:: GOBOARD7 semantic presentation begin theorem Th1: :: GOBOARD7:1 for r1, r2, s being Real holds ( not abs (r1 - r2) > s or r1 + s < r2 or r2 + s < r1 ) proof let r1, r2, s be Real; ::_thesis: ( not abs (r1 - r2) > s or r1 + s < r2 or r2 + s < r1 ) assume A1: abs (r1 - r2) > s ; ::_thesis: ( r1 + s < r2 or r2 + s < r1 ) now__::_thesis:_(_(_r1_<_r2_&_r1_+_s_<_r2_)_or_(_r2_<=_r1_&_r2_+_s_<_r1_)_) percases ( r1 < r2 or r2 <= r1 ) ; case r1 < r2 ; ::_thesis: r1 + s < r2 then r1 - r2 < 0 by XREAL_1:49; then abs (r1 - r2) = - (r1 - r2) by ABSVALUE:def_1 .= r2 - r1 ; hence r1 + s < r2 by A1, XREAL_1:20; ::_thesis: verum end; case r2 <= r1 ; ::_thesis: r2 + s < r1 then r1 - r2 >= 0 by XREAL_1:48; then abs (r1 - r2) = r1 - r2 by ABSVALUE:def_1; hence r2 + s < r1 by A1, XREAL_1:20; ::_thesis: verum end; end; end; hence ( r1 + s < r2 or r2 + s < r1 ) ; ::_thesis: verum end; theorem Th2: :: GOBOARD7:2 for r, s being Real holds ( abs (r - s) = 0 iff r = s ) proof let r, s be Real; ::_thesis: ( abs (r - s) = 0 iff r = s ) hereby ::_thesis: ( r = s implies abs (r - s) = 0 ) assume abs (r - s) = 0 ; ::_thesis: r = s then r - s = 0 by ABSVALUE:2; hence r = s ; ::_thesis: verum end; assume r = s ; ::_thesis: abs (r - s) = 0 hence abs (r - s) = 0 by ABSVALUE:2; ::_thesis: verum end; theorem Th3: :: GOBOARD7:3 for n being Element of NAT for p, p1, q being Point of (TOP-REAL n) st p + p1 = q + p1 holds p = q proof let n be Element of NAT ; ::_thesis: for p, p1, q being Point of (TOP-REAL n) st p + p1 = q + p1 holds p = q let p, p1, q be Point of (TOP-REAL n); ::_thesis: ( p + p1 = q + p1 implies p = q ) assume A1: p + p1 = q + p1 ; ::_thesis: p = q thus p = p + (0. (TOP-REAL n)) by EUCLID:27 .= p + (p1 - p1) by EUCLID:42 .= (p + p1) - p1 by EUCLID:45 .= q + (p1 - p1) by A1, EUCLID:45 .= q + (0. (TOP-REAL n)) by EUCLID:42 .= q by EUCLID:27 ; ::_thesis: verum end; theorem :: GOBOARD7:4 for n being Element of NAT for p, p1, q being Point of (TOP-REAL n) st p1 + p = p1 + q holds p = q by Th3; theorem Th5: :: GOBOARD7:5 for p1, p, q being Point of (TOP-REAL 2) st p1 in LSeg (p,q) & p `1 = q `1 holds p1 `1 = q `1 proof let p1, p, q be Point of (TOP-REAL 2); ::_thesis: ( p1 in LSeg (p,q) & p `1 = q `1 implies p1 `1 = q `1 ) assume p1 in LSeg (p,q) ; ::_thesis: ( not p `1 = q `1 or p1 `1 = q `1 ) then consider r being Real such that A1: p1 = ((1 - r) * p) + (r * q) and 0 <= r and r <= 1 ; assume A2: p `1 = q `1 ; ::_thesis: p1 `1 = q `1 p1 `1 = (((1 - r) * p) `1) + ((r * q) `1) by A1, TOPREAL3:2 .= (((1 - r) * p) `1) + (r * (q `1)) by TOPREAL3:4 .= ((1 - r) * (p `1)) + (r * (q `1)) by TOPREAL3:4 ; hence p1 `1 = q `1 by A2; ::_thesis: verum end; theorem Th6: :: GOBOARD7:6 for p1, p, q being Point of (TOP-REAL 2) st p1 in LSeg (p,q) & p `2 = q `2 holds p1 `2 = q `2 proof let p1, p, q be Point of (TOP-REAL 2); ::_thesis: ( p1 in LSeg (p,q) & p `2 = q `2 implies p1 `2 = q `2 ) assume p1 in LSeg (p,q) ; ::_thesis: ( not p `2 = q `2 or p1 `2 = q `2 ) then consider r being Real such that A1: p1 = ((1 - r) * p) + (r * q) and 0 <= r and r <= 1 ; assume A2: p `2 = q `2 ; ::_thesis: p1 `2 = q `2 p1 `2 = (((1 - r) * p) `2) + ((r * q) `2) by A1, TOPREAL3:2 .= (((1 - r) * p) `2) + (r * (q `2)) by TOPREAL3:4 .= ((1 - r) * (p `2)) + (r * (q `2)) by TOPREAL3:4 ; hence p1 `2 = q `2 by A2; ::_thesis: verum end; theorem Th7: :: GOBOARD7:7 for p, q, p1 being Point of (TOP-REAL 2) st p `1 = q `1 & q `1 = p1 `1 & p `2 <= q `2 & q `2 <= p1 `2 holds q in LSeg (p,p1) proof let p, q, p1 be Point of (TOP-REAL 2); ::_thesis: ( p `1 = q `1 & q `1 = p1 `1 & p `2 <= q `2 & q `2 <= p1 `2 implies q in LSeg (p,p1) ) assume that A1: p `1 = q `1 and A2: q `1 = p1 `1 and A3: ( p `2 <= q `2 & q `2 <= p1 `2 ) ; ::_thesis: q in LSeg (p,p1) A4: p `2 <= p1 `2 by A3, XXREAL_0:2; percases ( p `2 = p1 `2 or p `2 < p1 `2 ) by A4, XXREAL_0:1; supposeA5: p `2 = p1 `2 ; ::_thesis: q in LSeg (p,p1) then p `2 = q `2 by A3, XXREAL_0:1; then A6: q = |[(p `1),(p `2)]| by A1, EUCLID:53 .= p by EUCLID:53 ; p = |[(p1 `1),(p1 `2)]| by A1, A2, A5, EUCLID:53 .= p1 by EUCLID:53 ; then LSeg (p,p1) = {p} by RLTOPSP1:70; hence q in LSeg (p,p1) by A6, TARSKI:def_1; ::_thesis: verum end; supposeA7: p `2 < p1 `2 ; ::_thesis: q in LSeg (p,p1) A8: q in { q1 where q1 is Point of (TOP-REAL 2) : ( q1 `1 = q `1 & p `2 <= q1 `2 & q1 `2 <= p1 `2 ) } by A3; ( p = |[(q `1),(p `2)]| & p1 = |[(q `1),(p1 `2)]| ) by A1, A2, EUCLID:53; hence q in LSeg (p,p1) by A7, A8, TOPREAL3:9; ::_thesis: verum end; end; end; theorem Th8: :: GOBOARD7:8 for p, q, p1 being Point of (TOP-REAL 2) st p `1 <= q `1 & q `1 <= p1 `1 & p `2 = q `2 & q `2 = p1 `2 holds q in LSeg (p,p1) proof let p, q, p1 be Point of (TOP-REAL 2); ::_thesis: ( p `1 <= q `1 & q `1 <= p1 `1 & p `2 = q `2 & q `2 = p1 `2 implies q in LSeg (p,p1) ) assume that A1: ( p `1 <= q `1 & q `1 <= p1 `1 ) and A2: p `2 = q `2 and A3: q `2 = p1 `2 ; ::_thesis: q in LSeg (p,p1) A4: p `1 <= p1 `1 by A1, XXREAL_0:2; percases ( p `1 = p1 `1 or p `1 < p1 `1 ) by A4, XXREAL_0:1; supposeA5: p `1 = p1 `1 ; ::_thesis: q in LSeg (p,p1) then p `1 = q `1 by A1, XXREAL_0:1; then A6: q = |[(p `1),(p `2)]| by A2, EUCLID:53 .= p by EUCLID:53 ; p = |[(p1 `1),(p1 `2)]| by A2, A3, A5, EUCLID:53 .= p1 by EUCLID:53 ; then LSeg (p,p1) = {p} by RLTOPSP1:70; hence q in LSeg (p,p1) by A6, TARSKI:def_1; ::_thesis: verum end; supposeA7: p `1 < p1 `1 ; ::_thesis: q in LSeg (p,p1) A8: q in { q1 where q1 is Point of (TOP-REAL 2) : ( q1 `2 = q `2 & p `1 <= q1 `1 & q1 `1 <= p1 `1 ) } by A1; ( p = |[(p `1),(q `2)]| & p1 = |[(p1 `1),(q `2)]| ) by A2, A3, EUCLID:53; hence q in LSeg (p,p1) by A7, A8, TOPREAL3:10; ::_thesis: verum end; end; end; theorem :: GOBOARD7:9 for i, j being Element of NAT for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j))) proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j))) let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j))) ) assume that A1: 1 <= i and A2: i + 1 <= len G and A3: 1 <= j and A4: j + 1 <= width G ; ::_thesis: (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j))) A5: j < width G by A4, NAT_1:13; A6: 1 <= j + 1 by NAT_1:11; A7: 1 <= i + 1 by NAT_1:11; then A8: (G * ((i + 1),j)) `1 = (G * ((i + 1),1)) `1 by A2, A3, A5, GOBOARD5:2 .= (G * ((i + 1),(j + 1))) `1 by A2, A4, A7, A6, GOBOARD5:2 ; A9: i < len G by A2, NAT_1:13; then A10: (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A3, A5, GOBOARD5:2 .= (G * (i,(j + 1))) `1 by A1, A4, A9, A6, GOBOARD5:2 ; A11: (G * ((i + 1),(j + 1))) `2 = (G * (1,(j + 1))) `2 by A2, A4, A7, A6, GOBOARD5:1 .= (G * (i,(j + 1))) `2 by A1, A4, A9, A6, GOBOARD5:1 ; A12: (G * (i,j)) `2 = (G * (1,j)) `2 by A1, A3, A9, A5, GOBOARD5:1 .= (G * ((i + 1),j)) `2 by A2, A3, A7, A5, GOBOARD5:1 ; A13: ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `2 = (1 / 2) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2) by TOPREAL3:4 .= (1 / 2) * (((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2)) by TOPREAL3:2 .= (1 / 2) * (((G * (i,(j + 1))) + (G * ((i + 1),j))) `2) by A12, A11, TOPREAL3:2 .= ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `2 by TOPREAL3:4 ; ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `1 = (1 / 2) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1) by TOPREAL3:4 .= (1 / 2) * (((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1)) by TOPREAL3:2 .= (1 / 2) * (((G * (i,(j + 1))) + (G * ((i + 1),j))) `1) by A10, A8, TOPREAL3:2 .= ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `1 by TOPREAL3:4 ; hence (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = |[(((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `1),(((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `2)]| by A13, EUCLID:53 .= (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j))) by EUCLID:53 ; ::_thesis: verum end; theorem Th10: :: GOBOARD7:10 for f being non empty FinSequence of (TOP-REAL 2) for k being Element of NAT st LSeg (f,k) is horizontal holds ex j being Element of NAT st ( 1 <= j & j <= width (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,j)) `2 ) ) proof let f be non empty FinSequence of (TOP-REAL 2); ::_thesis: for k being Element of NAT st LSeg (f,k) is horizontal holds ex j being Element of NAT st ( 1 <= j & j <= width (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,j)) `2 ) ) let k be Element of NAT ; ::_thesis: ( LSeg (f,k) is horizontal implies ex j being Element of NAT st ( 1 <= j & j <= width (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,j)) `2 ) ) ) assume A1: LSeg (f,k) is horizontal ; ::_thesis: ex j being Element of NAT st ( 1 <= j & j <= width (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,j)) `2 ) ) percases ( ( 1 <= k & k + 1 <= len f ) or not 1 <= k or not k + 1 <= len f ) ; supposeA2: ( 1 <= k & k + 1 <= len f ) ; ::_thesis: ex j being Element of NAT st ( 1 <= j & j <= width (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,j)) `2 ) ) k <= k + 1 by NAT_1:11; then k <= len f by A2, XXREAL_0:2; then k in dom f by A2, FINSEQ_3:25; then consider i, j being Element of NAT such that A3: [i,j] in Indices (GoB f) and A4: f /. k = (GoB f) * (i,j) by GOBOARD2:14; take j ; ::_thesis: ( 1 <= j & j <= width (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,j)) `2 ) ) thus A5: ( 1 <= j & j <= width (GoB f) ) by A3, MATRIX_1:38; ::_thesis: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,j)) `2 A6: f /. k in LSeg (f,k) by A2, TOPREAL1:21; let p be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (f,k) implies p `2 = ((GoB f) * (1,j)) `2 ) A7: ( 1 <= i & i <= len (GoB f) ) by A3, MATRIX_1:38; assume p in LSeg (f,k) ; ::_thesis: p `2 = ((GoB f) * (1,j)) `2 hence p `2 = (f /. k) `2 by A1, A6, SPPOL_1:def_2 .= ((GoB f) * (1,j)) `2 by A4, A5, A7, GOBOARD5:1 ; ::_thesis: verum end; supposeA8: ( not 1 <= k or not k + 1 <= len f ) ; ::_thesis: ex j being Element of NAT st ( 1 <= j & j <= width (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,j)) `2 ) ) take 1 ; ::_thesis: ( 1 <= 1 & 1 <= width (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,1)) `2 ) ) width (GoB f) <> 0 by GOBOARD1:def_3; hence ( 1 <= 1 & 1 <= width (GoB f) ) by NAT_1:14; ::_thesis: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,1)) `2 thus for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,1)) `2 by A8, TOPREAL1:def_3; ::_thesis: verum end; end; end; theorem Th11: :: GOBOARD7:11 for f being non empty FinSequence of (TOP-REAL 2) for k being Element of NAT st LSeg (f,k) is vertical holds ex i being Element of NAT st ( 1 <= i & i <= len (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (i,1)) `1 ) ) proof let f be non empty FinSequence of (TOP-REAL 2); ::_thesis: for k being Element of NAT st LSeg (f,k) is vertical holds ex i being Element of NAT st ( 1 <= i & i <= len (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (i,1)) `1 ) ) let k be Element of NAT ; ::_thesis: ( LSeg (f,k) is vertical implies ex i being Element of NAT st ( 1 <= i & i <= len (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (i,1)) `1 ) ) ) assume A1: LSeg (f,k) is vertical ; ::_thesis: ex i being Element of NAT st ( 1 <= i & i <= len (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (i,1)) `1 ) ) percases ( ( 1 <= k & k + 1 <= len f ) or not 1 <= k or not k + 1 <= len f ) ; supposeA2: ( 1 <= k & k + 1 <= len f ) ; ::_thesis: ex i being Element of NAT st ( 1 <= i & i <= len (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (i,1)) `1 ) ) k <= k + 1 by NAT_1:11; then k <= len f by A2, XXREAL_0:2; then k in dom f by A2, FINSEQ_3:25; then consider i, j being Element of NAT such that A3: [i,j] in Indices (GoB f) and A4: f /. k = (GoB f) * (i,j) by GOBOARD2:14; take i ; ::_thesis: ( 1 <= i & i <= len (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (i,1)) `1 ) ) thus A5: ( 1 <= i & i <= len (GoB f) ) by A3, MATRIX_1:38; ::_thesis: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (i,1)) `1 A6: f /. k in LSeg (f,k) by A2, TOPREAL1:21; let p be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (f,k) implies p `1 = ((GoB f) * (i,1)) `1 ) A7: ( 1 <= j & j <= width (GoB f) ) by A3, MATRIX_1:38; assume p in LSeg (f,k) ; ::_thesis: p `1 = ((GoB f) * (i,1)) `1 hence p `1 = (f /. k) `1 by A1, A6, SPPOL_1:def_3 .