses C by A146,A185,A186,A187,A188,A192,A282,BINARITH:39,GOBRD13:22; hence left_cell(F.(k+1),m,G) misses C by A188,A201,A205,A250,A251,A252,A278,A281,A282,A283,GOBRD13:26; front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i1-'1,j2) by A146,A185,A186,A187,A188,A282,GOBRD13:35; hence right_cell(F.(k+1),m,G) meets C by A188,A201,A205,A250,A251,A252,A278,A281,A282,A283,GOBRD13:27; suppose that A284: i1+1 = i2 & j1 = j2 and A285: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2+1); i2-'1 = i1 & cell(G,i1,j1) misses C by A146,A185,A186,A187,A188,A192,A284,BINARITH:39,GOBRD13:24; hence left_cell(F.(k+1),m,G) misses C by A188,A202,A205,A250,A251,A278,A281,A284,A285,GOBRD13:22; front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2) by A146,A185,A186,A187,A188,A284,GOBRD13:37; hence right_cell(F.(k+1),m,G) meets C by A188,A202,A205,A250,A251,A278,A281,A284,A285,GOBRD13:23; suppose that A286: i1 = i2+1 & j1 = j2 and A287: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2-'1); cell(G,i2,j2-'1) misses C by A146,A185,A186,A187,A188,A192,A286,GOBRD13:26; hence left_cell(F.(k+1),m,G) misses C by A188,A203,A205,A250,A251,A253,A278,A281,A286,A287,GOBRD13:28; front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1,j2-'1) by A146,A185,A186,A187,A188,A286,GOBRD13:39; hence right_cell(F.(k+1),m,G) meets C by A188,A203,A205,A250,A251,A253,A278,A281,A286,A287,GOBRD13:29; suppose that A288: i1 = i2 & j1 = j2+1 and A289: (F.(k+1))/.(len(F.k)+1) = G*(i2+1,j2); cell(G,i2,j2) misses C by A146,A185,A186,A187,A188,A192,A288, GOBRD13:28; hence left_cell(F.(k+1),m,G) misses C by A188,A204,A205,A250,A251,A278,A281,A288,A289,GOBRD13:24; front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2-'1) by A146,A185,A186,A187,A188,A288,GOBRD13:41; hence right_cell(F.(k+1),m,G) meets C by A188,A204,A205,A250,A251,A278,A281,A288,A289,GOBRD13:25; end; hence thesis; end; suppose m+1 <> len(F.(k+1)); then m+1 < len(F.(k+1)) by A250,REAL_1:def 5; then A290: m+1 <= len(F.k)by A148,A149,NAT_1:38; then consider i1,j1,i2,j2 being Nat such that A291: [i1,j1] in Indices G & (F.k)/.m = G*(i1,j1) and A292: [i2,j2] in Indices G & (F.k)/.(m+1) = G*(i2,j2) and A293: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A146,A250,JORDAN8:6; A294: 1 <= m+1 by NAT_1:37; m <= len(F.k) by A290,NAT_1:38; then A295: m in dom(F.k) & m+1 in dom(F.k) by A250,A290,A294,FINSEQ_3:27; A296: left_cell(F.k,m,G) misses C & right_cell(F.k,m,G) meets C by A147,A250,A290; now per cases; suppose front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; then consider i,j such that (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G and A297: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A146,A148,A183,A192; take i,j; thus F.(k+1) = (F.k)^<*G*(i,j)*> by A297; suppose front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G and A298: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A146,A148,A183,A192; take i,j; thus F.(k+1) = (F.k)^<*G*(i,j)*> by A298; suppose front_left_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G and A299: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A146,A148,A183,A192; take i,j; thus F.(k+1) = (F.k)^<*G*(i,j)*> by A299; end; then consider i,j such that A300: F.(k+1) = (F.k)^<*G*(i,j)*>; A301: (F.(k+1))/.m = G*(i1,j1) & (F.(k+1))/.(m+1) = G*(i2,j2) by A291,A292,A295,A300,GROUP_5:95; now per cases by A293; suppose A302: i1 = i2 & j1+1 = j2; then left_cell(F.k,m,G) = cell(G,i1-'1,j1) & right_cell(F.k,m,G) = cell(G,i1,j1) by A146,A250,A290,A291,A292,GOBRD13:22,23; hence thesis by A205,A250,A291,A292,A296,A301,A302,GOBRD13:22,23; suppose A303: i1+1 = i2 & j1 = j2; then left_cell(F.k,m,G) = cell(G,i1,j1) & right_cell(F.k,m,G) = cell(G,i1,j1-'1) by A146,A250,A290,A291,A292,GOBRD13:24,25; hence thesis by A205,A250,A291,A292,A296,A301,A303,GOBRD13:24,25; suppose A304: i1 = i2+1 & j1 = j2; then left_cell(F.k,m,G) = cell(G,i2,j2-'1) & right_cell(F.k,m,G) = cell(G,i2,j2) by A146,A250,A290,A291,A292,GOBRD13:26,27; hence thesis by A205,A250,A291,A292,A296,A301,A304,GOBRD13:26,27; suppose A305: i1 = i2 & j1 = j2+1; then left_cell(F.k,m,G) = cell(G,i2,j2) & right_cell(F.k,m,G) = cell(G,i1-'1,j2) by A146,A250,A290,A291,A292,GOBRD13:28,29; hence thesis by A205,A250,A291,A292,A296,A301,A305,GOBRD13:28,29; end; hence thesis; end; hence thesis; end; A306: for k holds P[k] from Ind(A108,A145); A307: for k,i1,i2,j1,j2 st k > 1 & [i1,j1] in Indices G & (F.k)/.((len(F.k)) -'1) = G*(i1,j1) & [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2) holds (front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C implies F.(k+1) turns_right (len(F.k))-'1,G & (i1 = i2 & j1+1 = j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*>) & (i1+1 = i2 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*>) & (i1 = i2+1 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*>)& (i1 = i2 & j1 = j2+1 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*>)) & (front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C implies F.(k+1) goes_straight (len(F.k))-'1,G & (i1 = i2 & j1+1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*>) & (i1+1 = i2 & j1 = j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*>) & (i1 = i2+1 & j1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*>)& (i1 = i2 & j1 = j2+1 implies [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*>)) & (front_left_cell(F.k,(len(F.k))-'1,G) meets C implies F.(k+1) turns_left (len(F.k))-'1,G & (i1 = i2 & j1+1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*>) & (i1+1 = i2 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*>) & (i1 = i2+1 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*>)& (i1 = i2 & j1 = j2+1 implies [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*>)) proof let k,i1,i2,j1,j2 such that A308: k > 1 and A309: [i1,j1] in Indices G & (F.k)/.((len(F.k)) -'1) = G*(i1,j1) and A310: [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2); A311: (len(F.k))-'1 <= len(F.k) by GOBOARD9:2; A312: F.k is_sequence_on G by A306; A313: len(F.k) = k by A107; then A314: 1 <= (len(F.k))-'1 by A308,JORDAN3:12; A315: (len(F.k)) -'1 +1 = len(F.k) by A308,A313,AMI_5:4; then A316: right_cell(F.k,(len(F.k))-'1,G) meets C by A306,A314; A317: (len(F.k))-'1 in dom(F.k) & len(F.k) in dom(F.k) by A308,A311,A313,A314,FINSEQ_3:27; A318: (len(F.k))-'1+(1+1) = (len(F.k))+1 by A315,XCMPLX_1:1; A319: i1+1 > i1 & i2+1 > i2 & j1+1 > j1 & j2+1 > j2 by NAT_1:38; hereby assume front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; then consider i,j such that A320: (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G and A321: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A308,A312,A313,A316; A322: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) & (F.(k+1))/.len(F.k) = G* (i2,j2) by A309,A310,A317,A321,GROUP_5:95; A323: (F.(k+1))/.(len(F.k)+1) = G*(i,j) by A321,TOPREAL4:1; thus F.(k+1) turns_right (len(F.k))-'1,G by A320,A321; thus i1 = i2 & j1+1 = j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A309,A310,A315,A318,A319,A320,A321,A322,A323,GOBRD13:def 6; thus i1+1 = i2 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*> by A309,A310,A315,A318,A319,A320,A321,A322,A323,GOBRD13:def 6; thus i1 = i2+1 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A309,A310,A315,A318,A319,A320,A321,A322,A323,GOBRD13:def 6; thus i1 = i2 & j1 = j2+1 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*> by A309,A310,A315,A318,A319,A320,A321,A322,A323,GOBRD13:def 6; end; hereby assume front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that A324: (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G and A325: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A308,A312,A313,A316; A326: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) & (F.(k+1))/.len(F.k) = G* (i2,j2) by A309,A310,A317,A325,GROUP_5:95; A327: (F.(k+1))/.(len(F.k)+1) = G*(i,j) by A325,TOPREAL4:1; thus F.(k+1) goes_straight (len(F.k))-'1,G by A324,A325; thus i1 = i2 & j1+1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A309,A310,A315,A318,A319,A324,A325,A326,A327,GOBRD13:def 8; thus i1+1 = i2 & j1 = j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A309,A310,A315,A318,A319,A324,A325,A326,A327,GOBRD13:def 8; thus i1 = i2+1 & j1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*> by A309,A310,A315,A318,A319,A324,A325,A326,A327,GOBRD13:def 8; thus i1 = i2 & j1 = j2+1 implies [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*> by A309,A310,A315,A318,A319,A324,A325,A326,A327,GOBRD13:def 8; end; assume front_left_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that A328: (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G and A329: F.(k+1) = (F.k)^<*G*(i,j)*> by A92,A308,A312,A313,A316; A330: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) & (F.(k+1))/.len(F.k) = G* (i2,j2) by A309,A310,A317,A329,GROUP_5:95; A331: (F.(k+1))/.(len(F.k)+1) = G*(i,j) by A329,TOPREAL4:1; thus F.(k+1) turns_left (len(F.k))-'1,G by A328,A329; thus i1 = i2 & j1+1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*> by A309,A310,A315,A318,A319,A328,A329,A330,A331,GOBRD13:def 7; thus i1+1 = i2 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A309,A310,A315,A318,A319,A328,A329,A330,A331,GOBRD13:def 7; thus i1 = i2+1 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*> by A309,A310,A315,A318,A319,A328,A329,A330,A331,GOBRD13:def 7; thus i1 = i2 & j1 = j2+1 implies [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A309,A310,A315,A318,A319,A328,A329,A330,A331,GOBRD13:def 7; end; A332: for k holds ex i,j st [i,j] in Indices G & F.(k+1) = (F.k)^<*G*(i,j)*> proof let k; A333: F.k is_sequence_on G by A306; A334: len(F.k) = k by A107; A335: len G = 2|^n+3 & len G = width G by JORDAN8:def 1; len G >= 4 by JORDAN8:13; then A336: 1 < len G by AXIOMS:22; per cases by REAL_1:def 5; suppose k < 1; then A337: k = 0 by RLVECT_1:98; consider i such that A338: 1 <= i & i+1 <= len G and N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) and A339: F.(0+1) = <*G*(i,width G)*> by A92; take i, j = width G; i < len G by A338,NAT_1:38; hence [i,j] in Indices G by A335,A336,A338,GOBOARD7:10; thus thesis by A91,A337,A339,FINSEQ_1:47; suppose A340: k = 1; consider i such that A341: 1 <= i & i+1 <= len G and A342: N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) and A343: F.(0+1) = <*G*(i,width G)*> by A92; consider i' being Nat such that A344: 1 <= i' & i'+1 <= len G & N-min C in cell(G,i',width G-'1) & N-min C <> G*(i',width G-'1) and A345: F.(1+1) = <*G*(i',width G),G*(i'+1,width G)*> by A92; A346: i = i' by A341,A342,A344,Th31; take i+1,j = width G; 1 <= i+1 & i+1 <= len G by A341,NAT_1:38; hence [i+1,j] in Indices G by A335,A336,GOBOARD7:10; thus thesis by A340,A343,A345,A346,FINSEQ_1:def 9; suppose A347: k > 1; then A348: 1 <= (len(F.k))-'1 by A334,JORDAN3:12; (len(F.k)) -'1 +1 = len(F.k) by A334,A347,AMI_5:4; then consider i1,j1,i2,j2 being Nat such that A349: [i1,j1] in Indices G & (F.k)/.