= ((GoB f) * (i,1)) `1 by A4, A5, A7, GOBOARD5:2 ; ::_thesis: verum end; supposeA8: ( not 1 <= k or not k + 1 <= len f ) ; ::_thesis: ex i being Element of NAT st ( 1 <= i & i <= len (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (i,1)) `1 ) ) take 1 ; ::_thesis: ( 1 <= 1 & 1 <= len (GoB f) & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (1,1)) `1 ) ) 0 <> len (GoB f) by GOBOARD1:def_3; hence ( 1 <= 1 & 1 <= len (GoB f) ) by NAT_1:14; ::_thesis: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (1,1)) `1 thus for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (1,1)) `1 by A8, TOPREAL1:def_3; ::_thesis: verum end; end; end; theorem :: GOBOARD7:12 for f being non empty FinSequence of (TOP-REAL 2) for i, j being Element of NAT st f is special & i <= len (GoB f) & j <= width (GoB f) holds Int (cell ((GoB f),i,j)) misses L~ f proof let f be non empty FinSequence of (TOP-REAL 2); ::_thesis: for i, j being Element of NAT st f is special & i <= len (GoB f) & j <= width (GoB f) holds Int (cell ((GoB f),i,j)) misses L~ f let i, j be Element of NAT ; ::_thesis: ( f is special & i <= len (GoB f) & j <= width (GoB f) implies Int (cell ((GoB f),i,j)) misses L~ f ) assume that A1: f is special and A2: i <= len (GoB f) and A3: j <= width (GoB f) ; ::_thesis: Int (cell ((GoB f),i,j)) misses L~ f A4: Int (cell ((GoB f),i,j)) = Int ((v_strip ((GoB f),i)) /\ (h_strip ((GoB f),j))) by GOBOARD5:def_3 .= (Int (v_strip ((GoB f),i))) /\ (Int (h_strip ((GoB f),j))) by TOPS_1:17 ; assume Int (cell ((GoB f),i,j)) meets L~ f ; ::_thesis: contradiction then consider x being set such that A5: x in Int (cell ((GoB f),i,j)) and A6: x in L~ f by XBOOLE_0:3; L~ f = union { (LSeg (f,k)) where k is Element of NAT : ( 1 <= k & k + 1 <= len f ) } by TOPREAL1:def_4; then consider X being set such that A7: x in X and A8: X in { (LSeg (f,k)) where k is Element of NAT : ( 1 <= k & k + 1 <= len f ) } by A6, TARSKI:def_4; consider k being Element of NAT such that A9: X = LSeg (f,k) and 1 <= k and k + 1 <= len f by A8; reconsider p = x as Point of (TOP-REAL 2) by A7, A9; percases ( LSeg (f,k) is horizontal or LSeg (f,k) is vertical ) by A1, SPPOL_1:19; suppose LSeg (f,k) is horizontal ; ::_thesis: contradiction then consider j0 being Element of NAT such that A10: 1 <= j0 and A11: j0 <= width (GoB f) and A12: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `2 = ((GoB f) * (1,j0)) `2 by Th10; now__::_thesis:_not_p_in_Int_(h_strip_((GoB_f),j)) A13: ( j0 > j implies j0 >= j + 1 ) by NAT_1:13; assume A14: p in Int (h_strip ((GoB f),j)) ; ::_thesis: contradiction percases ( j0 < j or j0 = j or j0 > j + 1 or j0 = j + 1 ) by A13, XXREAL_0:1; supposeA15: j0 < j ; ::_thesis: contradiction 0 <> len (GoB f) by GOBOARD1:def_3; then 1 <= len (GoB f) by NAT_1:14; then A16: ((GoB f) * (1,j)) `2 > ((GoB f) * (1,j0)) `2 by A3, A10, A15, GOBOARD5:4; j >= 1 by A10, A15, XXREAL_0:2; then p `2 > ((GoB f) * (1,j)) `2 by A3, A14, GOBOARD6:27; hence contradiction by A7, A9, A12, A16; ::_thesis: verum end; suppose j0 = j ; ::_thesis: contradiction then p `2 > ((GoB f) * (1,j0)) `2 by A10, A11, A14, GOBOARD6:27; hence contradiction by A7, A9, A12; ::_thesis: verum end; supposeA17: j0 > j + 1 ; ::_thesis: contradiction then j + 1 <= width (GoB f) by A11, XXREAL_0:2; then j < width (GoB f) by NAT_1:13; then A18: p `2 < ((GoB f) * (1,(j + 1))) `2 by A14, GOBOARD6:28; 0 <> len (GoB f) by GOBOARD1:def_3; then A19: 1 <= len (GoB f) by NAT_1:14; j + 1 >= 1 by NAT_1:11; then ((GoB f) * (1,(j + 1))) `2 < ((GoB f) * (1,j0)) `2 by A11, A17, A19, GOBOARD5:4; hence contradiction by A7, A9, A12, A18; ::_thesis: verum end; supposeA20: j0 = j + 1 ; ::_thesis: contradiction then j < width (GoB f) by A11, NAT_1:13; then p `2 < ((GoB f) * (1,j0)) `2 by A14, A20, GOBOARD6:28; hence contradiction by A7, A9, A12; ::_thesis: verum end; end; end; hence contradiction by A5, A4, XBOOLE_0:def_4; ::_thesis: verum end; suppose LSeg (f,k) is vertical ; ::_thesis: contradiction then consider i0 being Element of NAT such that A21: 1 <= i0 and A22: i0 <= len (GoB f) and A23: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds p `1 = ((GoB f) * (i0,1)) `1 by Th11; now__::_thesis:_not_p_in_Int_(v_strip_((GoB_f),i)) A24: ( i0 > i implies i0 >= i + 1 ) by NAT_1:13; assume A25: p in Int (v_strip ((GoB f),i)) ; ::_thesis: contradiction percases ( i0 < i or i0 = i or i0 > i + 1 or i0 = i + 1 ) by A24, XXREAL_0:1; supposeA26: i0 < i ; ::_thesis: contradiction 0 <> width (GoB f) by GOBOARD1:def_3; then 1 <= width (GoB f) by NAT_1:14; then A27: ((GoB f) * (i,1)) `1 > ((GoB f) * (i0,1)) `1 by A2, A21, A26, GOBOARD5:3; i >= 1 by A21, A26, XXREAL_0:2; then p `1 > ((GoB f) * (i,1)) `1 by A2, A25, GOBOARD6:29; hence contradiction by A7, A9, A23, A27; ::_thesis: verum end; suppose i0 = i ; ::_thesis: contradiction then p `1 > ((GoB f) * (i0,1)) `1 by A21, A22, A25, GOBOARD6:29; hence contradiction by A7, A9, A23; ::_thesis: verum end; supposeA28: i0 > i + 1 ; ::_thesis: contradiction then i + 1 <= len (GoB f) by A22, XXREAL_0:2; then i < len (GoB f) by NAT_1:13; then A29: p `1 < ((GoB f) * ((i + 1),1)) `1 by A25, GOBOARD6:30; 0 <> width (GoB f) by GOBOARD1:def_3; then A30: 1 <= width (GoB f) by NAT_1:14; i + 1 >= 1 by NAT_1:11; then ((GoB f) * ((i + 1),1)) `1 < ((GoB f) * (i0,1)) `1 by A22, A28, A30, GOBOARD5:3; hence contradiction by A7, A9, A23, A29; ::_thesis: verum end; supposeA31: i0 = i + 1 ; ::_thesis: contradiction then i < len (GoB f) by A22, NAT_1:13; then p `1 < ((GoB f) * (i0,1)) `1 by A25, A31, GOBOARD6:30; hence contradiction by A7, A9, A23; ::_thesis: verum end; end; end; hence contradiction by A5, A4, XBOOLE_0:def_4; ::_thesis: verum end; end; end; begin theorem Th13: :: GOBOARD7:13 for i, j being Element of NAT for G being Go-board st 1 <= i & i <= len G & 1 <= j & j + 2 <= width G holds (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i <= len G & 1 <= j & j + 2 <= width G holds (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))} let G be Go-board; ::_thesis: ( 1 <= i & i <= len G & 1 <= j & j + 2 <= width G implies (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))} ) assume that A1: ( 1 <= i & i <= len G ) and A2: 1 <= j and A3: j + 2 <= width G ; ::_thesis: (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))} now__::_thesis:_for_x_being_set_holds_ (_(_x_in_(LSeg_((G_*_(i,j)),(G_*_(i,(j_+_1)))))_/\_(LSeg_((G_*_(i,(j_+_1))),(G_*_(i,(j_+_2)))))_implies_x_=_G_*_(i,(j_+_1))_)_&_(_x_=_G_*_(i,(j_+_1))_implies_x_in_(LSeg_((G_*_(i,j)),(G_*_(i,(j_+_1)))))_/\_(LSeg_((G_*_(i,(j_+_1))),(G_*_(i,(j_+_2)))))_)_) let x be set ; ::_thesis: ( ( x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) implies x = G * (i,(j + 1)) ) & ( x = G * (i,(j + 1)) implies x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) ) ) hereby ::_thesis: ( x = G * (i,(j + 1)) implies x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) ) assume A4: x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) ; ::_thesis: x = G * (i,(j + 1)) then reconsider p = x as Point of (TOP-REAL 2) ; A5: x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) by A4, XBOOLE_0:def_4; A6: p in LSeg ((G * (i,(j + 1))),(G * (i,(j + 2)))) by A4, XBOOLE_0:def_4; j <= j + 2 by NAT_1:11; then A7: j <= width G by A3, XXREAL_0:2; A8: j + 1 < j + 2 by XREAL_1:6; then A9: j + 1 <= width G by A3, XXREAL_0:2; A10: 1 <= j + 1 by NAT_1:11; then (G * (i,(j + 1))) `1 = (G * (i,1)) `1 by A1, A9, GOBOARD5:2 .= (G * (i,j)) `1 by A1, A2, A7, GOBOARD5:2 ; then A11: p `1 = (G * (i,(j + 1))) `1 by A5, Th5; j < j + 1 by XREAL_1:29; then (G * (i,j)) `2 < (G * (i,(j + 1))) `2 by A1, A2, A9, GOBOARD5:4; then A12: p `2 <= (G * (i,(j + 1))) `2 by A5, TOPREAL1:4; (G * (i,(j + 1))) `2 < (G * (i,(j + 2))) `2 by A1, A3, A8, A10, GOBOARD5:4; then p `2 >= (G * (i,(j + 1))) `2 by A6, TOPREAL1:4; then p `2 = (G * (i,(j + 1))) `2 by A12, XXREAL_0:1; hence x = G * (i,(j + 1)) by A11, TOPREAL3:6; ::_thesis: verum end; assume x = G * (i,(j + 1)) ; ::_thesis: x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) then ( x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) & x in LSeg ((G * (i,(j + 1))),(G * (i,(j + 2)))) ) by RLTOPSP1:68; hence x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) by XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))} by TARSKI:def_1; ::_thesis: verum end; theorem Th14: :: GOBOARD7:14 for i, j being Element of NAT for G being Go-board st 1 <= i & i + 2 <= len G & 1 <= j & j <= width G holds (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) = {(G * ((i + 1),j))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 2 <= len G & 1 <= j & j <= width G holds (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) = {(G * ((i + 1),j))} let G be Go-board; ::_thesis: ( 1 <= i & i + 2 <= len G & 1 <= j & j <= width G implies (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) = {(G * ((i + 1),j))} ) assume that A1: 1 <= i and A2: i + 2 <= len G and A3: ( 1 <= j & j <= width G ) ; ::_thesis: (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) = {(G * ((i + 1),j))} now__::_thesis:_for_x_being_set_holds_ (_(_x_in_(LSeg_((G_*_(i,j)),(G_*_((i_+_1),j))))_/\_(LSeg_((G_*_((i_+_1),j)),(G_*_((i_+_2),j))))_implies_x_=_G_*_((i_+_1),j)_)_&_(_x_=_G_*_((i_+_1),j)_implies_x_in_(LSeg_((G_*_(i,j)),(G_*_((i_+_1),j))))_/\_(LSeg_((G_*_((i_+_1),j)),(G_*_((i_+_2),j))))_)_) let x be set ; ::_thesis: ( ( x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) implies x = G * ((i + 1),j) ) & ( x = G * ((i + 1),j) implies x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) ) ) hereby ::_thesis: ( x = G * ((i + 1),j) implies x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) ) assume A4: x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) ; ::_thesis: x = G * ((i + 1),j) then reconsider p = x as Point of (TOP-REAL 2) ; A5: x in LSeg ((G * (i,j)),(G * ((i + 1),j))) by A4, XBOOLE_0:def_4; A6: p in LSeg ((G * ((i + 1),j)),(G * ((i + 2),j))) by A4, XBOOLE_0:def_4; i <= i + 2 by NAT_1:11; then A7: i <= len G by A2, XXREAL_0:2; A8: i + 1 < i + 2 by XREAL_1:6; then A9: i + 1 <= len G by A2, XXREAL_0:2; A10: 1 <= i + 1 by NAT_1:11; then (G * ((i + 1),j)) `2 = (G * (1,j)) `2 by A3, A9, GOBOARD5:1 .= (G * (i,j)) `2 by A1, A3, A7, GOBOARD5:1 ; then A11: p `2 = (G * ((i + 1),j)) `2 by A5, Th6; i < i + 1 by XREAL_1:29; then (G * (i,j)) `1 < (G * ((i + 1),j)) `1 by A1, A3, A9, GOBOARD5:3; then A12: p `1 <= (G * ((i + 1),j)) `1 by A5, TOPREAL1:3; (G * ((i + 1),j)) `1 < (G * ((i + 2),j)) `1 by A2, A3, A8, A10, GOBOARD5:3; then p `1 >= (G * ((i + 1),j)) `1 by A6, TOPREAL1:3; then p `1 = (G * ((i + 1),j)) `1 by A12, XXREAL_0:1; hence x = G * ((i + 1),j) by A11, TOPREAL3:6; ::_thesis: verum end; assume x = G * ((i + 1),j) ; ::_thesis: x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) then ( x in LSeg ((G * (i,j)),(G * ((i + 1),j))) & x in LSeg ((G * ((i + 1),j)),(G * ((i + 2),j))) ) by RLTOPSP1:68; hence x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) by XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) = {(G * ((i + 1),j))} by TARSKI:def_1; ::_thesis: verum end; theorem Th15: :: GOBOARD7:15 for i, j being Element of NAT for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) = {(G * (i,(j + 1)))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) = {(G * (i,(j + 1)))} let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) = {(G * (i,(j + 1)))} ) assume that A1: 1 <= i and A2: i + 1 <= len G and A3: 1 <= j and A4: j + 1 <= width G ; ::_thesis: (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) = {(G * (i,(j + 1)))} now__::_thesis:_for_x_being_set_holds_ (_(_x_in_(LSeg_((G_*_(i,j)),(G_*_(i,(j_+_1)))))_/\_(LSeg_((G_*_(i,(j_+_1))),(G_*_((i_+_1),(j_+_1)))))_implies_x_=_G_*_(i,(j_+_1))_)_&_(_x_=_G_*_(i,(j_+_1))_implies_x_in_(LSeg_((G_*_(i,j)),(G_*_(i,(j_+_1)))))_/\_(LSeg_((G_*_(i,(j_+_1))),(G_*_((i_+_1),(j_+_1)))))_)_) let x be set ; ::_thesis: ( ( x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) implies x = G * (i,(j + 1)) ) & ( x = G * (i,(j + 1)) implies x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) ) ) hereby ::_thesis: ( x = G * (i,(j + 1)) implies x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) ) assume A5: x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) ; ::_thesis: x = G * (i,(j + 1)) then reconsider p = x as Point of (TOP-REAL 2) ; A6: x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) by A5, XBOOLE_0:def_4; A7: p in LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1)))) by A5, XBOOLE_0:def_4; A8: 1 <= i + 1 by NAT_1:11; i <= i + 1 by NAT_1:11; then A9: i <= len G by A2, XXREAL_0:2; A10: 1 <= j + 1 by NAT_1:11; then (G * (i,(j + 1))) `2 = (G * (1,(j + 1))) `2 by A1, A4, A9, GOBOARD5:1 .= (G * ((i + 1),(j + 1))) `2 by A2, A4, A10, A8, GOBOARD5:1 ; then A11: p `2 = (G * (i,(j + 1))) `2 by A7, Th6; j < j + 1 by XREAL_1:29; then j <= width G by A4, XXREAL_0:2; then (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A3, A9, GOBOARD5:2 .= (G * (i,(j + 1))) `1 by A1, A4, A9, A10, GOBOARD5:2 ; then p `1 = (G * (i,(j + 1))) `1 by A6, Th5; hence x = G * (i,(j + 1)) by A11, TOPREAL3:6; ::_thesis: verum end; assume x = G * (i,(j + 1)) ; ::_thesis: x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) then ( x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) & x in LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1)))) ) by RLTOPSP1:68; hence x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) by XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) = {(G * (i,(j + 1)))} by TARSKI:def_1; ::_thesis: verum end; theorem Th16: :: GOBOARD7:16 for i, j being Element of NAT for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) = {(G * ((i + 1),(j + 1)))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) = {(G * ((i + 1),(j + 1)))} let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) = {(G * ((i + 1),(j + 1)))} ) assume that A1: 1 <= i and A2: i + 1 <= len G and A3: 1 <= j and A4: j + 1 <= width G ; ::_thesis: (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) = {(G * ((i + 1),(j + 1)))} now__::_thesis:_for_x_being_set_holds_ (_(_x_in_(LSeg_((G_*_(i,(j_+_1))),(G_*_((i_+_1),(j_+_1)))))_/\_(LSeg_((G_*_((i_+_1),j)),(G_*_((i_+_1),(j_+_1)))))_implies_x_=_G_*_((i_+_1),(j_+_1))_)_&_(_x_=_G_*_((i_+_1),(j_+_1))_implies_x_in_(LSeg_((G_*_(i,(j_+_1))),(G_*_((i_+_1),(j_+_1)))))_/\_(LSeg_((G_*_((i_+_1),j)),(G_*_((i_+_1),(j_+_1)))))_)_) let x be set ; ::_thesis: ( ( x in (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) implies x = G * ((i + 1),(j + 1)) ) & ( x = G * ((i + 1),(j + 1)) implies x in (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) ) ) hereby ::_thesis: ( x = G * ((i + 1),(j + 1)) implies x in (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) ) assume A5: x in (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) ; ::_thesis: x = G * ((i + 1),(j + 1)) then reconsider p = x as Point of (TOP-REAL 2) ; A6: x in LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1)))) by A5, XBOOLE_0:def_4; A7: p in LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1)))) by A5, XBOOLE_0:def_4; A8: 1 <= i + 1 by NAT_1:11; A9: 1 <= i + 1 by NAT_1:11; A10: 1 <= j + 1 by NAT_1:11; j < j + 1 by XREAL_1:29; then j <= width G by A4, XXREAL_0:2; then (G * ((i + 1),j)) `1 = (G * ((i + 1),1)) `1 by A2, A3, A8, GOBOARD5:2 .= (G * ((i + 1),(j + 1))) `1 by A2, A4, A10, A9, GOBOARD5:2 ; then A11: p `1 = (G * ((i + 1),(j + 1))) `1 by A7, Th5; i <= i + 1 by NAT_1:11; then i <= len G by A2, XXREAL_0:2; then (G * (i,(j + 1))) `2 = (G * (1,(j + 1))) `2 by A1, A4, A10, GOBOARD5:1 .= (G * ((i + 1),(j + 1))) `2 by A2, A4, A10, A8, GOBOARD5:1 ; then p `2 = (G * ((i + 1),(j + 1))) `2 by A6, Th6; hence x = G * ((i + 1),(j + 1)) by A11, TOPREAL3:6; ::_thesis: verum end; assume x = G * ((i + 1),(j + 1)) ; ::_thesis: x in (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) then ( x in LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1)))) & x in LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1)))) ) by RLTOPSP1:68; hence x in (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) by XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) = {(G * ((i + 1),(j + 1)))} by TARSKI:def_1; ::_thesis: verum end; theorem Th17: :: GOBOARD7:17 for i, j being Element of NAT for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))} let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))} ) assume that A1: 1 <= i and A2: i + 1 <= len G and A3: 1 <= j and A4: j + 1 <= width G ; ::_thesis: (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))} now__::_thesis:_for_x_being_set_holds_ (_(_x_in_(LSeg_((G_*_(i,j)),(G_*_((i_+_1),j))))_/\_(LSeg_((G_*_(i,j)),(G_*_(i,(j_+_1)))))_implies_x_=_G_*_(i,j)_)_&_(_x_=_G_*_(i,j)_implies_x_in_(LSeg_((G_*_(i,j)),(G_*_((i_+_1),j))))_/\_(LSeg_((G_*_(i,j)),(G_*_(i,(j_+_1)))))_)_) let x be set ; ::_thesis: ( ( x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) implies x = G * (i,j) ) & ( x = G * (i,j) implies x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) ) ) hereby ::_thesis: ( x = G * (i,j) implies x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) ) assume A5: x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) ; ::_thesis: x = G * (i,j) then reconsider p = x as Point of (TOP-REAL 2) ; A6: x in LSeg ((G * (i,j)),(G * ((i + 1),j))) by A5, XBOOLE_0:def_4; A7: p in LSeg ((G * (i,j)),(G * (i,(j + 1)))) by A5, XBOOLE_0:def_4; A8: 1 <= i + 1 by NAT_1:11; A9: 1 <= j + 1 by NAT_1:11; j < j + 1 by XREAL_1:29; then A10: j <= width G by A4, XXREAL_0:2; i <= i + 1 by NAT_1:11; then A11: i <= len G by A2, XXREAL_0:2; then (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A3, A10, GOBOARD5:2 .= (G * (i,(j + 1))) `1 by A1, A4, A11, A9, GOBOARD5:2 ; then A12: p `1 = (G * (i,j)) `1 by A7, Th5; (G * (i,j)) `2 = (G * (1,j)) `2 by A1, A3, A11, A10, GOBOARD5:1 .= (G * ((i + 1),j)) `2 by A2, A3, A8, A10, GOBOARD5:1 ; then p `2 = (G * (i,j)) `2 by A6, Th6; hence x = G * (i,j) by A12, TOPREAL3:6; ::_thesis: verum end; assume x = G * (i,j) ; ::_thesis: x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) then ( x in LSeg ((G * (i,j)),(G * ((i + 1),j))) & x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) ) by RLTOPSP1:68; hence x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) by XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))} by TARSKI:def_1; ::_thesis: verum end; theorem Th18: :: GOBOARD7:18 for i, j being Element of NAT for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) = {(G * ((i + 1),j))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) = {(G * ((i + 1),j))} let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) = {(G * ((i + 1),j))} ) assume that A1: 1 <= i and A2: i + 1 <= len G and A3: 1 <= j and A4: j + 1 <= width G ; ::_thesis: (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) = {(G * ((i + 1),j))} now__::_thesis:_for_x_being_set_holds_ (_(_x_in_(LSeg_((G_*_(i,j)),(G_*_((i_+_1),j))))_/\_(LSeg_((G_*_((i_+_1),j)),(G_*_((i_+_1),(j_+_1)))))_implies_x_=_G_*_((i_+_1),j)_)_&_(_x_=_G_*_((i_+_1),j)_implies_x_in_(LSeg_((G_*_(i,j)),(G_*_((i_+_1),j))))_/\_(LSeg_((G_*_((i_+_1),j)),(G_*_((i_+_1),(j_+_1)))))_)_) let x be set ; ::_thesis: ( ( x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) implies x = G * ((i + 1),j) ) & ( x = G * ((i + 1),j) implies x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) ) ) hereby ::_thesis: ( x = G * ((i + 1),j) implies x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) ) assume A5: x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) ; ::_thesis: x = G * ((i + 1),j) then reconsider p = x as Point of (TOP-REAL 2) ; A6: x in LSeg ((G * (i,j)),(G * ((i + 1),j))) by A5, XBOOLE_0:def_4; A7: p in LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1)))) by A5, XBOOLE_0:def_4; A8: ( 1 <= j + 1 & 1 <= i + 1 ) by NAT_1:11; j < j + 1 by XREAL_1:29; then A9: j <= width G by A4, XXREAL_0:2; A10: 1 <= i + 1 by NAT_1:11; then (G * ((i + 1),j)) `1 = (G * ((i + 1),1)) `1 by A2, A3, A9, GOBOARD5:2 .= (G * ((i + 1),(j + 1))) `1 by A2, A4, A8, GOBOARD5:2 ; then A11: p `1 = (G * ((i + 1),j)) `1 by A7, Th5; i <= i + 1 by NAT_1:11; then i <= len G by A2, XXREAL_0:2; then (G * (i,j)) `2 = (G * (1,j)) `2 by A1, A3, A9, GOBOARD5:1 .= (G * ((i + 1),j)) `2 by A2, A3, A10, A9, GOBOARD5:1 ; then p `2 = (G * ((i + 1),j)) `2 by A6, Th6; hence x = G * ((i + 1),j) by A11, TOPREAL3:6; ::_thesis: verum end; assume x = G * ((i + 1),j) ; ::_thesis: x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) then ( x in LSeg ((G * (i,j)),(G * ((i + 1),j))) & x in LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1)))) ) by RLTOPSP1:68; hence x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) by XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) = {(G * ((i + 1),j))} by TARSKI:def_1; ::_thesis: verum end; theorem Th19: :: GOBOARD7:19 for G being Go-board for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) holds ( i1 = i2 & abs (j1 - j2) <= 1 ) proof let G be Go-board; ::_thesis: for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) holds ( i1 = i2 & abs (j1 - j2) <= 1 ) let i1, j1, i2, j2 be Element of NAT ; ::_thesis: ( 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) implies ( i1 = i2 & abs (j1 - j2) <= 1 ) ) assume that A1: ( 1 <= i1 & i1 <= len G ) and A2: 1 <= j1 and A3: j1 + 1 <= width G and A4: ( 1 <= i2 & i2 <= len G ) and A5: 1 <= j2 and A6: j2 + 1 <= width G ; ::_thesis: ( not LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) or ( i1 = i2 & abs (j1 - j2) <= 1 ) ) A7: 1 <= j1 + 1 by A2, NAT_1:13; A8: j1 < width G by A3, NAT_1:13; assume LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ; ::_thesis: ( i1 = i2 & abs (j1 - j2) <= 1 ) then consider x being set such that A9: x in LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) and A10: x in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) by XBOOLE_0:3; reconsider p = x as Point of (TOP-REAL 2) by A9; consider r1 being Real such that A11: p = ((1 - r1) * (G * (i1,j1))) + (r1 * (G * (i1,(j1 + 1)))) and A12: r1 >= 0 and A13: r1 <= 1 by A9; consider r2 being Real such that A14: p = ((1 - r2) * (G * (i2,j2))) + (r2 * (G * (i2,(j2 + 1)))) and A15: r2 >= 0 and A16: r2 <= 1 by A10; A17: 1 <= j2 + 1 by A5, NAT_1:13; A18: j2 < width G by A6, NAT_1:13; assume A19: ( not i1 = i2 or not abs (j1 - j2) <= 1 ) ; ::_thesis: contradiction percases ( i1 <> i2 or abs (j1 - j2) > 1 ) by A19; suppose i1 <> i2 ; ::_thesis: contradiction then A20: ( i1 < i2 or i2 < i1 ) by XXREAL_0:1; A21: (G * (i2,j2)) `1 = (G * (i2,1)) `1 by A4, A5, A18, GOBOARD5:2 .