((len(F.k)) -'1) = G*(i1,j1) and A350: [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2) and A351: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A333,A348,JORDAN8:6; now per cases; suppose A352: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; now per cases by A351; suppose i1 = i2 & j1+1 = j2; then [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A307,A347,A349,A350,A352; hence thesis; suppose i1+1 = i2 & j1 = j2; then [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*> by A307,A347,A349,A350,A352; hence thesis; suppose i1 = i2+1 & j1 = j2; then [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A307,A347,A349,A350,A352; hence thesis; suppose i1 = i2 & j1 = j2+1; then [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*> by A307,A347,A349,A350,A352; hence thesis; end; hence thesis; suppose A353: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; now per cases by A351; suppose i1 = i2 & j1+1 = j2; then [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A307,A347,A349,A350,A353; hence thesis; suppose i1+1 = i2 & j1 = j2; then [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A307,A347,A349,A350,A353; hence thesis; suppose i1 = i2+1 & j1 = j2; then [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*> by A307,A347,A349,A350,A353; hence thesis; suppose i1 = i2 & j1 = j2+1; then [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*> by A307,A347,A349,A350,A353; hence thesis; end; hence thesis; suppose A354: front_left_cell(F.k,(len(F.k))-'1,G) meets C; now per cases by A351; suppose i1 = i2 & j1+1 = j2; then [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*> by A307,A347,A349,A350,A354; hence thesis; suppose i1+1 = i2 & j1 = j2; then [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A307,A347,A349,A350,A354; hence thesis; suppose i1 = i2+1 & j1 = j2; then [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*> by A307,A347,A349,A350,A354; hence thesis; suppose i1 = i2 & j1 = j2+1; then [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A307,A347,A349,A350,A354; hence thesis; end; hence thesis; end; hence thesis; end; defpred P[Nat] means for m st m <= $1 holds (F.$1)|m = F.m; A355: P[0] proof let m; assume A356: m <= 0; A357: 0 <= m by NAT_1:18; then (F.0)|0 = (F.0)|m by A356,AXIOMS:21; hence thesis by A91,A356,A357,AXIOMS:21; end; A358: for k st P[k] holds P[k+1] proof let k such that A359: for m st m <= k holds (F.k)|m = F.m; let m such that A360: m <= k+1; per cases by A360,REAL_1:def 5; suppose m < k+1; then A361: m <= k by NAT_1:38; A362: len(F.k) = k by A107; consider i,j such that [i,j] in Indices G and A363: F.(k+1) = F.k^<*G*(i,j)*> by A332; (F.(k+1))|m = (F.k)|m by A361,A362,A363,FINSEQ_5:25; hence (F.(k+1))|m = F.m by A359,A361; suppose A364: m = k+1; len(F.(k+1)) = k+1 by A107; hence (F.(k+1))|m = F.m by A364,TOPREAL1:2; end; A365: for k holds P[k] from Ind(A355,A358); A366: for k st k > 1 holds (front_left_cell(F.k,k-'1,Gauge(C,n)) misses C & front_right_cell(F.k,k-'1,Gauge(C,n)) misses C implies F.(k+1) turns_right k-'1,Gauge(C,n)) & (front_left_cell(F.k,k-'1,Gauge(C,n)) misses C & front_right_cell(F.k,k-'1,Gauge(C,n)) meets C implies F.(k+1) goes_straight k-'1,Gauge(C,n)) & (front_left_cell(F.k,k-'1,Gauge(C,n)) meets C implies F.(k+1) turns_left k-'1,Gauge(C,n)) proof let k such that A367: k > 1; A368: F.k is_sequence_on G by A306; A369: len(F.k) = k by A107; then A370: 1 <= (len(F.k))-'1 by A367,JORDAN3:12; (len(F.k)) -'1 +1 = len(F.k) by A367,A369,AMI_5:4; then ex i1,j1,i2,j2 being Nat st [i1,j1] in Indices G & (F.k)/.((len(F.k)) -'1) = G*(i1,j1) & [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2) & (i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1) by A368,A370,JORDAN8:6; hence thesis by A307,A367,A369; end; defpred P[Nat] means F.$1 is unfolded; len(F.0) = 0 by A91,FINSEQ_1:25; then A371: P[0] by SPPOL_2:27; A372: for k st P[k] holds P[k+1] proof let k such that A373: F.k is unfolded; A374: F.k is_sequence_on G by A306; per cases; suppose k <= 1; then k+1 <= 1+1 by AXIOMS:24; then len(F.(k+1)) <= 2 by A107; hence F.(k+1) is unfolded by SPPOL_2:27; suppose A375: k > 1; set m = k-'1; A376: 1 <= m by A375,JORDAN3:12; A377: m+1 = k by A375,AMI_5:4; A378: len(F.k) = k by A107; then consider i1,j1,i2,j2 being Nat such that A379: [i1,j1] in Indices G & (F.k)/.m = G*(i1,j1) and A380: [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2) and A381: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A374,A376,A377,JORDAN8:6; A382: 1 <= i1 & i1 <= len G & 1 <= j1 & j1 <= width G by A379,GOBOARD5:1; A383: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A380,GOBOARD5:1; then A384: (i2-'1)+1 = i2 by AMI_5:4; A385: (j2-'1)+1 = j2 by A383,AMI_5:4; A386: LSeg(F.k,m) = LSeg(G*(i1,j1),G*(i2,j2)) by A376,A377,A378,A379,A380, TOPREAL1:def 5; now per cases; suppose A387: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; now per cases by A381; suppose A388: i1 = i2 & j1+1 = j2; then A389: [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A307,A375,A378,A379,A380,A387; then i2+1 <= len G by GOBOARD5:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2+1,j2)) = {(F.k)/.len(F.k)} by A380,A382,A383,A386,A388,GOBOARD7:17; hence F.(k+1) is unfolded by A373,A377,A378,A389,SPPOL_2:31; suppose A390: i1+1 = i2 & j1 = j2; then A391: [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*> by A307,A375,A378,A379,A380,A387; then 1 <= j2-'1 & LSeg((F.k)/.len(F.k),G*(i2,j2-'1))=LSeg(G*(i2,j2-'1), (F.k)/.len(F.k)) by GOBOARD5:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2,j2-'1)) = {(F.k)/.len(F.k)} by A380,A382,A383,A385,A386,A390,GOBOARD7:18; hence F.(k+1) is unfolded by A373,A377,A378,A391,SPPOL_2:31; suppose A392: i1 = i2+1 & j1 = j2; then A393: [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A307,A375,A378,A379,A380,A387; then j2+1 <= width G by GOBOARD5:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2,j2+1)) = {(F.k)/.len(F.k)} by A380,A382,A383,A386,A392,GOBOARD7:19; hence F.(k+1) is unfolded by A373,A377,A378,A393,SPPOL_2:31; suppose A394: i1 = i2 & j1 = j2+1; then A395: [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*> by A307,A375,A378,A379,A380,A387; then 1 <= i2-'1 by GOBOARD5:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2-'1,j2)) = {(F.k)/.len(F.k)} by A380,A382,A383,A384,A386,A394,GOBOARD7:20; hence F.(k+1) is unfolded by A373,A377,A378,A395,SPPOL_2:31; end; hence F.(k+1) is unfolded; suppose A396: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; now per cases by A381; suppose A397: i1 = i2 & j1+1 = j2; then A398: [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A307,A375,A378,A379,A380,A396; then 1 <= j2+1 & j2+1 <= width G & j2+1 = j1+(1+1) by A397,GOBOARD5:1,XCMPLX_1:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2,j2+1)) = {(F.k)/.len(F.k)} by A380,A382,A386,A397,GOBOARD7:15; hence F.(k+1) is unfolded by A373,A377,A378,A398,SPPOL_2:31; suppose A399: i1+1 = i2 & j1 = j2; then A400: [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A307,A375,A378,A379,A380,A396; then 1 <= i2+1 & i2+1 <= len G & i2+1 = i1+(1+1) by A399,GOBOARD5:1,XCMPLX_1:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2+1,j2)) = {(F.k)/.len(F.k)} by A380,A382,A386,A399,GOBOARD7:16; hence F.(k+1) is unfolded by A373,A377,A378,A400,SPPOL_2:31; suppose A401: i1 = i2+1 & j1 = j2; then A402: [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*> by A307,A375,A378,A379,A380,A396; then 1 <= i2-'1 & i2-'1+1+1 = i2-'1+(1+1) by GOBOARD5:1,XCMPLX_1:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2-'1,j2)) = {(F.k)/.len(F.k)} by A380,A382,A384,A386,A401,GOBOARD7:16; hence F.(k+1) is unfolded by A373,A377,A378,A402,SPPOL_2:31; suppose A403: i1 = i2 & j1 = j2+1; then A404: [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*> by A307,A375,A378,A379,A380,A396; then 1 <= j2-'1 & j2-'1+1+1 = j2-'1+(1+1) by GOBOARD5:1,XCMPLX_1:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2,j2-'1)) = {(F.k)/.len(F.k)} by A380,A382,A385,A386,A403,GOBOARD7:15; hence F.(k+1) is unfolded by A373,A377,A378,A404,SPPOL_2:31; end; hence F.(k+1) is unfolded; suppose A405: front_left_cell(F.k,(len(F.k))-'1,G) meets C; now per cases by A381; suppose A406: i1 = i2 & j1+1 = j2; then A407: [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*> by A307,A375,A378,A379,A380,A405; then 1 <= i2-'1 & LSeg((F.k)/.len(F.k),G*(i2-'1,j2))=LSeg(G*(i2-'1,j2), (F.k)/.len(F.k)) by GOBOARD5:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2-'1,j2)) = {(F.k)/.len(F.k)} by A380,A382,A383,A384,A386,A406,GOBOARD7:18; hence F.(k+1) is unfolded by A373,A377,A378,A407,SPPOL_2:31; suppose A408: i1+1 = i2 & j1 = j2; then A409: [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A307,A375,A378,A379,A380,A405; then 1 <= j2+1 & j2+1 <= width G by GOBOARD5:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2,j2+1)) = {(F.k)/.len(F.k)} by A380,A382,A383,A386,A408,GOBOARD7:20; hence F.(k+1) is unfolded by A373,A377,A378,A409,SPPOL_2:31; suppose A410: i1 = i2+1 & j1 = j2; then A411: [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*> by A307,A375,A378,A379,A380,A405; then 1 <= j2-'1 & LSeg((F.k)/.len(F.k),G*(i2,j2-'1))=LSeg(G*(i2,j2-'1), (F.k)/.len(F.k)) by GOBOARD5:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2,j2-'1)) = {(F.k)/.len(F.k)} by A380,A382,A383,A385,A386,A410,GOBOARD7:17; hence F.(k+1) is unfolded by A373,A377,A378,A411,SPPOL_2:31; suppose A412: i1 = i2 & j1 = j2+1; then A413: [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A307,A375,A378,A379,A380,A405; then 1 <= i2+1 & i2+1 <= len G & LSeg((F.k)/.len(F.k),G*(i2+1,j2))=LSeg(G* (i2+1,j2),(F.k)/.len(F.k)) by GOBOARD5:1; then LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2+1,j2)) = {(F.k)/.len(F.k)} by A380,A382,A383,A386,A412,GOBOARD7:19; hence F.(k+1) is unfolded by A373,A377,A378,A413,SPPOL_2:31; end; hence F.(k+1) is unfolded; end; hence F.(k+1) is unfolded; end; A414: for k holds P[k] from Ind(A371,A372); defpred Q[Nat] means $1 >= 1 & ex m st m in dom(F.$1) & m <> len(F.$1) & (F.$1)/.m = (F.$1)/.len(F.$1); now assume A415: for k st k >= 1 holds for m st m in dom(F.k) & m <> len(F.k) holds (F.k)/.m <> (F.k)/.len(F.k); defpred P[Nat] means F.$1 is one-to-one; A416: P[0] by A91; A417: for k st P[k] holds P[k+1] proof let k; assume A418: F.k is one-to-one; now let n,m such that A419: n in dom(F.(k+1)) & m in dom(F.(k+1)) and A420: (F.(k+1))/.n = (F.(k+1))/.m; consider i,j such that [i,j] in Indices G and A421: F.(k+1) = (F.k)^<*G*(i,j)*> by A332; A422: 1 <= n & n <= len(F.(k+1)) & 1 <= m & m <= len(F.(k+1)) by A419,FINSEQ_3:27; A423: len(F.k) = k & len(F.(k+1)) = k+1 by A107; A424: 1 <= k+1 by NAT_1:37; per cases by A422,A423,NAT_1:26; suppose n <= k & m <= k; then A425: n in dom(F.k) & m in dom(F.k) by A422,A423,FINSEQ_3:27; then (F.(k+1))/.n = (F.k)/.n & (F.(k+1))/.m = (F.k)/.m by A421,GROUP_5:95; hence n = m by A418,A420,A425,PARTFUN2:17; suppose n = k+1 & m <= k; hence n = m by A415,A419,A420,A423,A424; suppose n <= k & m = k+1; hence n = m by A415,A419,A420,A423,A424; suppose n = k+1 & m = k+1; hence n = m; end; hence F.(k+1) is one-to-one by PARTFUN2:16; end; A426: for k holds P[k] from Ind(A416,A417); A427: for k holds card rng(F.k) = k proof let k; F.k is one-to-one by A426; hence card rng(F.k) = len(F.k) by FINSEQ_4:77 .= k by A107; end; set k = (len G)*(width G)+1; A428: k > (len G)*(width G) by NAT_1:38; A429: card Values G <= (len G)*(width G) by GOBRD13:8; F.k is_sequence_on G by A306; then rng(F.k) c= Values G by GOBRD13:9; then card rng(F.k) <= card Values G by CARD_1:80; then card rng(F.k) <= (len G)*(width G) by A429,AXIOMS:22; hence contradiction by A427,A428; end; then A430: ex k st Q[k]; consider k such that A431: Q[k] and A432: for l st Q[l] holds k <= l from Min(A430); consider m such that A433: m in dom(F.k) and A434: m <> len(F.k) and A435: (F.k)/.m = (F.k)/.len(F.k) by A431; A436: len(F.k) = k by A107; 0<k by A431,AXIOMS:22; then reconsider f = F.k as non empty FinSequence of TOP-REAL 2 by A436, FINSEQ_1:25; A437: f is_sequence_on G by A306; A438: 1 <= m & m <= len f by A433,FINSEQ_3:27; then A439: m < len f by A434,REAL_1:def 5; then 1 < len f by A438,AXIOMS:22; then A440: len f >= 1+1 by NAT_1:38; then reconsider f as non constant special unfolded non empty FinSequence of TOP-REAL 2 by A414,A437,JORDAN8:7,8; set g = f/^(m-'1); A441: m+1 <= len f by A439,NAT_1:38; m-'1 <= m & m < m+1 by JORDAN3:7,NAT_1:38; then m-'1 < m+1 by AXIOMS:22; then A442: m-'1 < len f by A441,AXIOMS:22; then A443: len g = len f - (m-'1) by RFINSEQ:def 2; then (m-'1)-(m-'1) < len g by A442,REAL_1:54; then len g > 0 by XCMPLX_1:14; then reconsider g as non empty FinSequence of TOP-REAL 2 by FINSEQ_1:25; 1 in dom g by FINSEQ_5:6; then A444: g/.1 = f/.(m-'1+1) by FINSEQ_5:30 .= f/.m by A438,AMI_5:4; len g in dom g by FINSEQ_5:6; then A445: g/.len g = f/.(m-'1+len g) by FINSEQ_5:30 .= f/.len f by A443,XCMPLX_1:27; A446: g is_sequence_on G by A437,JORDAN8:5; then A447: g is standard by JORDAN8:7; A448: for i st 1 <= i & i < len g & 1 <= j & j < len g & g/.i = g/.j holds i = j proof let i such that A449: 1 <= i & i < len g and A450: 1 <= j & j < len g and A451: g/.i = g/.j and A452: i <> j; i in dom g by A449,FINSEQ_3:27; then A453: g/.i = f/.(m-'1+i) by FINSEQ_5:30; j in dom g by A450,FINSEQ_3:27; then A454: g/.j = f/.(m-'1+j) by FINSEQ_5:30; A455: 0 <= m-'1 by NAT_1:18; per cases by A452,REAL_1:def 5; suppose A456: i < j; set l = m-'1+j, m'= m-'1+i; A457: m' < l by A456,REAL_1:53; A458: l < k by A436,A443,A450,REAL_1:86; then A459: f|l = F.l by A365; 0+j <= l by A455,AXIOMS:24; then A460: 1 <= l by A450,AXIOMS:22; A461: len(F.l) = l by A107; 0+i <= m' by A455,AXIOMS:24; then 1 <= m' by A449,AXIOMS:22; then A462: m' in dom(F.l) by A457,A461,FINSEQ_3:27; then A463: (F.l)/.m' = f/.m' by A459,TOPREAL1:1; l in dom(F.l) by A460,A461,FINSEQ_3:27; then (F.l)/.l = f/.l by A459,TOPREAL1:1; hence contradiction by A432,A451,A453,A454,A457,A458,A460,A461,A462,A463; suppose A464: j < i; set l = m-'1+i, m'= m-'1+j; A465: m' < l by A464,REAL_1:53; A466: l < k by A436,A443,A449,REAL_1:86; then A467: f|l = F.l by A365; 0+i <= l by A455,AXIOMS:24; then A468: 1 <= l by A449,AXIOMS:22; A469: len(F.l) = l by A107; 0+j <= m' by A455,AXIOMS:24; then 1 <= m' by A450,AXIOMS:22; then A470: m' in dom(F.l) by A465,A469,FINSEQ_3:27; then A471: (F.l)/.m' = f/.m' by A467,TOPREAL1:1; l in dom(F.l) by A468,A469,FINSEQ_3:27; then (F.l)/.l = f/.l by A467,TOPREAL1:1; hence contradiction by A432,A451,A453,A454,A465,A466,A468,A469,A470,A471; end; A472: for i st 1 < i & i < j & j <= len g holds g/.i <> g/.j proof let i such that A473: 1 < i and A474: i < j and A475: j <= len g and A476: g/.i = g/.j; A477: 1 < j by A473,A474,AXIOMS:22; A478: i < len g by A474,A475,AXIOMS:22; then A479: 1 < len g by A473,AXIOMS:22; per cases; suppose j <> len g; then j < len g by A475,REAL_1:def 5; hence contradiction by A448,A473,A474,A476,A477,A478; suppose j = len g; hence contradiction by A435,A444,A445,A448,A473,A474,A476,A479; end; A480: for i st 1 <= i & i < j & j < len g holds g/.i <> g/.j proof let i such that A481: 1 <= i and A482: i < j and A483: j < len g and A484: g/.i = g/.j; A485: 1 < j by A481,A482,AXIOMS:22; i < len g by A482,A483,AXIOMS:22; hence contradiction by A448,A481,A482,A483,A484,A485; end; A486: for i st 1 <= i & i+1 <= len f holds right_cell(f,i,G) = Cl Int right_cell(f,i,G) proof let i such that A487: 1 <= i & i+1 <= len f; consider i1,j1,i2,j2 such that A488: [i1,j1] in Indices G and A489: f/.i = G*(i1,j1) and A490: [i2,j2] in Indices G and A491: f/.(i+1) = G*(i2,j2) and A492: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A437,A487,JORDAN8:6; A493: 1 <= i1 & i1 <= len G & 1 <= j1 & j1 <= width G by A488,GOBOARD5:1; A494: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A490,GOBOARD5:1; A495: i1+1 > i1 & j1+1 > j1 & i2+1 > i2 & j2+1 > j2 by NAT_1:38; per cases by A492; suppose i1 = i2 & j1+1 = j2; then right_cell(f,i,G) = cell(G,i1,j1) by A437,A487,A488,A489,A490,A491,A495,GOBRD13:def 2; hence thesis by A493,GOBRD11:35; suppose A496: i1+1 = i2 & j1 = j2; A497: j1-'1 <= width G by A493,JORDAN3:7; right_cell(f,i,G) = cell(G,i1,j1-'1) by A437,A487,A488,A489,A490,A491,A495,A496,GOBRD13:def 2; hence thesis by A493,A497,GOBRD11:35; suppose i1 = i2+1 & j1 = j2; then right_cell(f,i,G) = cell(G,i2,j2) by A437,A487,A488,A489,A490,A491,A495,GOBRD13:def 2; hence thesis by A494,GOBRD11:35; suppose A498: i1 = i2 & j1 = j2+1; A499: i1-'1 <= len G by A493,JORDAN3:7; right_cell(f,i,G) = cell(G,i1-'1,j2) by A437,A487,A488,A489,A490,A491,A495,A498,GOBRD13:def 2; hence thesis by A494,A499,GOBRD11:35; end; A500: g is s.c.c. proof let i,j such that A501: i+1 < j and A502: i > 1 & j < len g or j+1 < len g; A503: j <= j+1 by NAT_1:37; then A504: i+1 < j+1 by A501,AXIOMS:22; A505: 1 <= i+1 by NAT_1:37; A506: 1 < j by A501,NAT_1:37; i < j by A501,NAT_1:38; then A507: i < j+1 by A503,AXIOMS:22; per cases by A502,RLVECT_1:99; suppose A508: i > 1 & j < len g; then A509: i+1 < len g by A501,AXIOMS:22; then A510: LSeg(g,i) = LSeg(g/.i,g/.(i+1)) by A508,TOPREAL1:def 5; A511: 1 < i+1 by A508,NAT_1:38; A512: i < len g by A509,NAT_1:38; consider i1,j1,i2,j2 such that A513: [i1,j1] in Indices G and A514: g/.i = G*(i1,j1) and A515: [i2,j2] in Indices G and A516: g/.(i+1) = G*(i2,j2) and A517: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A446,A508,A509,JORDAN8:6; A518: 1 <= i1 & i1 <= len G & 1 <= j1 & j1 <= width G by A513,GOBOARD5:1; A519: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A515,GOBOARD5:1; A520: j+1 <= len g by A508,NAT_1:38; then A521: LSeg(g,j) = LSeg(g/.j,g/.(j+1)) by A506,TOPREAL1:def 5; consider i1',j1',i2',j2' being Nat such that A522: [i1',j1'] in Indices G and A523: g/.j = G*(i1',j1') and A524: [i2',j2'] in Indices G and A525: g/.(j+1) = G*(i2',j2') and A526: i1' = i2' & j1'+1 = j2' or i1'+1 = i2' & j1' = j2' or i1' = i2'+1 & j1' = j2' or i1' = i2' & j1' = j2'+1 by A446,A506,A520,JORDAN8:6; A527: 1 <= i1' & i1' <= len G & 1 <= j1' & j1' <= width G by A522,GOBOARD5:1 ; A528: 1 <= i2' & i2' <= len G & 1 <= j2' & j2' <= width G by A524,GOBOARD5:1 ; assume A529: LSeg(g,i) meets LSeg(g,j); now per cases by A517,A526; suppose A530: i1 = i2 & j1+1 = j2 & i1' = i2' & j1'+1 = j2'; then A531: i1 = i1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528, A529,GOBOARD7:21; j1 = j1' or j1 = j1'+1 or j1+1 = j1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,A530, GOBOARD7:24; hence contradiction by A448,A472,A501,A503,A505,A506,A507,A508,A509,A512,A514,A516,A520, A523,A525,A530,A531; suppose A532: i1 = i2 & j1+1 = j2 & i1'+1 = i2' & j1' = j2'; then i1 = i1' & j1 = j1' or i1 = i1' & j1+1 = j1' or i1 = i1'+1 & j1 = j1' or i1 = i1'+1 & j1+1 = j1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,GOBOARD7:23 ; hence contradiction by A448,A472,A501,A504,A506,A507,A508,A511,A512,A514,A516,A520,A523 ,A525,A532; suppose A533: i1 = i2 & j1+1 = j2 & i1' = i2'+1 & j1' = j2'; then i1 = i2' & j1' = j1 or i1 = i2' & j1+1 = j1' or i1 = i2'+1 & j1' = j1 or i1 = i2'+1 & j1+1 = j1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,GOBOARD7: 23; hence contradiction by A448,A472,A501,A504,A506,A507,A508,A511,A512,A514,A516,A520,A523, A525,A533; suppose A534: i1 = i2 & j1+1 = j2 & i1' = i2' & j1' = j2'+1; then A535: i1 = i1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528, A529,GOBOARD7:21; j1 = j2' or j1 = j2'+1 or j1+1 = j2' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,A534, GOBOARD7:24; hence contradiction by A448,A472,A501,A504,A506,A508,A511,A512,A514, A516,A520,A523,A525,A534,A535; suppose A536: i1+1 = i2 & j1 = j2 & i1' = i2' & j1'+1 = j2'; then i1' = i1 & j1 = j1' or i1' = i1 & j1'+1 = j1 or i1' = i1+1 & j1 = j1' or i1' = i1+1 & j1'+1 = j1 by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,GOBOARD7: 23; hence contradiction by A448,A472,A501,A504,A506,A507,A508,A511,A512,A514,A516,A520,A523 ,A525,A536; suppose A537: i1+1 = i2 & j1 = j2 & i1'+1 = i2' & j1' = j2'; then A538: j1 = j1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528, A529,GOBOARD7:22; i1 = i1' or i1 = i1'+1 or i1+1 = i1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,A537, GOBOARD7:25; hence contradiction by A448,A472,A501,A503,A505,A506,A507,A508,A509,A512,A514,A516,A520, A523,A525,A537,A538; suppose A539: i1+1 = i2 & j1 = j2 & i1' = i2'+1 & j1' = j2'; then A540: j1 = j1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528, A529,GOBOARD7:22; i1 = i2' or i1 = i2'+1 or i1+1 = i2' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,A539, GOBOARD7:25; hence contradiction by A448,A472,A504,A506,A507,A508,A511,A512,A514,A516,A520,A523,A525, A539,A540; suppose A541: i1+1 = i2 & j1 = j2 & i1' = i2' & j1' = j2'+1; then i1' = i1 & j1 = j2' or i1' = i1 & j2'+1 = j1 or i1' = i1+1 & j1 = j2' or i1' = i1+1 & j2'+1 = j1 by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,GOBOARD7 :23; hence contradiction by A448,A472,A501,A504,A506,A507,A508,A511,A512,A514,A516,A520,A523, A525,A541; suppose A542: i1 = i2+1 & j1 = j2 & i1' = i2' & j1'+1 = j2'; then i1' = i2 & j1' = j1 or i1' = i2 & j1'+1 = j1 or i1' = i2+1 & j1' = j1 or i1' = i2+1 & j1'+1 = j1 by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,GOBOARD7:23 ; hence contradiction by A448,A472,A501,A504,A506,A507,A508,A511,A512,A514,A516,A520,A523, A525,A542; suppose A543: i1 = i2+1 & j1 = j2 & i1'+1 = i2' & j1' = j2'; then A544: j1 = j1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528, A529,GOBOARD7:22; i2 = i1' or i2 = i1'+1 or i2+1 = i1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,A543, GOBOARD7:25; hence contradiction by A448,A472,A501,A504,A506,A508,A511,A512,A514,A516,A520,A523,A525, A543,A544; suppose A545: i1 = i2+1 & j1 = j2 & i1' = i2'+1 & j1' = j2'; then A546: j1 = j1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528, A529,GOBOARD7:22; i2 = i2' or i2 = i2'+1 or i2+1 = i2' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,A545, GOBOARD7:25; hence contradiction by A472,A501,A504,A507,A508,A511,A514,A516,A520, A523,A525,A545,A546; suppose A547: i1 = i2+1 & j1 = j2 & i1' = i2' & j1' = j2'+1; then i1' = i2 & j2' = j1 or i1' = i2 & j2'+1 = j1 or i1' = i2+1 & j2' = j1 or i1' = i2+1 & j2'+1 = j1 by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,GOBOARD7: 23; hence contradiction by A448,A472,A501,A504,A506,A507,A508,A511,A512,A514,A516,A520,A523, A525,A547; suppose A548: i1 = i2 & j1 = j2+1 & i1' = i2' & j1'+1 = j2'; then A549: i1 = i1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528, A529,GOBOARD7:21; j2 = j1' or j2 = j1'+1 or j2+1 = j1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,A548, GOBOARD7:24; hence contradiction by A448,A472,A501,A504,A506,A508,A511,A512,A514,A516,A520,A523,A525, A548,A549; suppose A550: i1 = i2 & j1 = j2+1 & i1'+1 = i2' & j1' = j2'; then i1 = i1' & j2 = j1' or i1 = i1' & j2+1 = j1' or i1 = i1'+1 & j2 = j1' or i1 = i1'+1 & j2+1 = j1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,GOBOARD7:23 ; hence contradiction by A448,A472,A501,A504,A506,A507,A508,A511,A512,A514,A516,A520,A523, A525,A550; suppose A551: i1 = i2 & j1 = j2+1 & i1' = i2'+1 & j1' = j2'; then i1 = i2' & j2 = j1' or i1 = i2' & j2+1 = j1' or i1 = i2'+1 & j2 = j1' or i1 = i2'+1 & j2+1 = j1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,GOBOARD7:23 ; hence contradiction by A448,A472,A501,A504,A506,A507,A508,A511,A512,A514,A516,A520,A523, A525,A551; suppose A552: i1 = i2 & j1 = j2+1 & i1' = i2' & j1' = j2'+1; then A553: i1 = i1' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528, A529,GOBOARD7:21; j2 = j2' or j2 = j2'+1 or j2+1 = j2' by A510,A514,A516,A518,A519,A521,A523,A525,A527,A528,A529,A552, GOBOARD7:24; hence contradiction by A472,A501,A504,A507,A508,A511,A514,A516,A520, A523,A525,A552,A553; end; hence contradiction; suppose i = 0 & j+1 < len g; then LSeg(g,i) = {} by TOPREAL1:def 5; hence LSeg(g,i) misses LSeg(g,j) by XBOOLE_1:65; suppose A554: 1 <= i & j+1 < len g; then A555: i+1 < len g by A504,AXIOMS:22; then A556: LSeg(g,i) = LSeg(g/.i,g/.(i+1)) by A554,TOPREAL1:def 5; A557: 1 < i+1 by A554,NAT_1:38; A558: i < len g by A555,NAT_1:38; A559: j < len g by A554,NAT_1:37; consider i1,j1,i2,j2 such that A560: [i1,j1] in Indices G and A561: g/.i = G*(i1,j1) and A562: [i2,j2] in Indices G and A563: g/.(i+1) = G*(i2,j2) and A564: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A446,A554,A555,JORDAN8:6; A565: 1 <= i1 & i1 <= len G & 1 <= j1 & j1 <= width G by A560,GOBOARD5:1; A566: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A562,GOBOARD5:1; A567: LSeg(g,j) = LSeg(g/.j,g/.(j+1)) by A506,A554,TOPREAL1:def 5; consider i1',j1',i2',j2' being Nat such that A568: [i1',j1'] in Indices G and A569: g/.j = G*(i1',j1') and A570: [i2',j2'] in Indices G and A571: g/.(j+1) = G*(i2',j2') and A572: i1' = i2' & j1'+1 = j2' or i1'+1 = i2' & j1' = j2' or i1' = i2'+1 & j1' = j2' or i1' = i2' & j1' = j2'+1 by A446,A506,A554,JORDAN8:6; A573: 1 <= i1' & i1' <= len G & 1 <= j1' & j1' <= width G by A568,GOBOARD5:1 ; A574: 1 <= i2' & i2' <= len G & 1 <= j2' & j2' <= width G by A570,GOBOARD5:1 ; assume A575: LSeg(g,i) meets LSeg(g,j); now per cases by A564,A572; suppose A576: i1 = i2 & j1+1 = j2 & i1' = i2' & j1'+1 = j2'; then A577: i1 = i1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574, A575,GOBOARD7:21; j1 = j1' or j1 = j1'+1 or j1+1 = j1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,A576, GOBOARD7:24; hence contradiction by A480,A501,A504,A505,A507,A554,A559,A561,A563, A569,A571,A576,A577; suppose A578: i1 = i2 & j1+1 = j2 & i1'+1 = i2' & j1' = j2'; then i1 = i1' & j1 = j1' or i1 = i1' & j1+1 = j1' or i1 = i1'+1 & j1 = j1' or i1 = i1'+1 & j1+1 = j1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,GOBOARD7: 23; hence contradiction by A448,A480,A501,A504,A506,A507,A554,A557,A558,A559,A561,A563,A569, A571,A578; suppose A579: i1 = i2 & j1+1 = j2 & i1' = i2'+1 & j1' = j2'; then i1 = i2' & j1' = j1 or i1 = i2' & j1+1 = j1' or i1 = i2'+1 & j1' = j1 or i1 = i2'+1 & j1+1 = j1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,GOBOARD7: 23; hence contradiction by A448,A480,A501,A504,A506,A507,A554,A557,A558,A559,A561,A563,A569, A571,A579; suppose A580: i1 = i2 & j1+1 = j2 & i1' = i2' & j1' = j2'+1; then A581: i1 = i1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574, A575,GOBOARD7:21; j1 = j2' or j1 = j2'+1 or j1+1 = j2' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,A580, GOBOARD7:24; hence contradiction by A448,A472,A501,A504,A506,A554,A557,A558,A559, A561,A563,A569,A571,A580,A581; suppose A582: i1+1 = i2 & j1 = j2 & i1' = i2' & j1'+1 = j2'; then i1' = i1 & j1 = j1' or i1' = i1 & j1'+1 = j1 or i1' = i1+1 & j1 = j1' or i1' = i1+1 & j1'+1 = j1 by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,GOBOARD7: 23; hence contradiction by A448,A480,A501,A504,A505,A506,A507,A554,A558, A559,A561,A563,A569,A571,A582; suppose A583: i1+1 = i2 & j1 = j2 & i1'+1 = i2' & j1' = j2'; then A584: j1 = j1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574, A575,GOBOARD7:22; i1 = i1' or i1 = i1'+1 or i1+1 = i1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,A583, GOBOARD7:25; hence contradiction by A480,A501,A504,A505,A507,A554,A559,A561,A563, A569,A571,A583,A584; suppose A585: i1+1 = i2 & j1 = j2 & i1' = i2'+1 & j1' = j2'; then A586: j1 = j1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574, A575,GOBOARD7:22; i1 = i2' or i1 = i2'+1 or i1+1 = i2' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,A585, GOBOARD7:25; hence contradiction by A448,A480,A504,A506,A507,A554,A557,A558,A559, A561,A563,A569,A571,A585,A586; suppose A587: i1+1 = i2 & j1 = j2 & i1' = i2' & j1' = j2'+1; then i1' = i1 & j1 = j2' or i1' = i1 & j2'+1 = j1 or i1' = i1+1 & j1 = j2' or i1' = i1+1 & j2'+1 = j1 by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,GOBOARD7: 23; hence contradiction by A448,A480,A501,A504,A506,A507,A554,A557,A558, A559,A561,A563,A569,A571,A587; suppose A588: i1 = i2+1 & j1 = j2 & i1' = i2' & j1'+1 = j2'; then i1' = i2 & j1' = j1 or i1' = i2 & j1'+1 = j1 or i1' = i2+1 & j1' = j1 or i1' = i2+1 & j1'+1 = j1 by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,GOBOARD7:23 ; hence contradiction by A448,A480,A501,A504,A506,A507,A554,A557,A558, A559,A561,A563,A569,A571,A588; suppose A589: i1 = i2+1 & j1 = j2 & i1'+1 = i2' & j1' = j2'; then A590: j1 = j1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574, A575,GOBOARD7:22; i2 = i1' or i2 = i1'+1 or i2+1 = i1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,A589, GOBOARD7:25; hence contradiction by A448,A472,A501,A504,A506,A554,A557,A558,A559, A561,A563,A569,A571,A589,A590; suppose A591: i1 = i2+1 & j1 = j2 & i1' = i2'+1 & j1' = j2'; then A592: j1 = j1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574, A575,GOBOARD7:22; i2 = i2' or i2 = i2'+1 or i2+1 = i2' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,A591, GOBOARD7:25; hence contradiction by A480,A501,A504,A507,A554,A557,A559,A561,A563, A569,A571,A591,A592; suppose A593: i1 = i2+1 & j1 = j2 & i1' = i2' & j1' = j2'+1; then i1' = i2 & j2' = j1 or i1' = i2 & j2'+1 = j1 or i1' = i2+1 & j2' = j1 or i1' = i2+1 & j2'+1 = j1 by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,GOBOARD7: 23; hence contradiction by A448,A480,A501,A504,A506,A507,A554,A557,A558, A559,A561,A563,A569,A571,A593; suppose A594: i1 = i2 & j1 = j2+1 & i1' = i2' & j1'+1 = j2'; then A595: i1 = i1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574, A575,GOBOARD7:21; j2 = j1' or j2 = j1'+1 or j2+1 = j1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,A594, GOBOARD7:24; hence contradiction by A448,A472,A501,A504,A506,A554,A557,A558,A559, A561,A563,A569,A571,A594,A595; suppose A596: i1 = i2 & j1 = j2+1 & i1'+1 = i2' & j1' = j2'; then i1 = i1' & j2 = j1' or i1 = i1' & j2+1 = j1' or i1 = i1'+1 & j2 = j1' or i1 = i1'+1 & j2+1 = j1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,GOBOARD7: 23; hence contradiction by A448,A480,A501,A504,A506,A507,A554,A557,A558, A559,A561,A563,A569,A571,A596; suppose A597: i1 = i2 & j1 = j2+1 & i1' = i2'+1 & j1' = j2'; then i1 = i2' & j2 = j1' or i1 = i2' & j2+1 = j1' or i1 = i2'+1 & j2 = j1' or i1 = i2'+1 & j2+1 = j1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,GOBOARD7: 23; hence contradiction by A448,A480,A501,A504,A506,A507,A554,A557,A558, A559,A561,A563,A569,A571,A597; suppose A598: i1 = i2 & j1 = j2+1 & i1' = i2' & j1' = j2'+1; then A599: i1 = i1' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574, A575,GOBOARD7:21; j2 = j2' or j2 = j2'+1 or j2+1 = j2' by A556,A561,A563,A565,A566,A567,A569,A571,A573,A574,A575,A598, GOBOARD7:24; hence contradiction by A480,A501,A504,A507,A554,A557,A559,A561,A563, A569,A571,A598,A599; end; hence contradiction; end; m+1-(m-'1) <= len g by A441,A443,REAL_1:49; then m+1-(m-1) <= len g by A438,SCMFSA_7:3; then 1+m-m+1 <= len g by XCMPLX_1:37; then A600: 1+1 <= len g by XCMPLX_1:26; g is non constant proof take 1,2; thus A601: 1 in dom g by FINSEQ_5:6; thus A602: 2 in dom g by A600,FINSEQ_3:27; then g/.1 <> g/.(1+1) by A447,A601,GOBOARD7:31; then g.1 <> g/.(1+1) by A601,FINSEQ_4:def 4; hence g.1 <> g.2 by A602,FINSEQ_4:def 4; end; then reconsider g as standard non constant special_circular_sequence by A435,A444,A445,A446,A500,FINSEQ_6:def 1,JORDAN8:7; now assume m <> 1; then A603: 1 < m by A438,REAL_1:def 5; A604: LeftComp g is_a_component_of (L~g)` & RightComp g is_a_component_of (L~g)` by GOBOARD9:def 1,def 2; reconsider Lg' = (L~g)` as Subset of TOP-REAL 2; A605: C c= Lg' proof let c be set; assume that A606: c in C and A607: not c in Lg'; reconsider c as Point of TOP-REAL 2 by A606; c in L~g by A607,SUBSET_1:50; then consider i such that A608: 1 <= i & i+1 <= len g and A609: c in LSeg(g/.i,g/.(i+1)) by SPPOL_2:14; i in dom g & i+1 in dom g by A608,GOBOARD2:3; then A610: g/.i = f/.(i+(m-'1)) & g/.(i+1) = f/.