= (G * (i2,(j2 + 1))) `1 by A4, A6, A17, GOBOARD5:2 ; (G * (i1,j1)) `1 = (G * (i1,1)) `1 by A1, A2, A8, GOBOARD5:2 .= (G * (i1,(j1 + 1))) `1 by A1, A3, A7, GOBOARD5:2 ; then 1 * ((G * (i1,j1)) `1) = ((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * (i1,(j1 + 1))) `1)) .= (((1 - r1) * (G * (i1,j1))) `1) + (r1 * ((G * (i1,(j1 + 1))) `1)) by TOPREAL3:4 .= (((1 - r1) * (G * (i1,j1))) `1) + ((r1 * (G * (i1,(j1 + 1)))) `1) by TOPREAL3:4 .= p `1 by A11, TOPREAL3:2 .= (((1 - r2) * (G * (i2,j2))) `1) + ((r2 * (G * (i2,(j2 + 1)))) `1) by A14, TOPREAL3:2 .= ((1 - r2) * ((G * (i2,j2)) `1)) + ((r2 * (G * (i2,(j2 + 1)))) `1) by TOPREAL3:4 .= ((1 - r2) * ((G * (i2,j2)) `1)) + (r2 * ((G * (i2,(j2 + 1))) `1)) by TOPREAL3:4 .= (G * (i2,1)) `1 by A4, A6, A17, A21, GOBOARD5:2 .= (G * (i2,j1)) `1 by A2, A4, A8, GOBOARD5:2 ; hence contradiction by A1, A2, A4, A8, A20, GOBOARD5:3; ::_thesis: verum end; supposeA22: abs (j1 - j2) > 1 ; ::_thesis: contradiction A23: (G * (i2,(j2 + 1))) `2 = (G * (1,(j2 + 1))) `2 by A4, A6, A17, GOBOARD5:1 .= (G * (i1,(j2 + 1))) `2 by A1, A6, A17, GOBOARD5:1 ; A24: (G * (i2,j2)) `2 = (G * (1,j2)) `2 by A4, A5, A18, GOBOARD5:1 .= (G * (i1,j2)) `2 by A1, A5, A18, GOBOARD5:1 ; A25: ((1 - r1) * ((G * (i1,j1)) `2)) + (r1 * ((G * (i1,(j1 + 1))) `2)) = (((1 - r1) * (G * (i1,j1))) `2) + (r1 * ((G * (i1,(j1 + 1))) `2)) by TOPREAL3:4 .= (((1 - r1) * (G * (i1,j1))) `2) + ((r1 * (G * (i1,(j1 + 1)))) `2) by TOPREAL3:4 .= p `2 by A11, TOPREAL3:2 .= (((1 - r2) * (G * (i2,j2))) `2) + ((r2 * (G * (i2,(j2 + 1)))) `2) by A14, TOPREAL3:2 .= ((1 - r2) * ((G * (i2,j2)) `2)) + ((r2 * (G * (i2,(j2 + 1)))) `2) by TOPREAL3:4 .= ((1 - r2) * ((G * (i1,j2)) `2)) + (r2 * ((G * (i1,(j2 + 1))) `2)) by A23, A24, TOPREAL3:4 ; now__::_thesis:_contradiction percases ( j1 + 1 < j2 or j2 + 1 < j1 ) by A22, Th1; supposeA26: j1 + 1 < j2 ; ::_thesis: contradiction j2 < j2 + 1 by XREAL_1:29; then (G * (i1,j2)) `2 < (G * (i1,(j2 + 1))) `2 by A1, A5, A6, GOBOARD5:4; then ( ((1 - r2) * ((G * (i1,j2)) `2)) + (r2 * ((G * (i1,j2)) `2)) = 1 * ((G * (i1,j2)) `2) & r2 * ((G * (i1,j2)) `2) <= r2 * ((G * (i1,(j2 + 1))) `2) ) by A15, XREAL_1:64; then A27: (G * (i1,j2)) `2 <= ((1 - r2) * ((G * (i1,j2)) `2)) + (r2 * ((G * (i1,(j2 + 1))) `2)) by XREAL_1:6; j1 < j1 + 1 by XREAL_1:29; then A28: (G * (i1,j1)) `2 <= (G * (i1,(j1 + 1))) `2 by A1, A2, A3, GOBOARD5:4; 1 - r1 >= 0 by A13, XREAL_1:48; then ( ((1 - r1) * ((G * (i1,(j1 + 1))) `2)) + (r1 * ((G * (i1,(j1 + 1))) `2)) = 1 * ((G * (i1,(j1 + 1))) `2) & (1 - r1) * ((G * (i1,j1)) `2) <= (1 - r1) * ((G * (i1,(j1 + 1))) `2) ) by A28, XREAL_1:64; then A29: ((1 - r1) * ((G * (i1,j1)) `2)) + (r1 * ((G * (i1,(j1 + 1))) `2)) <= (G * (i1,(j1 + 1))) `2 by XREAL_1:6; (G * (i1,(j1 + 1))) `2 < (G * (i1,j2)) `2 by A1, A7, A18, A26, GOBOARD5:4; hence contradiction by A25, A29, A27, XXREAL_0:2; ::_thesis: verum end; supposeA30: j2 + 1 < j1 ; ::_thesis: contradiction j1 < j1 + 1 by XREAL_1:29; then (G * (i1,j1)) `2 < (G * (i1,(j1 + 1))) `2 by A1, A2, A3, GOBOARD5:4; then ( ((1 - r1) * ((G * (i1,j1)) `2)) + (r1 * ((G * (i1,j1)) `2)) = 1 * ((G * (i1,j1)) `2) & r1 * ((G * (i1,j1)) `2) <= r1 * ((G * (i1,(j1 + 1))) `2) ) by A12, XREAL_1:64; then A31: (G * (i1,j1)) `2 <= ((1 - r1) * ((G * (i1,j1)) `2)) + (r1 * ((G * (i1,(j1 + 1))) `2)) by XREAL_1:6; j2 < j2 + 1 by XREAL_1:29; then A32: (G * (i1,j2)) `2 <= (G * (i1,(j2 + 1))) `2 by A1, A5, A6, GOBOARD5:4; 1 - r2 >= 0 by A16, XREAL_1:48; then ( ((1 - r2) * ((G * (i1,(j2 + 1))) `2)) + (r2 * ((G * (i1,(j2 + 1))) `2)) = 1 * ((G * (i1,(j2 + 1))) `2) & (1 - r2) * ((G * (i1,j2)) `2) <= (1 - r2) * ((G * (i1,(j2 + 1))) `2) ) by A32, XREAL_1:64; then A33: ((1 - r2) * ((G * (i1,j2)) `2)) + (r2 * ((G * (i1,(j2 + 1))) `2)) <= (G * (i1,(j2 + 1))) `2 by XREAL_1:6; (G * (i1,(j2 + 1))) `2 < (G * (i1,j1)) `2 by A1, A8, A17, A30, GOBOARD5:4; hence contradiction by A25, A33, A31, XXREAL_0:2; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; theorem Th20: :: GOBOARD7:20 for G being Go-board for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) holds ( j1 = j2 & abs (i1 - i2) <= 1 ) proof let G be Go-board; ::_thesis: for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) holds ( j1 = j2 & abs (i1 - i2) <= 1 ) let i1, j1, i2, j2 be Element of NAT ; ::_thesis: ( 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) implies ( j1 = j2 & abs (i1 - i2) <= 1 ) ) assume that A1: 1 <= i1 and A2: i1 + 1 <= len G and A3: ( 1 <= j1 & j1 <= width G ) and A4: 1 <= i2 and A5: i2 + 1 <= len G and A6: ( 1 <= j2 & j2 <= width G ) ; ::_thesis: ( not LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) or ( j1 = j2 & abs (i1 - i2) <= 1 ) ) A7: 1 <= i1 + 1 by A1, NAT_1:13; A8: i1 < len G by A2, NAT_1:13; assume LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ; ::_thesis: ( j1 = j2 & abs (i1 - i2) <= 1 ) then consider x being set such that A9: x in LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) and A10: x in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) by XBOOLE_0:3; reconsider p = x as Point of (TOP-REAL 2) by A9; consider r1 being Real such that A11: p = ((1 - r1) * (G * (i1,j1))) + (r1 * (G * ((i1 + 1),j1))) and A12: r1 >= 0 and A13: r1 <= 1 by A9; consider r2 being Real such that A14: p = ((1 - r2) * (G * (i2,j2))) + (r2 * (G * ((i2 + 1),j2))) and A15: r2 >= 0 and A16: r2 <= 1 by A10; A17: 1 <= i2 + 1 by A4, NAT_1:13; A18: i2 < len G by A5, NAT_1:13; assume A19: ( not j1 = j2 or not abs (i1 - i2) <= 1 ) ; ::_thesis: contradiction percases ( j1 <> j2 or abs (i1 - i2) > 1 ) by A19; suppose j1 <> j2 ; ::_thesis: contradiction then A20: ( j1 < j2 or j2 < j1 ) by XXREAL_0:1; A21: (G * (i2,j2)) `2 = (G * (1,j2)) `2 by A4, A6, A18, GOBOARD5:1 .= (G * ((i2 + 1),j2)) `2 by A5, A6, A17, GOBOARD5:1 ; (G * (i1,j1)) `2 = (G * (1,j1)) `2 by A1, A3, A8, GOBOARD5:1 .= (G * ((i1 + 1),j1)) `2 by A2, A3, A7, GOBOARD5:1 ; then 1 * ((G * (i1,j1)) `2) = ((1 - r1) * ((G * (i1,j1)) `2)) + (r1 * ((G * ((i1 + 1),j1)) `2)) .= (((1 - r1) * (G * (i1,j1))) `2) + (r1 * ((G * ((i1 + 1),j1)) `2)) by TOPREAL3:4 .= (((1 - r1) * (G * (i1,j1))) `2) + ((r1 * (G * ((i1 + 1),j1))) `2) by TOPREAL3:4 .= p `2 by A11, TOPREAL3:2 .= (((1 - r2) * (G * (i2,j2))) `2) + ((r2 * (G * ((i2 + 1),j2))) `2) by A14, TOPREAL3:2 .= ((1 - r2) * ((G * (i2,j2)) `2)) + ((r2 * (G * ((i2 + 1),j2))) `2) by TOPREAL3:4 .= ((1 - r2) * ((G * (i2,j2)) `2)) + (r2 * ((G * ((i2 + 1),j2)) `2)) by TOPREAL3:4 .= (G * (1,j2)) `2 by A5, A6, A17, A21, GOBOARD5:1 .= (G * (i1,j2)) `2 by A1, A6, A8, GOBOARD5:1 ; hence contradiction by A1, A3, A6, A8, A20, GOBOARD5:4; ::_thesis: verum end; supposeA22: abs (i1 - i2) > 1 ; ::_thesis: contradiction A23: (G * ((i2 + 1),j2)) `1 = (G * ((i2 + 1),1)) `1 by A5, A6, A17, GOBOARD5:2 .= (G * ((i2 + 1),j1)) `1 by A3, A5, A17, GOBOARD5:2 ; A24: (G * (i2,j2)) `1 = (G * (i2,1)) `1 by A4, A6, A18, GOBOARD5:2 .= (G * (i2,j1)) `1 by A3, A4, A18, GOBOARD5:2 ; A25: ((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * ((i1 + 1),j1)) `1)) = (((1 - r1) * (G * (i1,j1))) `1) + (r1 * ((G * ((i1 + 1),j1)) `1)) by TOPREAL3:4 .= (((1 - r1) * (G * (i1,j1))) `1) + ((r1 * (G * ((i1 + 1),j1))) `1) by TOPREAL3:4 .= p `1 by A11, TOPREAL3:2 .= (((1 - r2) * (G * (i2,j2))) `1) + ((r2 * (G * ((i2 + 1),j2))) `1) by A14, TOPREAL3:2 .= ((1 - r2) * ((G * (i2,j2)) `1)) + ((r2 * (G * ((i2 + 1),j2))) `1) by TOPREAL3:4 .= ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) by A23, A24, TOPREAL3:4 ; now__::_thesis:_contradiction percases ( i1 + 1 < i2 or i2 + 1 < i1 ) by A22, Th1; supposeA26: i1 + 1 < i2 ; ::_thesis: contradiction i2 < i2 + 1 by XREAL_1:29; then (G * (i2,j1)) `1 < (G * ((i2 + 1),j1)) `1 by A3, A4, A5, GOBOARD5:3; then ( ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * (i2,j1)) `1)) = 1 * ((G * (i2,j1)) `1) & r2 * ((G * (i2,j1)) `1) <= r2 * ((G * ((i2 + 1),j1)) `1) ) by A15, XREAL_1:64; then A27: (G * (i2,j1)) `1 <= ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) by XREAL_1:6; i1 < i1 + 1 by XREAL_1:29; then A28: (G * (i1,j1)) `1 <= (G * ((i1 + 1),j1)) `1 by A1, A2, A3, GOBOARD5:3; 1 - r1 >= 0 by A13, XREAL_1:48; then ( ((1 - r1) * ((G * ((i1 + 1),j1)) `1)) + (r1 * ((G * ((i1 + 1),j1)) `1)) = 1 * ((G * ((i1 + 1),j1)) `1) & (1 - r1) * ((G * (i1,j1)) `1) <= (1 - r1) * ((G * ((i1 + 1),j1)) `1) ) by A28, XREAL_1:64; then A29: ((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * ((i1 + 1),j1)) `1)) <= (G * ((i1 + 1),j1)) `1 by XREAL_1:6; (G * ((i1 + 1),j1)) `1 < (G * (i2,j1)) `1 by A3, A7, A18, A26, GOBOARD5:3; hence contradiction by A25, A29, A27, XXREAL_0:2; ::_thesis: verum end; supposeA30: i2 + 1 < i1 ; ::_thesis: contradiction i1 < i1 + 1 by XREAL_1:29; then (G * (i1,j1)) `1 < (G * ((i1 + 1),j1)) `1 by A1, A2, A3, GOBOARD5:3; then ( ((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * (i1,j1)) `1)) = 1 * ((G * (i1,j1)) `1) & r1 * ((G * (i1,j1)) `1) <= r1 * ((G * ((i1 + 1),j1)) `1) ) by A12, XREAL_1:64; then A31: (G * (i1,j1)) `1 <= ((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * ((i1 + 1),j1)) `1)) by XREAL_1:6; i2 < i2 + 1 by XREAL_1:29; then A32: (G * (i2,j1)) `1 <= (G * ((i2 + 1),j1)) `1 by A3, A4, A5, GOBOARD5:3; 1 - r2 >= 0 by A16, XREAL_1:48; then ( ((1 - r2) * ((G * ((i2 + 1),j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) = 1 * ((G * ((i2 + 1),j1)) `1) & (1 - r2) * ((G * (i2,j1)) `1) <= (1 - r2) * ((G * ((i2 + 1),j1)) `1) ) by A32, XREAL_1:64; then A33: ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) <= (G * ((i2 + 1),j1)) `1 by XREAL_1:6; (G * ((i2 + 1),j1)) `1 < (G * (i1,j1)) `1 by A3, A8, A17, A30, GOBOARD5:3; hence contradiction by A25, A33, A31, XXREAL_0:2; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; theorem Th21: :: GOBOARD7:21 for G being Go-board for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) holds ( ( i1 = i2 or i1 = i2 + 1 ) & ( j1 = j2 or j1 + 1 = j2 ) ) proof let G be Go-board; ::_thesis: for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) holds ( ( i1 = i2 or i1 = i2 + 1 ) & ( j1 = j2 or j1 + 1 = j2 ) ) let i1, j1, i2, j2 be Element of NAT ; ::_thesis: ( 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) implies ( ( i1 = i2 or i1 = i2 + 1 ) & ( j1 = j2 or j1 + 1 = j2 ) ) ) assume that A1: 1 <= i1 and A2: i1 <= len G and A3: 1 <= j1 and A4: j1 + 1 <= width G and A5: 1 <= i2 and A6: i2 + 1 <= len G and A7: 1 <= j2 and A8: j2 <= width G ; ::_thesis: ( not LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) or ( ( i1 = i2 or i1 = i2 + 1 ) & ( j1 = j2 or j1 + 1 = j2 ) ) ) assume LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ; ::_thesis: ( ( i1 = i2 or i1 = i2 + 1 ) & ( j1 = j2 or j1 + 1 = j2 ) ) then consider x being set such that A9: x in LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) and A10: x in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) by XBOOLE_0:3; reconsider p = x as Point of (TOP-REAL 2) by A9; consider r1 being Real such that A11: p = ((1 - r1) * (G * (i1,j1))) + (r1 * (G * (i1,(j1 + 1)))) and A12: r1 >= 0 and A13: r1 <= 1 by A9; consider r2 being Real such that A14: p = ((1 - r2) * (G * (i2,j2))) + (r2 * (G * ((i2 + 1),j2))) and A15: r2 >= 0 and A16: r2 <= 1 by A10; A17: i2 < len G by A6, NAT_1:13; A18: 1 <= j1 + 1 by A3, NAT_1:13; then A19: (G * (i1,(j1 + 1))) `2 = (G * (1,(j1 + 1))) `2 by A1, A2, A4, GOBOARD5:1 .