(i+1+(m-'1)) by FINSEQ_5:30; A611: 1 <= i+(m-'1) by A608,NAT_1:37; A612: i+1+(m-'1) = i+(m-'1)+1 by XCMPLX_1:1; then i+(m-'1)+1 <= (len g)+(m-'1) by A608,AXIOMS:24; then A613: i+(m-'1)+1 <= len f by A443,XCMPLX_1:27; then c in LSeg(f,i+(m-'1)) by A609,A610,A611,A612,TOPREAL1:def 5; then c in left_cell(f,i+(m-'1),G) /\ right_cell(f,i+(m-'1),G) by A437,A611,A613,GOBRD13:30; then c in left_cell(f,i+(m-'1),G) by XBOOLE_0:def 3; then left_cell(f,i+(m-'1),G) meets C by A606,XBOOLE_0:3; hence contradiction by A306,A611,A613; end; A614: TOP-REAL 2 = TopSpaceMetr (Euclid 2) by EUCLID:def 8; A615: the carrier of TOP-REAL 2 = the carrier of Euclid 2 by TOPREAL3:13; L~g is closed by SPPOL_1:49; then (L~g)` is open by TOPS_1:29; then Lg' is open; then A616: (L~g)` = Int (L~g)` by TOPS_1:55; A617: for j,k st 1 <= j & j <= k holds (F.k)/.j = (F.j)/.j proof let j,k; assume that A618: 1 <= j and A619: j <= k; A620: (F.k)|j = F.j by A365,A619; j <= len(F.k) by A107,A619; then len(F.k|j) = j by TOPREAL1:3; then j in dom((F.k)|j) by A618,FINSEQ_3:27; hence thesis by A620,TOPREAL1:1; end; A621: L~g c= L~f by JORDAN3:75; A622: for n st 1 <= n & n <= m-'1 holds not f/.n in L~g proof let n such that A623: 1 <= n and A624: n <= m-'1; A625: n <= len f by A442,A624,AXIOMS:22; set p = f/.n; assume p in L~g; then consider j such that A626: m-'1+1 <= j and A627: j+1 <= len f and A628: p in LSeg(f,j) by A442,Th9; A629: n < m-'1+1 by A624,NAT_1:38; then A630: n < j by A626,AXIOMS:22; A631: m-'1+1=m by A438,AMI_5:4; then A632: 1 < j by A603,A626,AXIOMS:22; A633: j < k by A436,A627,NAT_1:38; A634: 2 <= len G & 2 <= width G by A2,NAT_1:37; A635: p in Values G by A437,A623,A625,Th8; per cases by A437,A627,A628,A632,A634,A635,Th25; suppose A636: p = f/.j; n <= len(F.j) by A107,A630; then A637: n in dom(F.j) by A623,FINSEQ_3:27; A638: n <> len(F.j) by A107,A626,A629; (F.j)/.n = (F.n)/.n by A617,A623,A630 .= p by A436,A617,A623,A625 .= (F.j)/.j by A617,A632,A633,A636 .= (F.j)/.len(F.j) by A107; hence contradiction by A432,A632,A633,A637,A638; suppose A639: p = f/.(j+1); now per cases by A436,A627,REAL_1:def 5; suppose A640: j+1 = k; n <= len(F.m) by A107,A629,A631; then A641: n in dom(F.m) by A623,FINSEQ_3:27; A642: n <> len(F.m) by A107,A629,A631; (F.m)/.n = (F.n)/.n by A617,A623,A629,A631 .= (F.k)/.k by A436,A617,A623,A625,A639,A640 .= (F.m)/.m by A435,A436,A438,A617 .= (F.m)/.len(F.m) by A107; hence contradiction by A432,A436,A438,A439,A641,A642; suppose A643: j+1 < k; set l = j+1; A644: 1 <= l by NAT_1:29; A645: n < n+1 & n+1 < l by A630,REAL_1:53,69; then A646: n < l by AXIOMS:22; then n <= len(F.l) by A107; then A647: n in dom(F.l) by A623,FINSEQ_3:27; A648: n <> len(F.l) by A107,A645; (F.l)/.n = (F.n)/.n by A617,A623,A646 .= p by A436,A617,A623,A625 .= (F.l)/.l by A617,A639,A643,A644 .= (F.l)/.len(F.l) by A107; hence contradiction by A432,A643,A644,A647,A648; end; hence contradiction; end; A649: for h being standard non constant special_circular_sequence st L~h c= L~f for Comp being Subset of TOP-REAL 2 st Comp is_a_component_of (L~h)` for n st 1 <= n & n+1 <= len f & f/.n in Comp & not f/.n in L~h holds C meets Comp proof let h be standard non constant special_circular_sequence such that A650: L~h c= L~f; let Comp be Subset of TOP-REAL 2 such that A651: Comp is_a_component_of (L~h)`; let n such that A652: 1 <= n & n+1 <= len f and A653: f/.n in Comp and A654: not f/.n in L~h; reconsider rc = right_cell(f,n,G)\L~h as Subset of TOP-REAL 2; A655: rc c= right_cell(f,n,G) by XBOOLE_1:36; A656: rc = right_cell(f,n,G) /\ (L~h)` by SUBSET_1:32; A657: rc meets C proof right_cell(f,n,G) meets C by A306,A652; then consider p being set such that A658: p in right_cell(f,n,G) & p in C by XBOOLE_0:3; reconsider p as Element of TOP-REAL 2 by A658; now take a = p; now assume p in L~h; then consider j such that A659: 1 <= j and A660: j+1 <= len f and A661: p in LSeg(f,j) by A650,SPPOL_2:13; p in left_cell(f,j,G) /\ right_cell(f,j,G) by A437,A659,A660,A661,GOBRD13:30; then A662: p in left_cell(f,j,G) by XBOOLE_0:def 3; left_cell(f,j,G) misses C by A306,A659,A660; hence contradiction by A658,A662,XBOOLE_0:3; end; hence a in rc by A658,XBOOLE_0:def 4; thus a in C by A658; end; hence thesis by XBOOLE_0:3; end; A663: Int right_cell(f,n,G) is connected by A437,A652,Th12; Int right_cell(f,n,G) misses L~f by A437,A652,Th17; then Int right_cell(f,n,G) misses L~h by A650,XBOOLE_1:63; then A664: Int right_cell(f,n,G) c= (L~h)` by SUBSET_1:43; Int right_cell(f,n,G) c= right_cell(f,n,G) by TOPS_1:44; then A665: Int right_cell(f,n,G) c= rc by A656,A664,XBOOLE_1:19; rc c= Cl Int right_cell(f,n,G) by A437,A652,A655,Th13; then A666: rc is connected by A663,A665,CONNSP_1:19; f/.n in right_cell(f,n,G) by A437,A652,Th10; then f/.n in rc by A654,XBOOLE_0:def 4; then A667: rc meets Comp by A653,XBOOLE_0:3; rc c= (L~h)` by A656,XBOOLE_1:17; then rc c= Comp by A651,A666,A667,GOBOARD9:6; hence C meets Comp by A657,XBOOLE_1:63; end; A668: C meets RightComp Rev g proof set rg=Rev g; A669: rg is_sequence_on G by A446,Th7; A670: 1+1 <= len rg by A600,FINSEQ_5:def 3; then 1+1-'1 <= len rg-'1 by JORDAN3:5; then A671: 1 <= len rg -'1 by BINARITH:39; 1 < len rg by A670,NAT_1:38; then A672: len rg -'1+1 = len rg by AMI_5:4; set p = rg/.1, q = rg/.2; set a = f/.(m-'1); set l = (m-'1)+(len g-'1); A673: p = f/.m by A435,A445,FINSEQ_5:68; A674: m-'1+1 = m by A438,AMI_5:4; then A675: 1 <= m-'1 by A603,NAT_1:38; 1+1-'1 <= len g-'1 by A600,JORDAN3:5; then A676: 1 <= len g-'1 by BINARITH:39; 1+1 - 1 <= len g - 1 by A600,REAL_1:49; then A677: 0 <= len g - 1 by AXIOMS:22; A678: 1 - 1 < m - 1 by A603,REAL_1:54; l = m+(len g -'1)-'1 by A438,JORDAN4:3 .= (len g -'1)+m - 1 by A676,JORDAN4:2 .= (len g - 1)+m - 1 by A677,BINARITH:def 3 .= ((k - (m - 1)) - 1)+m - 1 by A436,A443,A678,BINARITH:def 3 .= ((k - m+1) - 1)+m - 1 by XCMPLX_1:37 .= (k - m)+m - 1 by XCMPLX_1:26 .= (k+-m+m) - 1 by XCMPLX_0:def 8 .= k - 1 by XCMPLX_1:139; then A679: k = l+1 by XCMPLX_1:27; then A680: l < k by REAL_1:69; len g-'1 <= l by NAT_1:29; then A681: 1 <= l by A676,AXIOMS:22; len g-'1 <= len g by JORDAN3:7; then A682: len g-'1 in dom g by A676,FINSEQ_3:27; 1 <= len g by A600,SPPOL_1:5; then A683: len g-'1+2 = len g+1 by Lm1; then A684: q = g/.(len g-'1) by A682,FINSEQ_5:69 .= f/.l by A682,FINSEQ_5:30; A685: right_cell(f,m-'1,G) meets C by A306,A438,A674,A675; A686: right_cell(f,l,G) meets C by A306,A436,A679,A681; not a in L~g by A622,A675; then A687: not a in L~rg by SPPOL_2:22; A688: m-'1+2 = m+1 by A438,Lm1; A689: m-'1 <= l by NAT_1:29; then m-'1 <= len(F.l) by A107; then A690: m-'1 in dom(F.l) by A675,FINSEQ_3:27; (m-'1)+1 <= l by A676,AXIOMS:24; then m-'1 < l by NAT_1:38; then A691: m-'1 <> len(F.l) by A107; consider p1,p2,q1,q2 being Nat such that A692: [p1,p2] in Indices G and A693: p = G*(p1,p2) and A694: [q1,q2] in Indices G and A695: q = G*(q1,q2) and A696: p1 = q1 & p2+1 = q2 or p1+1 = q1 & p2 = q2 or p1 = q1+1 & p2 = q2 or p1 = q1 & p2 = q2+1 by A669,A670,JORDAN8:6; A697: 1 <= p1 & p1 <= len G & 1 <= p2 & p2 <= width G by A692,GOBOARD5:1; per cases by A696; suppose A698: p1 = q1 & p2+1 = q2; consider a1,a2,p'1,p'2 being Nat such that A699: [a1,a2] in Indices G and A700: a = G*(a1,a2) and A701: [p'1,p'2] in Indices G and A702: p = G*(p'1,p'2) and A703: a1 = p'1 & a2+1 = p'2 or a1+1 = p'1 & a2 = p'2 or a1 = p'1+1 & a2 = p'2 or a1 = p'1 & a2 = p'2+1 by A437,A438,A673,A674,A675,JORDAN8:6; A704: 1 <= a1 & a1 <= len G & 1 <= a2 & a2 <= width G by A699,GOBOARD5:1; thus thesis proof per cases by A703; suppose a1 = p'1 & a2+1 = p'2; then A705: a1 = p1 & a2+1 = p2 by A692,A693,A701,A702,GOBOARD1:21; A706: F.m is_sequence_on G by A306; A707: m-'1+1 <= len (F.m) by A107,A674; A708: m-'1 <= m by A674,NAT_1:29; A709: F.k|(m+1)=F.(m+1) by A365,A436,A441; A710: p2+1 > a2+1 & a2+1 > a2 by A705,NAT_1:38; A711: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A436,A442,A617,A675 .= (F.m)/.(m-'1) by A617,A675,A708; A712: f/.(m-'1+1) = (F.m)/.m by A436,A438,A617,A674; right_cell(f,l,G) = cell(G,p1-'1,p2) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A698 ,GOBRD13:29 .= front_left_cell(F.m,m-'1,G) by A673,A674,A675,A692,A693,A699, A700,A705,A706,A707,A711,A712,GOBRD13:35; then F.(m+1) turns_left m-'1,G by A366,A603,A686; then f turns_left m-'1,G by A441,A675,A688,A709,GOBRD13:45; then A713: f/.(m+1) = G*(p1-'1,p2) & [p1-'1,p2] in Indices G by A673,A674,A688,A692,A693,A699,A700,A710,GOBRD13:def 7; then A714: 1 <= p1-'1 & p1-'1 <= len G by GOBOARD5:1; set rc = right_cell(rg,len rg-'1,G)\L~rg; A715: rc c= RightComp rg by A669,A671,A672,Th29; A716: p2-'1 = a2 & p1-'1+1 = p1 by A697,A705,AMI_5:4,BINARITH:39; A717: 2 in dom g by A600,FINSEQ_3:27; len rg-'1+2 = len g +1 by A683,FINSEQ_5:def 3; then A718: rg/.(len rg-'1) = g/.2 by A717,FINSEQ_5:69 .= f/.(m+1) by A688,A717,FINSEQ_5:30; p = g/.1 by A435,A444,A445,FINSEQ_5:68 .= rg/.len g by FINSEQ_5:68 .= rg/.len rg by FINSEQ_5:def 3; then right_cell(rg,len rg-'1,G) = cell(G,p1-'1,a2) by A669,A671,A672,A692,A693,A713,A716,A718,GOBRD13:25; then a in right_cell(rg,len rg-'1,G) by A697,A700,A704,A705,A714,A716,Th22; then A719: a in rc by A687,XBOOLE_0:def 4; A720: L~rg c= L~f by A621,SPPOL_2:22; RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2; hence C meets RightComp Rev g by A438,A649,A674,A675,A687,A715,A719, A720; suppose A721: a1+1 = p'1 & a2 = p'2; then A722: a1+1 = p1 & a2 = p2 by A692,A693,A701,A702,GOBOARD1:21; then q1-'1 = a1 by A698,BINARITH:39; then right_cell(f,l,G) = cell(G,a1,a2) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A698 ,A722,GOBRD13:29 .= left_cell(f,m-'1,G) by A437,A438,A673,A674,A675,A699,A700,A701,A702,A721,GOBRD13: 24; hence C meets RightComp Rev g by A306,A438,A674,A675,A686; suppose a1 = p'1+1 & a2 = p'2; then a1 = p1+1 & a2 = p2 by A692,A693,A701,A702,GOBOARD1:21; then right_cell(f,m-'1,G) = cell(G,p1,p2) by A437,A438,A673,A674,A675,A692,A693,A699,A700,GOBRD13:27 .= left_cell(f,l,G) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A698 ,GOBRD13:28; hence C meets RightComp Rev g by A306,A436,A679,A681,A685; suppose a1 = p'1 & a2 = p'2+1; then A723: a1 = q1 & a2 = q2 by A692,A693,A698,A701,A702,GOBOARD1:21; (F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A617,A675,A689 .= q by A436,A442,A617,A675,A695,A700,A723 .= (F.l)/.l by A617,A680,A681,A684 .= (F.l)/.len(F.l) by A107; hence C meets RightComp Rev g by A432,A680,A681,A690,A691; end; suppose A724: p1+1 = q1 & p2 = q2; consider a1,a2,p'1,p'2 being Nat such that A725: [a1,a2] in Indices G and A726: a = G*(a1,a2) and A727: [p'1,p'2] in Indices G and A728: p = G*(p'1,p'2) and A729: a1 = p'1 & a2+1 = p'2 or a1+1 = p'1 & a2 = p'2 or a1 = p'1+1 & a2 = p'2 or a1 = p'1 & a2 = p'2+1 by A437,A438,A673,A674,A675,JORDAN8:6; A730: 1 <= a1 & a1 <= len G & 1 <= a2 & a2 <= width G by A725,GOBOARD5:1; thus thesis proof per cases by A729; suppose A731: a1 = p'1 & a2+1 = p'2; then A732: a1 = p1 & a2+1 = p2 by A692,A693,A727,A728,GOBOARD1:21; then A733: q2-'1 = a2 by A724,BINARITH:39; right_cell(f,m-'1,G) = cell(G,a1,a2) by A437,A438,A673,A674,A675,A725,A726,A727,A728,A731,GOBRD13: 23 .= left_cell(f,l,G) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A724, A732,A733,GOBRD13:26; hence C meets RightComp Rev g by A306,A436,A679,A681,A685; suppose a1+1 = p'1 & a2 = p'2; then A734: a1+1 = p1 & a2 = p2 by A692,A693,A727,A728,GOBOARD1:21; A735: F.m is_sequence_on G by A306; A736: m-'1+1 <= len (F.m) by A107,A674; A737: m-'1 <= m by A674,NAT_1:29; A738: F.k|(m+1)=F.(m+1) by A365,A436,A441; A739: a1 < a1+1 & p1 < p1+1 by REAL_1:69; A740: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A436,A442,A617,A675 .= (F.m)/.(m-'1) by A617,A675,A737; A741: f/.(m-'1+1) = (F.m)/.m by A436,A438,A617,A674; right_cell(f,l,G) = cell(G,p1,p2) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A724 ,GOBRD13:27 .= front_left_cell(F.m,m-'1,G) by A673,A674,A675,A692,A693,A725, A726,A734,A735,A736,A740,A741,GOBRD13:37; then F.(m+1) turns_left m-'1,G by A366,A603,A686; then f turns_left m-'1,G by A441,A675,A688,A738,GOBRD13:45; then A742: f/.(m+1) = G*(p1,p2+1) & [p1,p2+1] in Indices G by A673,A674,A688,A692,A693,A725,A726,A734,A739,GOBRD13:def 7; then A743: p2+1 <= width G by GOBOARD5:1; set rc = right_cell(rg,len rg-'1,G)\L~rg; A744: rc c= RightComp rg by A669,A671,A672,Th29; A745: a1 = p1-'1 by A734,BINARITH:39; A746: 2 in dom g by A600,FINSEQ_3:27; len rg-'1+2 = len g +1 by A683,FINSEQ_5:def 3; then A747: rg/.(len rg-'1) = g/.2 by A746,FINSEQ_5:69 .= f/.(m+1) by A688,A746,FINSEQ_5:30; p = g/.1 by A435,A444,A445,FINSEQ_5:68 .= rg/.len g by FINSEQ_5:68 .= rg/.len rg by FINSEQ_5:def 3; then right_cell(rg,len rg-'1,G) = cell(G,p1-'1,a2) by A669,A671,A672,A692,A693,A734,A742,A747,GOBRD13:29; then a in right_cell(rg,len rg-'1,G) by A697,A726,A730,A734,A743,A745,Th22; then A748: a in rc by A687,XBOOLE_0:def 4; A749: L~rg c= L~f by A621,SPPOL_2:22; RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2; hence C meets RightComp Rev g by A438,A649,A674,A675,A687,A744,A748, A749; suppose a1 = p'1+1 & a2 = p'2; then A750: a1 = q1 & a2 = q2 by A692,A693,A724,A727,A728,GOBOARD1:21; (F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A617,A675,A689 .= q by A436,A442,A617,A675,A695,A726,A750 .= (F.l)/.l by A617,A680,A681,A684 .= (F.l)/.len(F.l) by A107; hence C meets RightComp Rev g by A432,A680,A681,A690,A691; suppose a1 = p'1 & a2 = p'2+1; then A751: a1 = p1 & a2 = p2+1 by A692,A693,A727,A728,GOBOARD1:21; right_cell(f,l,G) = cell(G,p1,p2) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A724 ,GOBRD13:27 .= left_cell(f,m-'1,G) by A437,A438,A673,A674,A675,A692,A693,A725,A726,A751,GOBRD13: 28; hence C meets RightComp Rev g by A306,A438,A674,A675,A686; end; suppose A752: p1 = q1+1 & p2 = q2; consider a1,a2,p'1,p'2 being Nat such that A753: [a1,a2] in Indices G and A754: a = G*(a1,a2) and A755: [p'1,p'2] in Indices G and A756: p = G*(p'1,p'2) and A757: a1 = p'1 & a2+1 = p'2 or a1+1 = p'1 & a2 = p'2 or a1 = p'1+1 & a2 = p'2 or a1 = p'1 & a2 = p'2+1 by A437,A438,A673,A674,A675,JORDAN8:6; A758: 1 <= a1 & a1 <= len G & 1 <= a2 & a2 <= width G by A753,GOBOARD5:1; thus thesis proof per cases by A757; suppose A759: a1 = p'1 & a2+1 = p'2; then A760: a1 = p1 & a2+1 = p2 by A692,A693,A755,A756,GOBOARD1:21; then A761: q1 = a1-'1 by A752,BINARITH:39; A762: q2-'1 = a2 by A752,A760,BINARITH:39; right_cell(f,l,G) = cell(G,q1,q2-'1) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A752, GOBRD13:25 .= left_cell(f,m-'1,G) by A437,A438,A673,A674,A675,A753,A754,A755,A756,A759,A761,A762, GOBRD13:22; hence C meets RightComp Rev g by A306,A438,A674,A675,A686; suppose a1+1 = p'1 & a2 = p'2; then a1+1 = p1 & a2 = p2 by A692,A693,A755,A756,GOBOARD1:21; then A763: a1 = q1 & a2 = q2 by A752,XCMPLX_1:2; (F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A617,A675,A689 .= q by A436,A442,A617,A675,A695,A754,A763 .= (F.l)/.l by A617,A680,A681,A684 .= (F.l)/.len(F.l) by A107; hence C meets RightComp Rev g by A432,A680,A681,A690,A691; suppose a1 = p'1+1 & a2 = p'2; then A764: a1 = p1+1 & a2 = p2 by A692,A693,A755,A756,GOBOARD1:21; A765: p1-'1 = q1 by A752,BINARITH:39; A766: F.m is_sequence_on G by A306; A767: m-'1+1 <= len (F.m) by A107,A674; A768: m-'1 <= m by A674,NAT_1:29; A769: F.k|(m+1)=F.(m+1) by A365,A436,A441; p1+1>p1 by REAL_1:69; then A770: a2+1 > p2 & a1+1 > p1 & a2 < p2+1 by A764,NAT_1:38; A771: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A436,A442,A617,A675 .= (F.m)/.(m-'1) by A617,A675,A768; A772: f/.(m-'1+1) = (F.m)/.m by A436,A438,A617,A674; right_cell(f,l,G) = cell(G,q1,q2-'1) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A752 ,GOBRD13:25 .= front_left_cell(F.m,m-'1,G) by A673,A674,A675,A692,A693,A752, A753,A754,A764,A765,A766,A767,A771,A772,GOBRD13:39 ; then F.(m+1) turns_left m-'1,G by A366,A603,A686; then f turns_left m-'1,G by A441,A675,A688,A769,GOBRD13:45; then A773: f/.(m+1) = G*(p1,p2-'1) & [p1,p2-'1] in Indices G by A673,A674,A688,A692,A693,A753,A754,A770,GOBRD13:def 7; then A774: 1 <= p2-'1 & p2-'1 <= width G by GOBOARD5:1; set rc = right_cell(rg,len rg-'1,G)\L~rg; A775: rc c= RightComp rg by A669,A671,A672,Th29; A776: p2-'1+1 = p2 by A697,AMI_5:4; A777: 2 in dom g by A600,FINSEQ_3:27; len rg-'1+2 = len g +1 by A683,FINSEQ_5:def 3; then A778: rg/.(len rg-'1) = g/.2 by A777,FINSEQ_5:69 .= f/.(m+1) by A688,A777,FINSEQ_5:30; p = g/.1 by A435,A444,A445,FINSEQ_5:68 .= rg/.len g by FINSEQ_5:68 .= rg/.len rg by FINSEQ_5:def 3; then right_cell(rg,len rg-'1,G) = cell(G,p1,p2-'1) by A669,A671,A672,A692,A693,A773,A776,A778,GOBRD13:23; then a in right_cell(rg,len rg-'1,G) by A697,A754,A758,A764,A774,A776,Th22; then A779: a in rc by A687,XBOOLE_0:def 4; A780: L~rg c= L~f by A621,SPPOL_2:22; RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2; hence C meets RightComp Rev g by A438,A649,A674,A675,A687,A775,A779, A780; suppose a1 = p'1 & a2 = p'2+1; then A781: a1 = p1 & a2 = p2+1 by A692,A693,A755,A756,GOBOARD1:21; then q1 = a1-'1 by A752,BINARITH:39; then right_cell(f,m-'1,G) = cell(G,q1,q2) by A437,A438,A673,A674,A675,A692,A693,A752,A753,A754,A781, GOBRD13:29 .= left_cell(f,l,G) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A752 ,GOBRD13:24; hence C meets RightComp Rev g by A306,A436,A679,A681,A685; end; suppose A782: p1 = q1 & p2 = q2+1; consider a1,a2,p'1,p'2 being Nat such that A783: [a1,a2] in Indices G and A784: a = G*(a1,a2) and A785: [p'1,p'2] in Indices G and A786: p = G*(p'1,p'2) and A787: a1 = p'1 & a2+1 = p'2 or a1+1 = p'1 & a2 = p'2 or a1 = p'1+1 & a2 = p'2 or a1 = p'1 & a2 = p'2+1 by A437,A438,A673,A674,A675,JORDAN8:6; A788: 1 <= a1 & a1 <= len G & 1 <= a2 & a2 <= width G by A783,GOBOARD5:1; thus thesis proof per cases by A787; suppose a1 = p'1 & a2+1 = p'2; then a1 = p1 & a2+1 = p2 by A692,A693,A785,A786,GOBOARD1:21; then A789: a1 = q1 & a2 = q2 by A782,XCMPLX_1:2; (F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A617,A675,A689 .= q by A436,A442,A617,A675,A695,A784,A789 .= (F.l)/.l by A617,A680,A681,A684 .= (F.l)/.len(F.l) by A107; hence C meets RightComp Rev g by A432,A680,A681,A690,A691; suppose A790: a1+1 = p'1 & a2 = p'2; then A791: a1+1 = p1 & a2 = p2 by A692,A693,A785,A786,GOBOARD1:21; then A792: a1 = q1-'1 by A782,BINARITH:39; A793: a2-'1 = q2 by A782,A791,BINARITH:39; right_cell(f,m-'1,G) = cell(G,a1,a2-'1) by A437,A438,A673,A674,A675,A783,A784,A785,A786,A790,GOBRD13: 25 .= left_cell(f,l,G) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A782, A792,A793,GOBRD13:22; hence C meets RightComp Rev g by A306,A436,A679,A681,A685; suppose a1 = p'1+1 & a2 = p'2; then A794: a1 = p1+1 & a2 = p2 by A692,A693,A785,A786,GOBOARD1:21; then A795: a2-'1 = q2 by A782,BINARITH:39; right_cell(f,l,G) = cell(G,q1,q2) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A782 ,GOBRD13:23 .= left_cell(f,m-'1,G) by A437,A438,A673,A674,A675,A692,A693,A782,A783,A784,A794,A795 ,GOBRD13:26; hence C meets RightComp Rev g by A306,A438,A674,A675,A686; suppose a1 = p'1 & a2 = p'2+1; then A796: a1 = p1 & a2 = p2+1 by A692,A693,A785,A786,GOBOARD1:21; A797: p2-'1 = q2 by A782,BINARITH:39; A798: F.m is_sequence_on G by A306; A799: m-'1+1 <= len (F.m) by A107,A674; A800: m-'1 <= m by A674,NAT_1:29; A801: F.k|(m+1)=F.(m+1) by A365,A436,A441; A802: a2+1>p2+1 & p2+1>p2 by A796,NAT_1:38; A803: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A436,A442,A617,A675 .= (F.m)/.(m-'1) by A617,A675,A800; A804: f/.(m-'1+1) = (F.m)/.m by A436,A438,A617,A674; right_cell(f,l,G) = cell(G,q1,q2) by A435,A436,A437,A673,A679,A681,A684,A692,A693,A694,A695,A782 ,GOBRD13:23 .= front_left_cell(F.m,m-'1,G) by A673,A674,A675,A692,A693,A782, A783,A784,A796,A797,A798,A799,A803,A804,GOBRD13:41 ; then F.(m+1) turns_left m-'1,G by A366,A603,A686; then f turns_left m-'1,G by A441,A675,A688,A801,GOBRD13:45; then A805: f/.(m+1) = G*(p1+1,p2) & [p1+1,p2] in Indices G by A673,A674,A688,A692,A693,A783,A784,A802,GOBRD13:def 7; then A806: p1+1 <= len G by GOBOARD5:1; set rc = right_cell(rg,len rg-'1,G)\L~rg; A807: rc c= RightComp rg by A669,A671,A672,Th29; A808: 2 in dom g by A600,FINSEQ_3:27; len rg-'1+2 = len g +1 by A683,FINSEQ_5:def 3; then A809: rg/.(len rg-'1) = g/.2 by A808,FINSEQ_5:69 .= f/.(m+1) by A688,A808,FINSEQ_5:30; p = g/.1 by A435,A444,A445,FINSEQ_5:68 .= rg/.len g by FINSEQ_5:68 .= rg/.len rg by FINSEQ_5:def 3; then right_cell(rg,len rg-'1,G) = cell(G,p1,p2) by A669,A671,A672,A692,A693,A805,A809,GOBRD13:27; then a in right_cell(rg,len rg-'1,G) by A697,A784,A788,A796,A806,Th22; then A810: a in rc by A687,XBOOLE_0:def 4; A811: L~rg c= L~f by A621,SPPOL_2:22; RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2; hence C meets RightComp Rev g by A438,A649,A674,A675,A687,A807,A810, A811; end; end; A812: C meets RightComp g proof right_cell(f,m,G) meets C by A306,A438,A441; then consider p being set such that A813: p in right_cell(f,m,G) & p in C by XBOOLE_0:3; reconsider p as Element of TOP-REAL 2 by A813; now take a = p; thus a in C by A813; reconsider u = p as Element of Euclid 2 by A615; consider r being real number such that A814: r > 0 and A815: Ball(u,r) c= (L~g)` by A605,A616,A813,GOBOARD6:8; reconsider r as Real by XREAL_0:def 1; A816: p in Ball(u,r) by A814,GOBOARD6:4; Ball(u,r) is Subset of TOP-REAL 2 by A614,TOPMETR:16; then reconsider B = Ball(u,r) as non empty Subset of TOP-REAL 2 by A814,GOBOARD6:4; A817: B is connected by SPRECT_3:17; right_cell(g,1,G) c= right_cell(g,1) by A446,A600,GOBRD13:34; then A818: Int right_cell(g,1,G) c= Int right_cell(g,1) by TOPS_1:48; Int right_cell(g,1) c= RightComp g by A600,GOBOARD9:28; then Int right_cell(g,1,G) c= RightComp g by A818,XBOOLE_1:1; then Int right_cell(f,m-'1+1,G) c= RightComp g by A437,A442,A600,GOBRD13:33 ; then A819: Int right_cell(f,m,G) c= RightComp g by A438,AMI_5:4; A820: right_cell(f,m,G) = Cl Int right_cell(f,m,G) by A438,A441,A486; B is open by GOBOARD6:6; then A821: Int right_cell(f,m,G) meets B by A813,A816,A820,TOPS_1:39; A822: p in B by A814,GOBOARD6:4; B c= RightComp g by A604,A815,A817,A819,A821,GOBOARD9:6; hence a in RightComp g by A822; end; hence thesis by XBOOLE_0:3; end; C meets LeftComp g by A668,GOBOARD9:27; then LeftComp g = RightComp g by A1,A604,A605,A812,Th3; hence contradiction by SPRECT_4:7; end; then A823: g = f/^0 by GOBOARD9:1 .= f by FINSEQ_5:31; A824: now consider i such that A825: 1 <= i & i+1 <= len G and A826: N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) and A827: F.(1+1) = <*G*(i,width G),G*(i+1,width G)*> by A92; take i; thus 1 <= i & i+1 <= len G by A825; A828: f|2 = F.2 by A365,A436,A440; A829: len(f|2) = 2 by A440,TOPREAL1:3; then 1 in dom(f|2) by FINSEQ_3:27; hence f/.1 = (f|2)/.1 by TOPREAL1:1 .= G*(i,width G) by A827,A828,FINSEQ_4:26; 2 in dom(f|2) by A829,FINSEQ_3:27; hence f/.2 = (f|2)/.2 by TOPREAL1:1 .