= (G * (i2,(j1 + 1))) `2 by A4, A5, A18, A17, GOBOARD5:1 ; A20: j1 < width G by A4, NAT_1:13; then A21: (G * (i1,j1)) `2 = (G * (1,j1)) `2 by A1, A2, A3, GOBOARD5:1 .= (G * (i2,j1)) `2 by A3, A5, A20, A17, GOBOARD5:1 ; A22: ((1 - r2) * ((G * (i2,j2)) `2)) + (r2 * ((G * ((i2 + 1),j2)) `2)) = (((1 - r2) * (G * (i2,j2))) `2) + (r2 * ((G * ((i2 + 1),j2)) `2)) by TOPREAL3:4 .= (((1 - r2) * (G * (i2,j2))) `2) + ((r2 * (G * ((i2 + 1),j2))) `2) by TOPREAL3:4 .= p `2 by A14, TOPREAL3:2 .= (((1 - r1) * (G * (i1,j1))) `2) + ((r1 * (G * (i1,(j1 + 1)))) `2) by A11, TOPREAL3:2 .= ((1 - r1) * ((G * (i1,j1)) `2)) + ((r1 * (G * (i1,(j1 + 1)))) `2) by TOPREAL3:4 .= ((1 - r1) * ((G * (i2,j1)) `2)) + (r1 * ((G * (i2,(j1 + 1))) `2)) by A19, A21, TOPREAL3:4 ; A23: 1 <= i2 + 1 by A5, NAT_1:13; thus ( i1 = i2 or i1 = i2 + 1 ) ::_thesis: ( j1 = j2 or j1 + 1 = j2 ) proof A24: (G * (i2,j2)) `1 = (G * (i2,1)) `1 by A5, A7, A8, A17, GOBOARD5:2 .= (G * (i2,j1)) `1 by A3, A5, A20, A17, GOBOARD5:2 ; A25: (G * ((i2 + 1),j2)) `1 = (G * ((i2 + 1),1)) `1 by A6, A7, A8, A23, GOBOARD5:2 .= (G * ((i2 + 1),j1)) `1 by A3, A6, A20, A23, GOBOARD5:2 ; A26: ((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * (i1,(j1 + 1))) `1)) = (((1 - r1) * (G * (i1,j1))) `1) + (r1 * ((G * (i1,(j1 + 1))) `1)) by TOPREAL3:4 .= (((1 - r1) * (G * (i1,j1))) `1) + ((r1 * (G * (i1,(j1 + 1)))) `1) by TOPREAL3:4 .= p `1 by A11, TOPREAL3:2 .= (((1 - r2) * (G * (i2,j2))) `1) + ((r2 * (G * ((i2 + 1),j2))) `1) by A14, TOPREAL3:2 .= ((1 - r2) * ((G * (i2,j2)) `1)) + ((r2 * (G * ((i2 + 1),j2))) `1) by TOPREAL3:4 .= ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) by A25, A24, TOPREAL3:4 ; A27: (G * (i1,j1)) `1 = (G * (i1,1)) `1 by A1, A2, A3, A20, GOBOARD5:2 .= (G * (i1,(j1 + 1))) `1 by A1, A2, A4, A18, GOBOARD5:2 ; assume A28: ( not i1 = i2 & not i1 = i2 + 1 ) ; ::_thesis: contradiction percases ( ( i1 < i2 & i1 < i2 + 1 ) or ( i1 < i2 & i2 + 1 < i1 ) or ( i2 < i1 & i1 < i2 + 1 ) or i2 + 1 < i1 ) by A28, XXREAL_0:1; supposeA29: ( i1 < i2 & i1 < i2 + 1 ) ; ::_thesis: contradiction i2 < i2 + 1 by XREAL_1:29; then (G * (i2,j1)) `1 < (G * ((i2 + 1),j1)) `1 by A3, A5, A6, A20, GOBOARD5:3; then A30: ( ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * (i2,j1)) `1)) = 1 * ((G * (i2,j1)) `1) & r2 * ((G * (i2,j1)) `1) <= r2 * ((G * ((i2 + 1),j1)) `1) ) by A15, XREAL_1:64; (G * (i1,j1)) `1 < (G * (i2,j1)) `1 by A1, A3, A20, A17, A29, GOBOARD5:3; hence contradiction by A26, A27, A30, XREAL_1:6; ::_thesis: verum end; suppose ( i1 < i2 & i2 + 1 < i1 ) ; ::_thesis: contradiction hence contradiction by NAT_1:13; ::_thesis: verum end; suppose ( i2 < i1 & i1 < i2 + 1 ) ; ::_thesis: contradiction hence contradiction by NAT_1:13; ::_thesis: verum end; supposeA31: i2 + 1 < i1 ; ::_thesis: contradiction i2 < i2 + 1 by XREAL_1:29; then A32: (G * (i2,j1)) `1 <= (G * ((i2 + 1),j1)) `1 by A3, A5, A6, A20, GOBOARD5:3; 1 - r2 >= 0 by A16, XREAL_1:48; then ( ((1 - r2) * ((G * ((i2 + 1),j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) = 1 * ((G * ((i2 + 1),j1)) `1) & (1 - r2) * ((G * (i2,j1)) `1) <= (1 - r2) * ((G * ((i2 + 1),j1)) `1) ) by A32, XREAL_1:64; then ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) <= (G * ((i2 + 1),j1)) `1 by XREAL_1:6; hence contradiction by A2, A3, A20, A23, A26, A27, A31, GOBOARD5:3; ::_thesis: verum end; end; end; A33: (G * (i2,j2)) `2 = (G * (1,j2)) `2 by A5, A7, A8, A17, GOBOARD5:1 .= (G * ((i2 + 1),j2)) `2 by A6, A7, A8, A23, GOBOARD5:1 ; assume A34: ( not j1 = j2 & not j1 + 1 = j2 ) ; ::_thesis: contradiction percases ( ( j2 < j1 & j2 < j1 + 1 ) or ( j2 < j1 & j1 + 1 < j2 ) or ( j1 < j2 & j2 < j1 + 1 ) or j1 + 1 < j2 ) by A34, XXREAL_0:1; supposeA35: ( j2 < j1 & j2 < j1 + 1 ) ; ::_thesis: contradiction j1 < j1 + 1 by XREAL_1:29; then (G * (i2,j1)) `2 < (G * (i2,(j1 + 1))) `2 by A3, A4, A5, A17, GOBOARD5:4; then A36: ( ((1 - r1) * ((G * (i2,j1)) `2)) + (r1 * ((G * (i2,j1)) `2)) = 1 * ((G * (i2,j1)) `2) & r1 * ((G * (i2,j1)) `2) <= r1 * ((G * (i2,(j1 + 1))) `2) ) by A12, XREAL_1:64; (G * (i2,j2)) `2 < (G * (i2,j1)) `2 by A5, A7, A20, A17, A35, GOBOARD5:4; hence contradiction by A22, A33, A36, XREAL_1:6; ::_thesis: verum end; suppose ( j2 < j1 & j1 + 1 < j2 ) ; ::_thesis: contradiction hence contradiction by NAT_1:13; ::_thesis: verum end; suppose ( j1 < j2 & j2 < j1 + 1 ) ; ::_thesis: contradiction hence contradiction by NAT_1:13; ::_thesis: verum end; supposeA37: j1 + 1 < j2 ; ::_thesis: contradiction j1 < j1 + 1 by XREAL_1:29; then A38: (G * (i2,j1)) `2 <= (G * (i2,(j1 + 1))) `2 by A3, A4, A5, A17, GOBOARD5:4; 1 - r1 >= 0 by A13, XREAL_1:48; then ( ((1 - r1) * ((G * (i2,(j1 + 1))) `2)) + (r1 * ((G * (i2,(j1 + 1))) `2)) = 1 * ((G * (i2,(j1 + 1))) `2) & (1 - r1) * ((G * (i2,j1)) `2) <= (1 - r1) * ((G * (i2,(j1 + 1))) `2) ) by A38, XREAL_1:64; then ((1 - r1) * ((G * (i2,j1)) `2)) + (r1 * ((G * (i2,(j1 + 1))) `2)) <= (G * (i2,(j1 + 1))) `2 by XREAL_1:6; hence contradiction by A5, A8, A18, A17, A22, A33, A37, GOBOARD5:4; ::_thesis: verum end; end; end; theorem :: GOBOARD7:22 for G being Go-board for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) & not ( j1 = j2 & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) = LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ) & not ( j1 = j2 + 1 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i1,j1))} ) holds ( j1 + 1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i2,j2))} ) proof let G be Go-board; ::_thesis: for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) & not ( j1 = j2 & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) = LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ) & not ( j1 = j2 + 1 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i1,j1))} ) holds ( j1 + 1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i2,j2))} ) let i1, j1, i2, j2 be Element of NAT ; ::_thesis: ( 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) & not ( j1 = j2 & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) = LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ) & not ( j1 = j2 + 1 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i1,j1))} ) implies ( j1 + 1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i2,j2))} ) ) assume that A1: ( 1 <= i1 & i1 <= len G ) and A2: 1 <= j1 and A3: j1 + 1 <= width G and A4: ( 1 <= i2 & i2 <= len G ) and A5: 1 <= j2 and A6: j2 + 1 <= width G and A7: LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ; ::_thesis: ( ( j1 = j2 & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) = LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ) or ( j1 = j2 + 1 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i1,j1))} ) or ( j1 + 1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i2,j2))} ) ) A8: i1 = i2 by A1, A2, A3, A4, A5, A6, A7, Th19; A9: (j1 + 1) + 1 = j1 + (1 + 1) ; A10: (j2 + 1) + 1 = j2 + (1 + 1) ; A11: ( abs (j1 - j2) = 0 or abs (j1 - j2) = 1 ) by A1, A2, A3, A4, A5, A6, A7, Th19, NAT_1:25; percases ( j1 = j2 or j1 = j2 + 1 or j1 + 1 = j2 ) by A11, Th2, SEQM_3:41; case j1 = j2 ; ::_thesis: LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) = LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) hence LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) = LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) by A8; ::_thesis: verum end; case j1 = j2 + 1 ; ::_thesis: (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i1,j1))} hence (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i1,j1))} by A1, A3, A5, A8, A10, Th13; ::_thesis: verum end; case j1 + 1 = j2 ; ::_thesis: (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i2,j2))} hence (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) = {(G * (i2,j2))} by A1, A2, A6, A8, A9, Th13; ::_thesis: verum end; end; end; theorem :: GOBOARD7:23 for G being Go-board for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) & not ( i1 = i2 & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) = LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ) & not ( i1 = i2 + 1 & (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} ) holds ( i1 + 1 = i2 & (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i2,j2))} ) proof let G be Go-board; ::_thesis: for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) & not ( i1 = i2 & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) = LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ) & not ( i1 = i2 + 1 & (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} ) holds ( i1 + 1 = i2 & (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i2,j2))} ) let i1, j1, i2, j2 be Element of NAT ; ::_thesis: ( 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) & not ( i1 = i2 & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) = LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ) & not ( i1 = i2 + 1 & (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} ) implies ( i1 + 1 = i2 & (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i2,j2))} ) ) assume that A1: 1 <= i1 and A2: i1 + 1 <= len G and A3: ( 1 <= j1 & j1 <= width G ) and A4: 1 <= i2 and A5: i2 + 1 <= len G and A6: ( 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ) ; ::_thesis: ( ( i1 = i2 & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) = LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ) or ( i1 = i2 + 1 & (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} ) or ( i1 + 1 = i2 & (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i2,j2))} ) ) A7: j1 = j2 by A1, A2, A3, A4, A5, A6, Th20; A8: (i1 + 1) + 1 = i1 + (1 + 1) ; A9: (i2 + 1) + 1 = i2 + (1 + 1) ; A10: ( abs (i1 - i2) = 0 or abs (i1 - i2) = 1 ) by A1, A2, A3, A4, A5, A6, Th20, NAT_1:25; percases ( i1 = i2 or i1 = i2 + 1 or i1 + 1 = i2 ) by A10, Th2, SEQM_3:41; case i1 = i2 ; ::_thesis: LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) = LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) hence LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) = LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) by A7; ::_thesis: verum end; case i1 = i2 + 1 ; ::_thesis: (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} hence (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} by A2, A3, A4, A7, A9, Th14; ::_thesis: verum end; case i1 + 1 = i2 ; ::_thesis: (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i2,j2))} hence (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i2,j2))} by A1, A3, A5, A7, A8, Th14; ::_thesis: verum end; end; end; theorem :: GOBOARD7:24 for G being Go-board for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) & not ( j1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} ) holds ( j1 + 1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,(j1 + 1)))} ) proof let G be Go-board; ::_thesis: for i1, j1, i2, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) & not ( j1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} ) holds ( j1 + 1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,(j1 + 1)))} ) let i1, j1, i2, j2 be Element of NAT ; ::_thesis: ( 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) & not ( j1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} ) implies ( j1 + 1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,(j1 + 1)))} ) ) assume that A1: ( 1 <= i1 & i1 <= len G ) and A2: ( 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 + 1 <= len G ) and A3: ( 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ) ; ::_thesis: ( ( j1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} ) or ( j1 + 1 = j2 & (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,(j1 + 1)))} ) ) percases ( j1 = j2 or j1 + 1 = j2 ) by A1, A2, A3, Th21; caseA4: j1 = j2 ; ::_thesis: (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} now__::_thesis:_(LSeg_((G_*_(i1,j1)),(G_*_(i1,(j1_+_1)))))_/\_(LSeg_((G_*_(i2,j2)),(G_*_((i2_+_1),j2))))_=_{(G_*_(i1,j1))} percases ( i1 = i2 or i1 = i2 + 1 ) by A1, A2, A3, Th21; suppose i1 = i2 ; ::_thesis: (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} hence (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} by A2, A4, Th17; ::_thesis: verum end; suppose i1 = i2 + 1 ; ::_thesis: (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} hence (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} by A2, A4, Th18; ::_thesis: verum end; end; end; hence (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,j1))} ; ::_thesis: verum end; caseA5: j1 + 1 = j2 ; ::_thesis: (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,(j1 + 1)))} now__::_thesis:_(LSeg_((G_*_(i1,j1)),(G_*_(i1,(j1_+_1)))))_/\_(LSeg_((G_*_(i2,j2)),(G_*_((i2_+_1),j2))))_=_{(G_*_(i1,(j1_+_1)))} percases ( i1 = i2 or i1 = i2 + 1 ) by A1, A2, A3, Th21; suppose i1 = i2 ; ::_thesis: (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,(j1 + 1)))} hence (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,(j1 + 1)))} by A2, A5, Th15; ::_thesis: verum end; suppose i1 = i2 + 1 ; ::_thesis: (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,(j1 + 1)))} hence (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,(j1 + 1)))} by A2, A5, Th16; ::_thesis: verum end; end; end; hence (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) = {(G * (i1,(j1 + 1)))} ; ::_thesis: verum end; end; end; Lm1: 1 - (1 / 2) = 1 / 2 ; theorem Th25: :: GOBOARD7:25 for i1, j1, i2, j2 being Element of NAT for G being Go-board st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) holds ( i1 = i2 & j1 = j2 ) proof let i1, j1, i2, j2 be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) holds ( i1 = i2 & j1 = j2 ) let G be Go-board; ::_thesis: ( 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) implies ( i1 = i2 & j1 = j2 ) ) assume that A1: ( 1 <= i1 & i1 <= len G ) and A2: 1 <= j1 and A3: j1 + 1 <= width G and A4: ( 1 <= i2 & i2 <= len G ) and A5: 1 <= j2 and A6: j2 + 1 <= width G ; ::_thesis: ( not (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) or ( i1 = i2 & j1 = j2 ) ) set mi = (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))); A7: ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1)))) = (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) by EUCLID:32; then A8: (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) by Lm1; assume A9: (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ; ::_thesis: ( i1 = i2 & j1 = j2 ) then A10: LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) by A8, XBOOLE_0:3; hence A11: i1 = i2 by A1, A2, A3, A4, A5, A6, Th19; ::_thesis: j1 = j2 now__::_thesis:_not_abs_(j1_-_j2)_=_1 j1 < j1 + 1 by XREAL_1:29; then A12: (G * (i1,(j1 + 1))) `2 > (G * (i1,j1)) `2 by A1, A2, A3, GOBOARD5:4; assume A13: abs (j1 - j2) = 1 ; ::_thesis: contradiction percases ( j1 = j2 + 1 or j1 + 1 = j2 ) by A13, SEQM_3:41; supposeA14: j1 = j2 + 1 ; ::_thesis: contradiction then (LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) /\ (LSeg ((G * (i2,(j2 + 1))),(G * (i2,(j2 + 2))))) = {(G * (i2,(j2 + 1)))} by A3, A4, A5, Th13; then (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in {(G * (i1,j1))} by A9, A8, A11, A14, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1)))) = G * (i1,j1) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * (i1,j1)) by EUCLID:29 .= ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1))) by EUCLID:33 ; then (1 / 2) * (G * (i1,j1)) = (1 / 2) * (G * (i1,(j1 + 1))) by Th3; hence contradiction by A12, EUCLID:34; ::_thesis: verum end; supposeA15: j1 + 1 = j2 ; ::_thesis: contradiction then (LSeg ((G * (i2,j1)),(G * (i2,(j1 + 1))))) /\ (LSeg ((G * (i2,(j1 + 1))),(G * (i2,(j1 + 2))))) = {(G * (i2,(j1 + 1)))} by A2, A4, A6, Th13; then (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in {(G * (i1,j2))} by A9, A8, A11, A15, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1)))) = G * (i1,j2) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * (i1,j2)) by EUCLID:29 .= ((1 / 2) * (G * (i1,j2))) + ((1 / 2) * (G * (i1,j2))) by EUCLID:33 ; then (1 / 2) * (G * (i1,j1)) = (1 / 2) * (G * (i1,(j1 + 1))) by A15, Th3; hence contradiction by A12, EUCLID:34; ::_thesis: verum end; end; end; then abs (j1 - j2) = 0 by A1, A2, A3, A4, A5, A6, A10, Th19, NAT_1:25; hence j1 = j2 by Th2; ::_thesis: verum end; theorem Th26: :: GOBOARD7:26 for i1, j1, i2, j2 being Element of NAT for G being Go-board st 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) holds ( i1 = i2 & j1 = j2 ) proof let i1, j1, i2, j2 be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) holds ( i1 = i2 & j1 = j2 ) let G be Go-board; ::_thesis: ( 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) implies ( i1 = i2 & j1 = j2 ) ) assume that A1: 1 <= i1 and A2: i1 + 1 <= len G and A3: ( 1 <= j1 & j1 <= width G ) and A4: 1 <= i2 and A5: i2 + 1 <= len G and A6: ( 1 <= j2 & j2 <= width G ) ; ::_thesis: ( not (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) or ( i1 = i2 & j1 = j2 ) ) set mi = (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))); A7: ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1))) = (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) by EUCLID:32; then A8: (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) by Lm1; assume A9: (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ; ::_thesis: ( i1 = i2 & j1 = j2 ) then A10: LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) by A8, XBOOLE_0:3; then A11: j1 = j2 by A1, A2, A3, A4, A5, A6, Th20; now__::_thesis:_not_abs_(i1_-_i2)_=_1 i1 < i1 + 1 by XREAL_1:29; then A12: (G * ((i1 + 1),j1)) `1 > (G * (i1,j1)) `1 by A1, A2, A3, GOBOARD5:3; assume A13: abs (i1 - i2) = 1 ; ::_thesis: contradiction percases ( i1 = i2 + 1 or i1 + 1 = i2 ) by A13, SEQM_3:41; supposeA14: i1 = i2 + 1 ; ::_thesis: contradiction then (LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) /\ (LSeg ((G * ((i2 + 1),j2)),(G * ((i2 + 2),j2)))) = {(G * ((i2 + 1),j2))} by A2, A4, A6, Th14; then (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in {(G * (i1,j1))} by A9, A8, A11, A14, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1))) = G * (i1,j1) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * (i1,j1)) by EUCLID:29 .= ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1))) by EUCLID:33 ; then (1 / 2) * (G * (i1,j1)) = (1 / 2) * (G * ((i1 + 1),j1)) by Th3; hence contradiction by A12, EUCLID:34; ::_thesis: verum end; supposeA15: i1 + 1 = i2 ; ::_thesis: contradiction then (LSeg ((G * (i1,j2)),(G * ((i1 + 1),j2)))) /\ (LSeg ((G * ((i1 + 1),j2)),(G * ((i1 + 2),j2)))) = {(G * ((i1 + 1),j2))} by A1, A5, A6, Th14; then (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in {(G * (i2,j1))} by A9, A8, A11, A15, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1))) = G * (i2,j1) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * (i2,j1)) by EUCLID:29 .= ((1 / 2) * (G * (i2,j1))) + ((1 / 2) * (G * (i2,j1))) by EUCLID:33 ; then (1 / 2) * (G * (i1,j1)) = (1 / 2) * (G * ((i1 + 1),j1)) by A15, Th3; hence contradiction by A12, EUCLID:34; ::_thesis: verum end; end; end; then abs (i1 - i2) = 0 by A1, A2, A3, A4, A5, A6, A10, Th20, NAT_1:25; hence i1 = i2 by Th2; ::_thesis: j1 = j2 thus j1 = j2 by A1, A2, A3, A4, A5, A6, A10, Th20; ::_thesis: verum end; theorem Th27: :: GOBOARD7:27 for i1, j1 being Element of NAT for G being Go-board st 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G holds for i2, j2 being Element of NAT holds ( not 1 <= i2 or not i2 <= len G or not 1 <= j2 or not j2 + 1 <= width G or not (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ) proof let i1, j1 be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G holds for i2, j2 being Element of NAT holds ( not 1 <= i2 or not i2 <= len G or not 1 <= j2 or not j2 + 1 <= width G or not (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ) let G be Go-board; ::_thesis: ( 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G implies for i2, j2 being Element of NAT holds ( not 1 <= i2 or not i2 <= len G or not 1 <= j2 or not j2 + 1 <= width G or not (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ) ) assume that A1: ( 1 <= i1 & i1 + 1 <= len G ) and A2: ( 1 <= j1 & j1 <= width G ) ; ::_thesis: for i2, j2 being Element of NAT holds ( not 1 <= i2 or not i2 <= len G or not 1 <= j2 or not j2 + 1 <= width G or not (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ) A3: i1 < i1 + 1 by XREAL_1:29; set mi = (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))); given i2, j2 being Element of NAT such that A4: ( 1 <= i2 & i2 <= len G ) and A5: ( 1 <= j2 & j2 + 1 <= width G ) and A6: (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) ; ::_thesis: contradiction A7: ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1))) = (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) by EUCLID:32; then A8: (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) by Lm1; then A9: LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))) by A6, XBOOLE_0:3; percases ( ( j1 = j2 & i1 = i2 ) or ( j1 = j2 & i1 + 1 = i2 ) or ( j1 = j2 + 1 & i1 = i2 ) or ( j1 = j2 + 1 & i1 + 1 = i2 ) ) by A1, A2, A4, A5, A9, Th21; supposeA10: ( j1 = j2 & i1 = i2 ) ; ::_thesis: contradiction then (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i1,(j1 + 1))),(G * (i1,j1)))) = {(G * (i1,j1))} by A1, A5, Th17; then (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in {(G * (i1,j1))} by A6, A8, A10, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1))) = G * (i1,j1) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * (i1,j1)) by EUCLID:29 .= ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1))) by EUCLID:33 ; then A11: (1 / 2) * (G * ((i1 + 1),j1)) = (1 / 2) * (G * (i1,j1)) by Th3; (G * ((i1 + 1),j1)) `1 > (G * (i1,j1)) `1 by A1, A2, A3, GOBOARD5:3; hence contradiction by A11, EUCLID:34; ::_thesis: verum end; supposeA12: ( j1 = j2 & i1 + 1 = i2 ) ; ::_thesis: contradiction then (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * ((i1 + 1),(j1 + 1))),(G * ((i1 + 1),j1)))) = {(G * ((i1 + 1),j1))} by A1, A5, Th18; then (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in {(G * ((i1 + 1),j1))} by A6, A8, A12, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1))) = G * ((i1 + 1),j1) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * ((i1 + 1),j1)) by EUCLID:29 .= ((1 / 2) * (G * ((i1 + 1),j1))) + ((1 / 2) * (G * ((i1 + 1),j1))) by EUCLID:33 ; then A13: (1 / 2) * (G * ((i1 + 1),j1)) = (1 / 2) * (G * (i1,j1)) by Th3; (G * ((i1 + 1),j1)) `1 > (G * (i1,j1)) `1 by A1, A2, A3, GOBOARD5:3; hence contradiction by A13, EUCLID:34; ::_thesis: verum end; supposeA14: ( j1 = j2 + 1 & i1 = i2 ) ; ::_thesis: contradiction then (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * (i1,j1)),(G * (i1,j2)))) = {(G * (i1,j1))} by A1, A5, Th15; then (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in {(G * (i1,j1))} by A6, A8, A14, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1))) = G * (i1,j1) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * (i1,j1)) by EUCLID:29 .= ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1))) by EUCLID:33 ; then A15: (1 / 2) * (G * ((i1 + 1),j1)) = (1 / 2) * (G * (i1,j1)) by Th3; (G * ((i1 + 1),j1)) `1 > (G * (i1,j1)) `1 by A1, A2, A3, GOBOARD5:3; hence contradiction by A15, EUCLID:34; ::_thesis: verum end; supposeA16: ( j1 = j2 + 1 & i1 + 1 = i2 ) ; ::_thesis: contradiction then (LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ (LSeg ((G * ((i1 + 1),j1)),(G * ((i1 + 1),j2)))) = {(G * ((i1 + 1),j1))} by A1, A5, Th16; then (1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))) in {(G * ((i1 + 1),j1))} by A6, A8, A16, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1))) = G * ((i1 + 1),j1) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * ((i1 + 1),j1)) by EUCLID:29 .= ((1 / 2) * (G * ((i1 + 1),j1))) + ((1 / 2) * (G * ((i1 + 1),j1))) by EUCLID:33 ; then A17: (1 / 2) * (G * ((i1 + 1),j1)) = (1 / 2) * (G * (i1,j1)) by Th3; (G * ((i1 + 1),j1)) `1 > (G * (i1,j1)) `1 by A1, A2, A3, GOBOARD5:3; hence contradiction by A17, EUCLID:34; ::_thesis: verum end; end; end; theorem Th28: :: GOBOARD7:28 for i1, j1 being Element of NAT for G being Go-board st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G holds for i2, j2 being Element of NAT holds ( not 1 <= i2 or not i2 + 1 <= len G or not 1 <= j2 or not j2 <= width G or not (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ) proof let i1, j1 be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G holds for i2, j2 being Element of NAT holds ( not 1 <= i2 or not i2 + 1 <= len G or not 1 <= j2 or not j2 <= width G or not (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ) let G be Go-board; ::_thesis: ( 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G implies for i2, j2 being Element of NAT holds ( not 1 <= i2 or not i2 + 1 <= len G or not 1 <= j2 or not j2 <= width G or not (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ) ) assume that A1: ( 1 <= i1 & i1 <= len G ) and A2: ( 1 <= j1 & j1 + 1 <= width G ) ; ::_thesis: for i2, j2 being Element of NAT holds ( not 1 <= i2 or not i2 + 1 <= len G or not 1 <= j2 or not j2 <= width G or not (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ) A3: j1 < j1 + 1 by XREAL_1:29; set mi = (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))); given i2, j2 being Element of NAT such that A4: ( 1 <= i2 & i2 + 1 <= len G ) and A5: ( 1 <= j2 & j2 <= width G ) and A6: (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ; ::_thesis: contradiction A7: ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1)))) = (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) by EUCLID:32; then A8: (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) by Lm1; then A9: LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) by A6, XBOOLE_0:3; percases ( ( i1 = i2 & j1 = j2 ) or ( i1 = i2 & j1 + 1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 + 1 & j1 + 1 = j2 ) ) by A1, A2, A4, A5, A9, Th21; supposeA10: ( i1 = i2 & j1 = j2 ) ; ::_thesis: contradiction then (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * ((i1 + 1),j1)),(G * (i1,j1)))) = {(G * (i1,j1))} by A2, A4, Th17; then (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in {(G * (i1,j1))} by A6, A8, A10, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1)))) = G * (i1,j1) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * (i1,j1)) by EUCLID:29 .= ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1))) by EUCLID:33 ; then A11: (1 / 2) * (G * (i1,(j1 + 1))) = (1 / 2) * (G * (i1,j1)) by Th3; (G * (i1,(j1 + 1))) `2 > (G * (i1,j1)) `2 by A1, A2, A3, GOBOARD5:4; hence contradiction by A11, EUCLID:34; ::_thesis: verum end; supposeA12: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: contradiction then (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * ((i1 + 1),(j1 + 1))),(G * (i1,(j1 + 1))))) = {(G * (i1,(j1 + 1)))} by A2, A4, Th15; then (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in {(G * (i1,(j1 + 1)))} by A6, A8, A12, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1)))) = G * (i1,(j1 + 1)) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * (i1,(j1 + 1))) by EUCLID:29 .= ((1 / 2) * (G * (i1,(j1 + 1)))) + ((1 / 2) * (G * (i1,(j1 + 1)))) by EUCLID:33 ; then A13: (1 / 2) * (G * (i1,(j1 + 1))) = (1 / 2) * (G * (i1,j1)) by Th3; (G * (i1,(j1 + 1))) `2 > (G * (i1,j1)) `2 by A1, A2, A3, GOBOARD5:4; hence contradiction by A13, EUCLID:34; ::_thesis: verum end; supposeA14: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: contradiction then (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i1,j1)),(G * (i2,j1)))) = {(G * (i1,j1))} by A2, A4, Th18; then (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in {(G * (i1,j1))} by A6, A8, A14, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1)))) = G * (i1,j1) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * (i1,j1)) by EUCLID:29 .= ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1))) by EUCLID:33 ; then A15: (1 / 2) * (G * (i1,(j1 + 1))) = (1 / 2) * (G * (i1,j1)) by Th3; (G * (i1,(j1 + 1))) `2 > (G * (i1,j1)) `2 by A1, A2, A3, GOBOARD5:4; hence contradiction by A15, EUCLID:34; ::_thesis: verum end; supposeA16: ( i1 = i2 + 1 & j1 + 1 = j2 ) ; ::_thesis: contradiction then (LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ (LSeg ((G * (i1,(j1 + 1))),(G * (i2,(j1 + 1))))) = {(G * (i1,(j1 + 1)))} by A2, A4, Th16; then (1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))) in {(G * (i1,(j1 + 1)))} by A6, A8, A16, XBOOLE_0:def_4; then ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1)))) = G * (i1,(j1 + 1)) by A7, TARSKI:def_1 .= ((1 / 2) + (1 / 2)) * (G * (i1,(j1 + 1))) by EUCLID:29 .= ((1 / 2) * (G * (i1,(j1 + 1)))) + ((1 / 2) * (G * (i1,(j1 + 1)))) by EUCLID:33 ; then A17: (1 / 2) * (G * (i1,(j1 + 1))) = (1 / 2) * (G * (i1,j1)) by Th3; (G * (i1,(j1 + 1))) `2 > (G * (i1,j1)) `2 by A1, A2, A3, GOBOARD5:4; hence contradiction by A17, EUCLID:34; ::_thesis: verum end; end; end; begin Lm2: for f being non constant standard special_circular_sequence holds len f > 1 proof let f be non constant standard special_circular_sequence; ::_thesis: len f > 1 consider n1, n2 being set such that A1: n1 in dom f and A2: ( n2 in dom f & f . n1 <> f . n2 ) by FUNCT_1:def_10; reconsider df = dom f as finite set ; A3: now__::_thesis:_not_card_df_<=_1 assume A4: card df <= 1 ; ::_thesis: contradiction percases ( card df = 0 or card df = 1 ) by A4, NAT_1:25; suppose card df = 0 ; ::_thesis: contradiction hence contradiction by A1; ::_thesis: verum end; suppose card df = 1 ; ::_thesis: contradiction then consider x being set such that A5: dom f = {x} by CARD_2:42; n1 = x by A1, A5, TARSKI:def_1; hence contradiction by A2, A5, TARSKI:def_1; ::_thesis: verum end; end; end; dom f = Seg (len f) by FINSEQ_1:def_3; hence len f > 1 by A3, FINSEQ_1:57; ::_thesis: verum end; theorem Th29: :: GOBOARD7:29 for i being Element of NAT for f being non empty standard FinSequence of (TOP-REAL 2) st i in dom f & i + 1 in dom f holds f /. i <> f /. (i + 1) proof let i be Element of NAT ; ::_thesis: for f being non empty standard FinSequence of (TOP-REAL 2) st i in dom f & i + 1 in dom f holds f /. i <> f /. (i + 1) A1: abs 0 = 0 by ABSVALUE:2; let f be non empty standard FinSequence of (TOP-REAL 2); ::_thesis: ( i in dom f & i + 1 in dom f implies f /. i <> f /. (i + 1) ) assume that A2: i in dom f and A3: i + 1 in dom f ; ::_thesis: f /. i <> f /. (i + 1) A4: f is_sequence_on GoB f by GOBOARD5:def_5; then consider i1, j1 being Element of NAT such that A5: ( [i1,j1] in Indices (GoB f) & f /. i = (GoB f) * (i1,j1) ) by A2, GOBOARD1:def_9; consider i2, j2 being Element of NAT such that A6: [i2,j2] in Indices (GoB f) and A7: f /. (i + 1) = (GoB f) * (i2,j2) by A3, A4, GOBOARD1:def_9; assume A8: f /. i = f /. (i + 1) ; ::_thesis: contradiction then j1 = j2 by A5, A6, A7, GOBOARD1:5; then A9: j1 - j2 = 0 ; i1 = i2 by A5, A6, A7, A8, GOBOARD1:5; then i1 - i2 = 0 ; then (abs 0) + (abs 0) = 1 by A2, A3, A4, A5, A7, A9, GOBOARD1:def_9; hence contradiction by A1; ::_thesis: verum end; theorem Th30: :: GOBOARD7:30 for f being non constant standard special_circular_sequence ex i being Element of NAT st ( i in dom f & (f /. i) `1 <> (f /. 1) `1 ) proof let f be non constant standard special_circular_sequence; ::_thesis: ex i being Element of NAT st ( i in dom f & (f /. i) `1 <> (f /. 1) `1 ) assume A1: for i being Element of NAT st i in dom f holds (f /. i) `1 = (f /. 1) `1 ; ::_thesis: contradiction A2: len f > 1 by Lm2; then A3: len f >= 1 + 1 by NAT_1:13; then A4: 1 + 1 in dom f by FINSEQ_3:25; A5: now__::_thesis:_not_(f_/._2)_`2_=_(f_/._1)_`2 assume A6: (f /. 2) `2 = (f /. 1) `2 ; ::_thesis: contradiction (f /. 2) `1 = (f /. 1) `1 by A1, A4; then f /. 2 = |[((f /. 1) `1),((f /. 1) `2)]| by A6, EUCLID:53 .= f /. 1 by EUCLID:53 ; hence contradiction by A4, Th29, FINSEQ_5:6; ::_thesis: verum end; ( len f = 2 implies f /. 2 = f /. 1 ) by FINSEQ_6:def_1; then A7: 2 < len f by A3, A5, XXREAL_0:1; percases ( (f /. 2) `2 < (f /. 1) `2 or (f /. 2) `2 > (f /. 1) `2 ) by A5, XXREAL_0:1; supposeA8: (f /. 2) `2 < (f /. 1) `2 ; ::_thesis: contradiction defpred S1[ Element of NAT ] means ( 2 <= $1 & $1 < len f implies ( (f /. $1) `2 <= (f /. 2) `2 & (f /. ($1 + 1)) `2 < (f /. $1) `2 ) ); A9: for j being Element of NAT st S1[j] holds S1[j + 1] proof let j be Element of NAT ; ::_thesis: ( S1[j] implies S1[j + 1] ) assume that A10: ( 2 <= j & j < len f implies ( (f /. j) `2 <= (f /. 2) `2 & (f /. (j + 1)) `2 < (f /. j) `2 ) ) and A11: 2 <= j + 1 and A12: j + 1 < len f ; ::_thesis: ( (f /. (j + 1)) `2 <= (f /. 2) `2 & (f /. ((j + 1) + 1)) `2 < (f /. (j + 1)) `2 ) 1 + 1 <= j + 1 by A11; then A13: 1 <= j by XREAL_1:6; thus (f /. (j + 1)) `2 <= (f /. 2) `2 ::_thesis: (f /. ((j + 1) + 1)) `2 < (f /. (j + 1)) `2 proof percases ( 2 = j + 1 or 2 < j + 1 ) by A11, XXREAL_0:1; suppose 2 = j + 1 ; ::_thesis: (f /. (j + 1)) `2 <= (f /. 