= G*(i+1,width G) by A827,A828,FINSEQ_4:26; thus N-min C in cell(G,i,width G-'1) & N-min C <> G* (i,width G-'1) by A826; end; reconsider f as standard non constant special_circular_sequence by A823; f is clockwise_oriented proof len G >= 3 by A2,NAT_1:37; then A830: 1 < len G by AXIOMS:22; for k st 1 <= k & k+1 <= len f holds left_cell(f,k,G) misses C & right_cell(f,k,G) meets C by A306; then A831: N-min L~f = f/.1 by A437,A824,Th32; f/.2 in LSeg(f/.1,f/.(1+1)) by TOPREAL1:6; then A832: f/.2 in L~f by A440,SPPOL_2:15; consider i such that A833: 1 <= i & i+1 <= len G and A834: f/.1 = G*(i,width G) and A835: f/.2 = G*(i+1,width G) and N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) by A824; A836: (NW-corner L~f)`2 = (NE-corner L~f)`2 by PSCOMP_1:82; A837: (NE-corner L~f)`2 = N-bound L~f & (N-min L~f)`2 = N-bound L~f by PSCOMP_1:77,94; i < len G by A833,NAT_1:38; then A838: G*(i,width G)`2 = G*(1,width G)`2 by A2,A830,A833,GOBOARD5:2; 1 <= i+1 by NAT_1:37; then A839: G*(i+1,width G)`2 = G*(1,width G)`2 by A2,A830,A833,GOBOARD5:2; (NW-corner L~f)`1 = W-bound L~f & (NE-corner L~f)`1 = E-bound L~f by PSCOMP_1:74,76; then (NW-corner L~f)`1 <= (f/.2)`1 & (f/.2)`1 <= (NE-corner L~f)`1 by A832,PSCOMP_1:71; then f/.2 in LSeg(NW-corner L~f, NE-corner L~f) by A831,A834,A835,A836,A837,A838,A839,GOBOARD7:9; then f/.2 in LSeg(NW-corner L~f, NE-corner L~f) /\ L~f by A832,XBOOLE_0:def 3; then f/.2 in N-most L~f by PSCOMP_1:def 39; hence thesis by A831,SPRECT_2:34; end; then reconsider f as clockwise_oriented (standard non constant special_circular_sequence); take f; thus f is_sequence_on G by A306; thus ex i st 1 <= i & i+1 <= len G & f/.1 = G*(i,width G) & f/.2 = G*(i+1,width G) & N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) by A824; let m such that A840: 1 <= m & m+2 <= len f; m+1 < m+2 by REAL_1:53; then A841: m+1 <= len f by A840,AXIOMS:22; then A842: f|(m+1) = F.(m+1) by A365,A436; A843: m = m+1-'1 by BINARITH:39; A844: m+1 > 1 by A840,NAT_1:38; A845: front_left_cell(F.(m+1),m,G) = front_left_cell(f,m,G) & front_right_cell(F.(m+1),m,G) = front_right_cell(f,m,G) by A437,A840,A841,A842,GOBRD13:43; A846: m+1+1 = m+(1+1) by XCMPLX_1:1; then A847: F.(m+1+1) = f|(m+1+1) by A365,A436,A840; hereby assume front_left_cell(f,m,G) misses C & front_right_cell(f,m,G) misses C; then F.(m+1+1) turns_right m,G by A366,A843,A844,A845; hence f turns_right m,G by A840,A846,A847,GOBRD13:44; end; hereby assume front_left_cell(f,m,G) misses C & front_right_cell(f,m,G) meets C; then F.(m+1+1) goes_straight m,G by A366,A843,A844,A845; hence f goes_straight m,G by A840,A846,A847,GOBRD13:46; end; assume front_left_cell(f,m,G) meets C; then F.(m+1+1) turns_left m,G by A366,A843,A844,A845; hence f turns_left m,G by A840,A846,A847,GOBRD13:45; end; uniqueness proof let f1,f2 be clockwise_oriented (standard non constant special_circular_sequence) such that A848: f1 is_sequence_on Gauge(C,n); given i1 such that A849: 1 <= i1 & i1+1 <= len Gauge(C,n) and A850: f1/.1 = Gauge(C,n)*(i1,width Gauge(C,n)) & f1/.2 = Gauge(C,n)*(i1+1,width Gauge(C,n)) and A851: N-min C in cell(Gauge(C,n),i1,width Gauge(C,n)-'1) and A852: N-min C <> Gauge(C,n)*(i1,width Gauge(C,n)-'1); assume that A853: for k st 1 <= k & k+2 <= len f1 holds (front_left_cell(f1,k,Gauge(C,n)) misses C & front_right_cell(f1,k,Gauge(C,n)) misses C implies f1 turns_right k,Gauge(C,n)) & (front_left_cell(f1,k,Gauge(C,n)) misses C & front_right_cell(f1,k,Gauge(C,n)) meets C implies f1 goes_straight k,Gauge(C,n)) & (front_left_cell(f1,k,Gauge(C,n)) meets C implies f1 turns_left k,Gauge(C,n)) and A854: f2 is_sequence_on Gauge(C,n); given i2 such that A855: 1 <= i2 & i2+1 <= len Gauge(C,n) and A856: f2/.1 = Gauge(C,n)*(i2,width Gauge(C,n)) & f2/.2 = Gauge(C,n)*(i2+1,width Gauge(C,n)) and A857: N-min C in cell(Gauge(C,n),i2,width Gauge(C,n)-'1) and A858: N-min C <> Gauge(C,n)*(i2,width Gauge(C,n)-'1); assume A859: for k st 1 <= k & k+2 <= len f2 holds (front_left_cell(f2,k,Gauge(C,n)) misses C & front_right_cell(f2,k,Gauge(C,n)) misses C implies f2 turns_right k,Gauge(C,n)) & (front_left_cell(f2,k,Gauge(C,n)) misses C & front_right_cell(f2,k,Gauge(C,n)) meets C implies f2 goes_straight k,Gauge(C,n)) & (front_left_cell(f2,k,Gauge(C,n)) meets C implies f2 turns_left k,Gauge(C,n)); defpred P[Nat] means f1|$1 = f2|$1; A860: P[0]; A861: for k st P[k] holds P[k+1] proof let k such that A862: f1|k = f2|k; A863: f1|1 = <*f1/.1*> & f2|1 = <*f2/.1*> by FINSEQ_5:23; A864: i1 = i2 by A849,A851,A852,A855,A857,A858,Th31; A865: len f1 > 4 & len f2 > 4 by GOBOARD7:36; per cases by CQC_THE1:2; suppose k = 0; hence f1|(k+1) = f2|(k+1) by A849,A850,A851,A852,A855,A856,A857,A858, A863,Th31; suppose A866: k = 1; len f1 > 2 & len f2 > 2 by A865,AXIOMS:22; then f1|2 = <*f1/.1,f1/.2*> & f2|2 = <*f2/.1,f2/.2*> by JORDAN8:1; hence f1|(k+1) = f2|(k+1) by A850,A856,A864,A866; suppose A867: k > 1; A868: f1/.1 = f1/.len f1 & f2/.1 = f2/.len f2 by FINSEQ_6:def 1; now per cases; suppose A869: len f1 > k; set m = k-'1; A870: 1 <= m by A867,JORDAN3:12; then A871: m+1 = k by JORDAN3:6; A872: now assume A873: len f2 <= k; then A874: f2 = f2|k by TOPREAL1:2; then 1 in dom(f2|k) by FINSEQ_5:6; then A875: (f1|k)/.1 = f1/.1 by A862,TOPREAL1:1; len f2 in dom(f2|k) by A874,FINSEQ_5:6; then A876: (f1|k)/.len f2 = f1/.len f2 by A862,TOPREAL1:1; 1 < len f2 & len f2 <= len f1 by A862,A865,A874,AXIOMS:22,FINSEQ_5:18 ; hence contradiction by A862,A868,A869,A873,A874,A875,A876,GOBOARD7:40 ; end; then A877: k+1 <= len f2 by NAT_1:38; A878: k+1 <= len f1 by A869,NAT_1:38; A879: front_left_cell(f1,m,Gauge(C,n))= front_left_cell(f1|k,m,Gauge(C,n)) by A848,A869,A870,A871,GOBRD13:43; A880: front_right_cell(f1,m,Gauge(C,n)) = front_right_cell(f1|k,m,Gauge(C,n)) by A848,A869,A870,A871,GOBRD13:43; A881: front_left_cell(f2,m,Gauge(C,n)) = front_left_cell(f2|k,m,Gauge(C,n)) by A854,A870,A871,A872,GOBRD13:43; A882: front_right_cell(f2,m,Gauge(C,n)) = front_right_cell(f2|k,m,Gauge(C,n)) by A854,A870,A871,A872,GOBRD13:43; A883: m+(1+1) = k+1 by A871,XCMPLX_1:1; now per cases; suppose front_left_cell(f1,m,Gauge(C,n)) misses C & front_right_cell(f1,m,Gauge(C,n)) misses C; then f1 turns_right m,Gauge(C,n) & f2 turns_right m,Gauge(C,n) by A853,A859,A862,A870,A877,A878,A879,A880,A881,A882,A883; hence f1|(k+1) = f2|(k+1) by A854,A862,A867,A877,A878,GOBRD13:47; suppose front_left_cell(f1,m,Gauge(C,n)) misses C & front_right_cell(f1,m,Gauge(C,n)) meets C; then f1 goes_straight m,Gauge(C,n) & f2 goes_straight m,Gauge(C,n) by A853,A859,A862,A870,A877,A878,A879,A880,A881,A882,A883; hence f1|(k+1) = f2|(k+1) by A854,A862,A867,A877,A878,GOBRD13:49; suppose front_left_cell(f1,m,Gauge(C,n)) meets C; then f1 turns_left m,Gauge(C,n) & f2 turns_left m,Gauge(C,n) by A853,A859,A862,A870,A877,A878,A879,A881,A883; hence f1|(k+1) = f2|(k+1) by A854,A862,A867,A877,A878,GOBRD13:48; end; hence f1|(k+1) = f2|(k+1); suppose A884: k >= len f1; then A885: f1 = f1|k by TOPREAL1:2; then 1 in dom(f1|k) by FINSEQ_5:6; then A886: (f2|k)/.1 = f2/.1 by A862,TOPREAL1:1; len f1 in dom(f1|k) by A885,FINSEQ_5:6; then A887: (f2|k)/.len f1 = f2/.len f1 by A862,TOPREAL1:1; 1 < len f1 & len f1 <= len f2 by A862,A865,A885,AXIOMS:22,FINSEQ_5: 18 ; then A888: len f2 = len f1 by A862,A868,A885,A886,A887,GOBOARD7:40; A889: k+1 > len f1 by A884,NAT_1:38; hence f1|(k+1) = f1 by TOPREAL1:2 .= f2 by A862,A884,A885,A888,TOPREAL1:2 .= f2|(k+1) by A888,A889,TOPREAL1:2; end; hence f1|(k+1) = f2|(k+1); end; for k holds P[k] from Ind(A860,A861); hence f1 = f2 by Th4; end; end; theorem Th33: C is connected & 1 <= k & k+1 <= len Cage(C,n) implies left_cell(Cage(C,n),k,Gauge(C,n)) misses C & right_cell(Cage(C,n),k,Gauge(C,n)) meets C proof set G = Gauge(C,n), f = Cage(C,n); set W = W-bound C, E = E-bound C, S = S-bound C, N = N-bound C; assume A1: C is connected; then A2: f is_sequence_on G by Def1; defpred P[Nat] means for m st 1 <= m & m+1 <= len(f|$1) holds left_cell(f|$1,m,G) misses C & right_cell(f|$1,m,G) meets C; A3: P[0] proof let m; assume A4: 1 <= m & m+1 <= len(f|0); 1 <= m+1 by NAT_1:37; then 1 <= len(f|0) by A4,AXIOMS:22; then 1 <= 0 by FINSEQ_1:25; hence left_cell(f|0,m,G) misses C & right_cell(f|0,m,G) meets C; end; A5: len G = 2|^n+3 & len G = width G by JORDAN8:def 1; A6: for k st P[k] holds P[k+1] proof let k such that A7: for m st 1 <= m & m+1 <= len(f|k) holds left_cell(f|k,m,G) misses C & right_cell(f|k,m,G) meets C; per cases; suppose A8: k >= len f; then k+1 >= len f by NAT_1:37; then f|k = f & f|(k+1) = f by A8,TOPREAL1:2; hence thesis by A7; suppose A9: k < len f; then A10: k+1 <= len f by NAT_1:38; A11: f|k is_sequence_on G by A2,GOBOARD1:38; A12: f|(k+1) is_sequence_on G by A2,GOBOARD1:38; consider i such that A13: 1 <= i & i+1 <= len G and A14: f/.1 = G*(i,width G) and A15: f/.2 = G*(i+1,width G) and A16: N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) by A1,Def1; A17: len(f|k) = k by A9,TOPREAL1:3; A18: len(f|(k+1)) = k+1 by A10,TOPREAL1:3; A19: 1 <= len G by A5,NAT_1:37; 2|^n >= n+1 & n+1 >= 1 by HEINE:7,NAT_1:37; then A20: 2|^n > 0 by AXIOMS:22; let m such that A21: 1 <= m & m+1 <= len(f|(k+1)); now per cases by CQC_THE1:2; suppose A22: k = 0; 1 <= m+1 by NAT_1:37; then m+1 = 0+1 by A18,A21,A22,AXIOMS:21; then m = 0 by XCMPLX_1:2; hence left_cell(f|(k+1),m,G) misses C & right_cell(f|(k+1),m,G) meets C by A21; suppose A23: k = 1; then f|(k+1) = <*G*(i,width G),G*(i+1,width G)*> by A10,A14,A15,JORDAN8:1; then A24: (f|(k+1))/.1 = G*(i,width G) & (f|(k+1))/.2 = G*(i+1,width G) by FINSEQ_4:26; A25: i < len G & 1 <= i+1 & i+1 <= len G by A13,NAT_1:38; then A26: [i,len G] in Indices G & [i+1,len G] in Indices G by A5,A13,A19,GOBOARD7:10; 1+1 <= m+1 by A21,AXIOMS:24; then m+1 = 1+1 by A18,A21,A23,AXIOMS:21; then A27: m = 1 by XCMPLX_1:2; A28: i < i+1 & i+1 < (i+1)+1 by NAT_1:38; then A29: left_cell(f|(k+1),m,G) = cell(G,i,len G) by A5,A12,A21,A24,A26,A27,GOBRD13:def 3; now assume left_cell(f|(k+1),m,G) meets C; then consider p being set such that A30: p in cell(G,i,len G) & p in C by A29,XBOOLE_0:3; reconsider p as Element of TOP-REAL 2 by A30; cell(G,i,len G) = { |[r,s]|: G*(i,1)`1 <= r & r <= G*(i+1,1)`1 & G*(1,len G)`2 <= s } by A5,A13,A25,GOBRD11:31; then consider r,s such that A31: p = |[r,s]| and G*(i,1)`1 <= r & r <= G*(i+1,1)`1 and A32: G*(1,len G)`2 <= s by A30; [1,len G] in Indices G by A5,A19,GOBOARD7:10; then A33: G*(1,len G) = |[W+((E-W)/(2|^n))*(1-2),S+((N-S)/(2|^n))*((len G)- 2)]| by JORDAN8:def 1; 2|^n+(2+1) = 2|^n+1+2 by XCMPLX_1:1; then (len G) - 2 = 2|^n+1 by A5,XCMPLX_1:26; then ((N-S)/(2|^n))*((len G)-2) = ((N-S)/(2|^n))*(2|^n)+((N-S)/(2|^n))*1 by XCMPLX_1:8 .= (N-S)+(N-S)/(2|^n) by A20,XCMPLX_1:88; then A34: S+((N-S)/(2|^n))*((len G) -2) = S+(N-S)+(N-S)/(2|^n) by XCMPLX_1:1 .= N+(N-S)/(2|^n) by XCMPLX_1:27; A35: G*(1,len G)`2 = S+((N-S)/(2|^n))*((len G)-2) by A33,EUCLID:56; N > S by JORDAN8:12; then N-S > S-S by REAL_1:54; then N-S > 0 by XCMPLX_1:14; then (N-S)/(2|^n) > 0 by A20,REAL_2:127; then N+0 < N+(N-S)/(2|^n) by REAL_1:53; then A36: N < s by A32,A34,A35,AXIOMS:22; p`2 <= N by A30,PSCOMP_1:71; hence contradiction by A31,A36,EUCLID:56; end; hence left_cell(f|(k+1),m,G) misses C; A37: N-min C in C by SPRECT_1:13; N-min C in right_cell(f|(k+1),m,G) by A5,A12,A16,A21,A24,A26,A27,A28,GOBRD13:def 2; hence right_cell(f|(k+1),m,G) meets C by A37,XBOOLE_0:3; suppose A38: k > 1; then A39: 1 <= (len(f|k))-'1 by A17,JORDAN3:12; A40: (len(f|k)) -'1 +1 = len(f|k) by A17,A38,AMI_5:4; now per cases; suppose A41: m+1 = len(f|(k+1)); then A42: m = len(f|k) by A17,A18,XCMPLX_1:2; A43: len(f|k) <= len f by FINSEQ_5:18; now consider i1,j1,i2,j2 being Nat such that A44: [i1,j1] in Indices G & f/.((len(f|k)) -'1) = G*(i1,j1) and A45: [i2,j2] in Indices G & f/.len(f|k) = G*(i2,j2) and i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A2,A9,A17,A39,A40,JORDAN8:6; A46: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A45,GOBOARD5:1; A47: (len(f|k))-'1+(1+1) = (len(f|k))+1 by A40,XCMPLX_1:1; A48: (i2-'1)+1 = i2 by A46,AMI_5:4; A49: (j2-'1)+1 = j2 by A46,AMI_5:4; left_cell(f|k,(len(f|k))-'1,G) misses C & right_cell(f|k,(len(f|k))-'1,G) meets C by A7,A39,A40; then A50: left_cell(f,(len(f|k))-'1,G) misses C & right_cell(f,(len(f|k))-'1,G) meets C by A2,A17,A39,A40,A43,GOBRD13:32; per cases; suppose A51: front_left_cell(f,(len(f|k))-'1,G) misses C & front_right_cell(f,(len(f|k))-'1,G) misses C; then A52: f turns_right (len(f|k))-'1,G by A1,A10,A17,A39,A47, Def1; now per cases by A40,A44,A45,A47,A52,GOBRD13:def 6; suppose that A53: i1 = i2 & j1+1 = j2 and A54: [i2+1,j2] in Indices G and A55: f/.((len(f|k))+1) = G*(i2+1,j2); front_right_cell(f,(len(f|k))-'1,G) = cell(G,i1,j2) by A2,A39,A40,A43,A44,A45,A53,GOBRD13:36; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A51,A53,A54,A55,GOBRD13:24; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; j2-'1 = j1 & cell(G,i1,j1) meets C by A2,A39,A40,A43,A44,A45,A50,A53,BINARITH:39,GOBRD13:23; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A53,A54,A55,GOBRD13:25; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; suppose that A56: i1+1 = i2 & j1 = j2 and A57: [i2,j2-'1] in Indices G and A58: f/.((len(f|k))+1) = G*(i2,j2-'1); front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2-'1) by A2,A39,A40,A43,A44,A45,A56,GOBRD13:38; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A49,A51,A57,A58,GOBRD13:28; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; i2-'1 = i1 & cell(G,i1,j1-'1) meets C by A2,A39,A40,A43,A44,A45,A50,A56,BINARITH:39,GOBRD13:25; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A49,A56,A57,A58,GOBRD13:29; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; suppose that A59: i1 = i2+1 & j1 = j2 and A60: [i2,j2+1] in Indices G and A61: f/.((len(f|k))+1) = G*(i2,j2+1); front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1,j2) by A2,A39,A40,A43,A44,A45,A59,GOBRD13:40; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A51,A60,A61,GOBRD13:22; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; cell(G,i2,j2) meets C by A2,A39,A40,A43,A44,A45,A50,A59,GOBRD13:27; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A60,A61,GOBRD13:23; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; suppose that A62: i1 = i2 & j1 = j2+1 and A63: [i2-'1,j2] in Indices G and A64: f/.((len(f|k))+1) = G*(i2-'1,j2); front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1,j2-'1) by A2,A39,A40,A43,A44,A45,A62,GOBRD13:42; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A48,A51,A63,A64,GOBRD13:26; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; cell(G,i2-'1,j2) meets C by A2,A39,A40,A43,A44,A45,A50,A62, GOBRD13:29; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A48,A63,A64,GOBRD13:27; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; end; hence thesis; suppose A65: front_left_cell(f,(len(f|k))-'1,G) misses C & front_right_cell(f,(len(f|k))-'1,G) meets C; then A66: f goes_straight (len(f|k))-'1,G by A1,A10,A17,A39, A47,Def1; now per cases by A40,A44,A45,A47,A66,GOBRD13:def 8; suppose that A67: i1 = i2 & j1+1 = j2 and A68: [i2,j2+1] in Indices G and A69: f/.(len(f|k)+1) = G*(i2,j2+1); front_left_cell(f,(len(f|k))-'1,G) = cell(G,i1-'1,j2) by A2,A39,A40,A43,A44,A45,A67,GOBRD13:35; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A65,A67,A68,A69,GOBRD13:22; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; front_right_cell(f,(len(f|k))-'1,G) = cell(G,i1,j2) by A2,A39,A40,A43,A44,A45,A67,GOBRD13:36; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A65,A67,A68,A69,GOBRD13:23; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; suppose that A70: i1+1 = i2 & j1 = j2 and A71: [i2+1,j2] in Indices G and A72: f/.(len(f|k)+1) = G*(i2+1,j2); front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2) by A2,A39,A40,A43,A44,A45,A70,GOBRD13:37; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A65,A71,A72,GOBRD13:24; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2-'1) by A2,A39,A40,A43,A44,A45,A70,GOBRD13:38; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A65,A71,A72,GOBRD13:25; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; suppose that A73: i1 = i2+1 & j1 = j2 and A74: [i2-'1,j2] in Indices G and A75: f/.(len(f|k)+1) = G*(i2-'1,j2); front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1,j2-'1) by A2,A39,A40,A43,A44,A45,A73,GOBRD13:39; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A48,A65,A74,A75,GOBRD13:26; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1,j2) by A2,A39,A40,A43,A44,A45,A73,GOBRD13:40; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A48,A65,A74,A75,GOBRD13:27; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; suppose that A76: i1 = i2 & j1 = j2+1 and A77: [i2,j2-'1] in Indices G and A78: f/.(len(f|k)+1) = G*(i2,j2-'1); front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2-'1) by A2,A39,A40,A43,A44,A45,A76,GOBRD13:41; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A49,A65,A77,A78,GOBRD13:28; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1,j2-'1) by A2,A39,A40,A43,A44,A45,A76,GOBRD13:42; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A49,A65,A77,A78,GOBRD13:29; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; end; hence thesis; suppose A79: front_left_cell(f,(len(f|k))-'1,G) meets C; then A80: f turns_left (len(f|k))-'1,G by A1,A10,A17,A39,A47,Def1; now per cases by A40,A44,A45,A47,A80,GOBRD13:def 7; suppose that A81: i1 = i2 & j1+1 = j2 and A82: [i2-'1,j2] in Indices G and A83: f/.(len(f|k)+1) = G*(i2-'1,j2); j2-'1 = j1 & cell(G,i1-'1,j1) misses C by A2,A39,A40,A43,A44,A45,A50,A81,BINARITH:39,GOBRD13:22; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A48,A81,A82,A83,GOBRD13:26; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; front_left_cell(f,(len(f|k))-'1,G) = cell(G,i1-'1,j2) by A2,A39,A40,A43,A44,A45,A81,GOBRD13:35; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A48,A79,A81,A82,A83,GOBRD13:27; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; suppose that A84: i1+1 = i2 & j1 = j2 and A85: [i2,j2+1] in Indices G and A86: f/.(len(f|k)+1) = G*(i2,j2+1); i2-'1 = i1 & cell(G,i1,j1) misses C by A2,A39,A40,A43,A44,A45,A50,A84,BINARITH:39,GOBRD13:24; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A84,A85,A86,GOBRD13:22; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2) by A2,A39,A40,A43,A44,A45,A84,GOBRD13:37; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A79,A85,A86,GOBRD13:23; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; suppose that A87: i1 = i2+1 & j1 = j2 and A88: [i2,j2-'1] in Indices G and A89: f/.(len(f|k)+1) = G*(i2,j2-'1); cell(G,i2,j2-'1) misses C by A2,A39,A40,A43,A44,A45,A50,A87,GOBRD13:26; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A49,A88,A89,GOBRD13:28; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1,j2-'1) by A2,A39,A40,A43,A44,A45,A87,GOBRD13:39; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A49,A79,A88,A89,GOBRD13:29; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; suppose that A90: i1 = i2 & j1 = j2+1 and A91: [i2+1,j2] in Indices G and A92: f/.(len(f|k)+1) = G*(i2+1,j2); cell(G,i2,j2) misses C by A2,A39,A40,A43,A44,A45,A50,A90,GOBRD13:28; then left_cell(f,m,G) misses C by A2,A10,A17,A21,A42,A45,A91,A92,GOBRD13:24; hence left_cell(f|(k+1),m,G) misses C by A2,A10,A18,A21,A41,GOBRD13:32; front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2-'1) by A2,A39,A40,A43,A44,A45,A90,GOBRD13:41; then right_cell(f,m,G) meets C by A2,A10,A17,A21,A42,A45,A79,A91,A92,GOBRD13:25; hence right_cell(f|(k+1),m,G) meets C by A2,A10,A18,A21,A41,GOBRD13:32; end; hence thesis; end; hence thesis; suppose m+1 <> len(f|(k+1)); then m+1 < len(f|(k+1)) by A21,REAL_1:def 5; then A93: m+1 <= len(f|k)by A17,A18,NAT_1:38; then consider i1,j1,i2,j2 being Nat such that A94: [i1,j1] in Indices G & (f|k)/.m = G*(i1,j1) and A95: [i2,j2] in Indices G & (f|k)/.(m+1) = G*(i2,j2) and A96: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A11,A21,JORDAN8:6; A97: 1 <= m+1 by NAT_1:37; m <= len(f|k) by A93,NAT_1:38; then A98: m in dom(f|k) & m+1 in dom(f|k) by A21,A93,A97,FINSEQ_3:27; A99: left_cell(f|k,m,G) misses C & right_cell(f|k,m,G) meets C by A7,A21,A93; f|(k+1) = (f|k)^<*f/.(k+1)*> by A10,JORDAN8:2; then A100: (f|(k+1))/.m = G*(i1,j1) & (f|(k+1))/.(m+1) = G*(i2,j2) by A94,A95,A98,GROUP_5:95; now per cases by A96; suppose A101: i1 = i2 & j1+1 = j2; then left_cell(f|k,m,G) = cell(G,i1-'1,j1) & right_cell(f|k,m,G) = cell(G,i1,j1) by A11,A21,A93,A94,A95,GOBRD13:22,23; hence thesis by A12,A21,A94,A95,A99,A100,A101,GOBRD13:22,23; suppose A102: i1+1 = i2 & j1 = j2; then left_cell(f|k,m,G) = cell(G,i1,j1) & right_cell(f|k,m,G) = cell(G,i1,j1-'1) by A11,A21,A93,A94,A95,GOBRD13:24,25; hence thesis by A12,A21,A94,A95,A99,A100,A102,GOBRD13:24,25; suppose A103: i1 = i2+1 & j1 = j2; then left_cell(f|k,m,G) = cell(G,i2,j2-'1) & right_cell(f|k,m,G) = cell(G,i2,j2) by A11,A21,A93,A94,A95,GOBRD13:26,27; hence thesis by A12,A21,A94,A95,A99,A100,A103,GOBRD13:26,27; suppose A104: i1 = i2 & j1 = j2+1; then left_cell(f|k,m,G) = cell(G,i2,j2) & right_cell(f|k,m,G) = cell(G,i1-'1,j2) by A11,A21,A93,A94,A95,GOBRD13:28,29; hence thesis by A12,A21,A94,A95,A99,A100,A104,GOBRD13:28,29; end; hence thesis; end; hence thesis; end; hence thesis; end; A105: for k holds P[k] from Ind(A3,A6); f|len f = f by TOPREAL1:2; hence thesis by A105; end; theorem C is connected implies N-min L~Cage(C,n) = (Cage(C,n))/.1 proof set f = Cage(C,n); assume A1: C is connected; then A2: f is_sequence_on Gauge(C,n) & (ex i st 1 <= i & i+1 <= len Gauge(C,n) & f/.1 = Gauge(C,n)*(i,width Gauge(C,n)) & f/.2 = Gauge(C,n)*(i+1,width Gauge(C,n)) & N-min C in cell(Gauge(C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i,width Gauge(C,n)-'1)) by Def1; for k st 1 <= k & k+1 <= len f holds left_cell(f,k,Gauge(C,n)) misses C & right_cell(f,k,Gauge(C,n)) meets C by A1,Th33; hence thesis by A2,Th32; end;

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