2) `2 hence (f /. (j + 1)) `2 <= (f /. 2) `2 ; ::_thesis: verum end; suppose 2 < j + 1 ; ::_thesis: (f /. (j + 1)) `2 <= (f /. 2) `2 hence (f /. (j + 1)) `2 <= (f /. 2) `2 by A10, A12, NAT_1:13, XXREAL_0:2; ::_thesis: verum end; end; end; A14: (j + 1) + 1 <= len f by A12, NAT_1:13; A15: now__::_thesis:_(f_/._(j_+_1))_`2_<_(f_/._j)_`2 percases ( 1 + 1 = j + 1 or 2 < j + 1 ) by A11, XXREAL_0:1; suppose 1 + 1 = j + 1 ; ::_thesis: (f /. (j + 1)) `2 < (f /. j) `2 hence (f /. (j + 1)) `2 < (f /. j) `2 by A8; ::_thesis: verum end; suppose 2 < j + 1 ; ::_thesis: (f /. (j + 1)) `2 < (f /. j) `2 hence (f /. (j + 1)) `2 < (f /. j) `2 by A10, A12, NAT_1:13; ::_thesis: verum end; end; end; A16: 1 <= j + 1 by NAT_1:11; then A17: j + 1 in dom f by A12, FINSEQ_3:25; then A18: (f /. (j + 1)) `1 = (f /. 1) `1 by A1; j < len f by A12, NAT_1:13; then A19: j in dom f by A13, FINSEQ_3:25; then A20: (f /. j) `1 = (f /. 1) `1 by A1; 1 <= (j + 1) + 1 by NAT_1:11; then A21: (j + 1) + 1 in dom f by A14, FINSEQ_3:25; then A22: (f /. ((j + 1) + 1)) `1 = (f /. 1) `1 by A1; assume A23: (f /. ((j + 1) + 1)) `2 >= (f /. (j + 1)) `2 ; ::_thesis: contradiction percases ( (f /. ((j + 1) + 1)) `2 > (f /. (j + 1)) `2 or (f /. ((j + 1) + 1)) `2 = (f /. (j + 1)) `2 ) by A23, XXREAL_0:1; supposeA24: (f /. ((j + 1) + 1)) `2 > (f /. (j + 1)) `2 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( (f /. j) `2 <= (f /. ((j + 1) + 1)) `2 or (f /. j) `2 >= (f /. ((j + 1) + 1)) `2 ) ; suppose (f /. j) `2 <= (f /. ((j + 1) + 1)) `2 ; ::_thesis: contradiction then f /. j in LSeg ((f /. (j + 1)),(f /. ((j + 1) + 1))) by A15, A20, A18, A22, Th7; then A25: f /. j in LSeg (f,(j + 1)) by A14, A16, TOPREAL1:def_3; (j + 1) + 1 = j + (1 + 1) ; then A26: (LSeg (f,j)) /\ (LSeg (f,(j + 1))) = {(f /. (j + 1))} by A14, A13, TOPREAL1:def_6; f /. j in LSeg (f,j) by A12, A13, TOPREAL1:21; then f /. j in (LSeg (f,j)) /\ (LSeg (f,(j + 1))) by A25, XBOOLE_0:def_4; then f /. j = f /. (j + 1) by A26, TARSKI:def_1; hence contradiction by A19, A17, Th29; ::_thesis: verum end; suppose (f /. j) `2 >= (f /. ((j + 1) + 1)) `2 ; ::_thesis: contradiction then f /. ((j + 1) + 1) in LSeg ((f /. j),(f /. (j + 1))) by A20, A18, A22, A24, Th7; then A27: f /. ((j + 1) + 1) in LSeg (f,j) by A12, A13, TOPREAL1:def_3; (j + 1) + 1 = j + (1 + 1) ; then A28: (LSeg (f,j)) /\ (LSeg (f,(j + 1))) = {(f /. (j + 1))} by A14, A13, TOPREAL1:def_6; f /. ((j + 1) + 1) in LSeg (f,(j + 1)) by A14, A16, TOPREAL1:21; then f /. ((j + 1) + 1) in (LSeg (f,j)) /\ (LSeg (f,(j + 1))) by A27, XBOOLE_0:def_4; then f /. ((j + 1) + 1) = f /. (j + 1) by A28, TARSKI:def_1; hence contradiction by A17, A21, Th29; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA29: (f /. ((j + 1) + 1)) `2 = (f /. (j + 1)) `2 ; ::_thesis: contradiction (f /. ((j + 1) + 1)) `1 = (f /. 1) `1 by A1, A21 .= (f /. (j + 1)) `1 by A1, A17 ; then f /. ((j + 1) + 1) = |[((f /. (j + 1)) `1),((f /. (j + 1)) `2)]| by A29, EUCLID:53 .= f /. (j + 1) by EUCLID:53 ; hence contradiction by A17, A21, Th29; ::_thesis: verum end; end; end; A30: ((len f) -' 1) + 1 = len f by A2, XREAL_1:235; then A31: ( 2 <= (len f) -' 1 & (len f) -' 1 < len f ) by A7, NAT_1:13; A32: S1[ 0 ] ; A33: for j being Element of NAT holds S1[j] from NAT_1:sch_1(A32, A9); then A34: (f /. ((len f) -' 1)) `2 <= (f /. 2) `2 by A31; (f /. (len f)) `2 < (f /. ((len f) -' 1)) `2 by A33, A30, A31; then (f /. (len f)) `2 < (f /. 2) `2 by A34, XXREAL_0:2; hence contradiction by A8, FINSEQ_6:def_1; ::_thesis: verum end; supposeA35: (f /. 2) `2 > (f /. 1) `2 ; ::_thesis: contradiction defpred S1[ Element of NAT ] means ( 2 <= $1 & $1 < len f implies ( (f /. $1) `2 >= (f /. 2) `2 & (f /. ($1 + 1)) `2 > (f /. $1) `2 ) ); A36: for j being Element of NAT st S1[j] holds S1[j + 1] proof let j be Element of NAT ; ::_thesis: ( S1[j] implies S1[j + 1] ) assume that A37: ( 2 <= j & j < len f implies ( (f /. j) `2 >= (f /. 2) `2 & (f /. (j + 1)) `2 > (f /. j) `2 ) ) and A38: 2 <= j + 1 and A39: j + 1 < len f ; ::_thesis: ( (f /. (j + 1)) `2 >= (f /. 2) `2 & (f /. ((j + 1) + 1)) `2 > (f /. (j + 1)) `2 ) 1 + 1 <= j + 1 by A38; then A40: 1 <= j by XREAL_1:6; thus (f /. (j + 1)) `2 >= (f /. 2) `2 ::_thesis: (f /. ((j + 1) + 1)) `2 > (f /. (j + 1)) `2 proof percases ( 2 = j + 1 or 2 < j + 1 ) by A38, XXREAL_0:1; suppose 2 = j + 1 ; ::_thesis: (f /. (j + 1)) `2 >= (f /. 2) `2 hence (f /. (j + 1)) `2 >= (f /. 2) `2 ; ::_thesis: verum end; suppose 2 < j + 1 ; ::_thesis: (f /. (j + 1)) `2 >= (f /. 2) `2 hence (f /. (j + 1)) `2 >= (f /. 2) `2 by A37, A39, NAT_1:13, XXREAL_0:2; ::_thesis: verum end; end; end; A41: (j + 1) + 1 <= len f by A39, NAT_1:13; A42: now__::_thesis:_(f_/._(j_+_1))_`2_>_(f_/._j)_`2 percases ( 1 + 1 = j + 1 or 2 < j + 1 ) by A38, XXREAL_0:1; suppose 1 + 1 = j + 1 ; ::_thesis: (f /. (j + 1)) `2 > (f /. j) `2 hence (f /. (j + 1)) `2 > (f /. j) `2 by A35; ::_thesis: verum end; suppose 2 < j + 1 ; ::_thesis: (f /. (j + 1)) `2 > (f /. j) `2 hence (f /. (j + 1)) `2 > (f /. j) `2 by A37, A39, NAT_1:13; ::_thesis: verum end; end; end; A43: 1 <= j + 1 by NAT_1:11; then A44: j + 1 in dom f by A39, FINSEQ_3:25; then A45: (f /. (j + 1)) `1 = (f /. 1) `1 by A1; j < len f by A39, NAT_1:13; then A46: j in dom f by A40, FINSEQ_3:25; then A47: (f /. j) `1 = (f /. 1) `1 by A1; 1 <= (j + 1) + 1 by NAT_1:11; then A48: (j + 1) + 1 in dom f by A41, FINSEQ_3:25; then A49: (f /. ((j + 1) + 1)) `1 = (f /. 1) `1 by A1; assume A50: (f /. ((j + 1) + 1)) `2 <= (f /. (j + 1)) `2 ; ::_thesis: contradiction percases ( (f /. ((j + 1) + 1)) `2 < (f /. (j + 1)) `2 or (f /. ((j + 1) + 1)) `2 = (f /. (j + 1)) `2 ) by A50, XXREAL_0:1; supposeA51: (f /. ((j + 1) + 1)) `2 < (f /. (j + 1)) `2 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( (f /. j) `2 >= (f /. ((j + 1) + 1)) `2 or (f /. j) `2 <= (f /. ((j + 1) + 1)) `2 ) ; suppose (f /. j) `2 >= (f /. ((j + 1) + 1)) `2 ; ::_thesis: contradiction then f /. j in LSeg ((f /. (j + 1)),(f /. ((j + 1) + 1))) by A42, A47, A45, A49, Th7; then A52: f /. j in LSeg (f,(j + 1)) by A41, A43, TOPREAL1:def_3; (j + 1) + 1 = j + (1 + 1) ; then A53: (LSeg (f,j)) /\ (LSeg (f,(j + 1))) = {(f /. (j + 1))} by A41, A40, TOPREAL1:def_6; f /. j in LSeg (f,j) by A39, A40, TOPREAL1:21; then f /. j in (LSeg (f,j)) /\ (LSeg (f,(j + 1))) by A52, XBOOLE_0:def_4; then f /. j = f /. (j + 1) by A53, TARSKI:def_1; hence contradiction by A46, A44, Th29; ::_thesis: verum end; suppose (f /. j) `2 <= (f /. ((j + 1) + 1)) `2 ; ::_thesis: contradiction then f /. ((j + 1) + 1) in LSeg ((f /. j),(f /. (j + 1))) by A47, A45, A49, A51, Th7; then A54: f /. ((j + 1) + 1) in LSeg (f,j) by A39, A40, TOPREAL1:def_3; (j + 1) + 1 = j + (1 + 1) ; then A55: (LSeg (f,j)) /\ (LSeg (f,(j + 1))) = {(f /. (j + 1))} by A41, A40, TOPREAL1:def_6; f /. ((j + 1) + 1) in LSeg (f,(j + 1)) by A41, A43, TOPREAL1:21; then f /. ((j + 1) + 1) in (LSeg (f,j)) /\ (LSeg (f,(j + 1))) by A54, XBOOLE_0:def_4; then f /. ((j + 1) + 1) = f /. (j + 1) by A55, TARSKI:def_1; hence contradiction by A44, A48, Th29; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA56: (f /. ((j + 1) + 1)) `2 = (f /. (j + 1)) `2 ; ::_thesis: contradiction (f /. ((j + 1) + 1)) `1 = (f /. 1) `1 by A1, A48 .= (f /. (j + 1)) `1 by A1, A44 ; then f /. ((j + 1) + 1) = |[((f /. (j + 1)) `1),((f /. (j + 1)) `2)]| by A56, EUCLID:53 .= f /. (j + 1) by EUCLID:53 ; hence contradiction by A44, A48, Th29; ::_thesis: verum end; end; end; A57: ((len f) -' 1) + 1 = len f by A2, XREAL_1:235; then A58: ( 2 <= (len f) -' 1 & (len f) -' 1 < len f ) by A7, NAT_1:13; A59: S1[ 0 ] ; A60: for j being Element of NAT holds S1[j] from NAT_1:sch_1(A59, A36); then A61: (f /. ((len f) -' 1)) `2 >= (f /. 2) `2 by A58; (f /. (len f)) `2 > (f /. ((len f) -' 1)) `2 by A60, A57, A58; then (f /. (len f)) `2 > (f /. 2) `2 by A61, XXREAL_0:2; hence contradiction by A35, FINSEQ_6:def_1; ::_thesis: verum end; end; end; theorem Th31: :: GOBOARD7:31 for f being non constant standard special_circular_sequence ex i being Element of NAT st ( i in dom f & (f /. i) `2 <> (f /. 1) `2 ) proof let f be non constant standard special_circular_sequence; ::_thesis: ex i being Element of NAT st ( i in dom f & (f /. i) `2 <> (f /. 1) `2 ) assume A1: for i being Element of NAT st i in dom f holds (f /. i) `2 = (f /. 1) `2 ; ::_thesis: contradiction A2: len f > 1 by Lm2; then A3: len f >= 1 + 1 by NAT_1:13; then A4: 1 + 1 in dom f by FINSEQ_3:25; A5: now__::_thesis:_not_(f_/._2)_`1_=_(f_/._1)_`1 assume A6: (f /. 2) `1 = (f /. 1) `1 ; ::_thesis: contradiction (f /. 2) `2 = (f /. 1) `2 by A1, A4; then f /. 2 = |[((f /. 1) `1),((f /. 1) `2)]| by A6, EUCLID:53 .= f /. 1 by EUCLID:53 ; hence contradiction by A4, Th29, FINSEQ_5:6; ::_thesis: verum end; ( len f = 2 implies f /. 2 = f /. 1 ) by FINSEQ_6:def_1; then A7: 2 < len f by A3, A5, XXREAL_0:1; percases ( (f /. 2) `1 < (f /. 1) `1 or (f /. 2) `1 > (f /. 1) `1 ) by A5, XXREAL_0:1; supposeA8: (f /. 2) `1 < (f /. 1) `1 ; ::_thesis: contradiction defpred S1[ Element of NAT ] means ( 2 <= $1 & $1 < len f implies ( (f /. $1) `1 <= (f /. 2) `1 & (f /. ($1 + 1)) `1 < (f /. $1) `1 ) ); A9: for j being Element of NAT st S1[j] holds S1[j + 1] proof let j be Element of NAT ; ::_thesis: ( S1[j] implies S1[j + 1] ) assume that A10: ( 2 <= j & j < len f implies ( (f /. j) `1 <= (f /. 2) `1 & (f /. (j + 1)) `1 < (f /. j) `1 ) ) and A11: 2 <= j + 1 and A12: j + 1 < len f ; ::_thesis: ( (f /. (j + 1)) `1 <= (f /. 2) `1 & (f /. ((j + 1) + 1)) `1 < (f /. (j + 1)) `1 ) 1 + 1 <= j + 1 by A11; then A13: 1 <= j by XREAL_1:6; thus (f /. (j + 1)) `1 <= (f /. 2) `1 ::_thesis: (f /. ((j + 1) + 1)) `1 < (f /. (j + 1)) `1 proof percases ( 2 = j + 1 or 2 < j + 1 ) by A11, XXREAL_0:1; suppose 2 = j + 1 ; ::_thesis: (f /. (j + 1)) `1 <= (f /. 2) `1 hence (f /. (j + 1)) `1 <= (f /. 2) `1 ; ::_thesis: verum end; suppose 2 < j + 1 ; ::_thesis: (f /. (j + 1)) `1 <= (f /. 2) `1 hence (f /. (j + 1)) `1 <= (f /. 2) `1 by A10, A12, NAT_1:13, XXREAL_0:2; ::_thesis: verum end; end; end; A14: (j + 1) + 1 <= len f by A12, NAT_1:13; A15: now__::_thesis:_(f_/._(j_+_1))_`1_<_(f_/._j)_`1 percases ( 1 + 1 = j + 1 or 2 < j + 1 ) by A11, XXREAL_0:1; suppose 1 + 1 = j + 1 ; ::_thesis: (f /. (j + 1)) `1 < (f /. j) `1 hence (f /. (j + 1)) `1 < (f /. j) `1 by A8; ::_thesis: verum end; suppose 2 < j + 1 ; ::_thesis: (f /. (j + 1)) `1 < (f /. j) `1 hence (f /. (j + 1)) `1 < (f /. j) `1 by A10, A12, NAT_1:13; ::_thesis: verum end; end; end; A16: 1 <= j + 1 by NAT_1:11; then A17: j + 1 in dom f by A12, FINSEQ_3:25; then A18: (f /. (j + 1)) `2 = (f /. 1) `2 by A1; j < len f by A12, NAT_1:13; then A19: j in dom f by A13, FINSEQ_3:25; then A20: (f /. j) `2 = (f /. 1) `2 by A1; 1 <= (j + 1) + 1 by NAT_1:11; then A21: (j + 1) + 1 in dom f by A14, FINSEQ_3:25; then A22: (f /. ((j + 1) + 1)) `2 = (f /. 1) `2 by A1; assume A23: (f /. ((j + 1) + 1)) `1 >= (f /. (j + 1)) `1 ; ::_thesis: contradiction percases ( (f /. ((j + 1) + 1)) `1 > (f /. (j + 1)) `1 or (f /. ((j + 1) + 1)) `1 = (f /. (j + 1)) `1 ) by A23, XXREAL_0:1; supposeA24: (f /. ((j + 1) + 1)) `1 > (f /. (j + 1)) `1 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( (f /. j) `1 <= (f /. ((j + 1) + 1)) `1 or (f /. j) `1 >= (f /. ((j + 1) + 1)) `1 ) ; suppose (f /. j) `1 <= (f /. ((j + 1) + 1)) `1 ; ::_thesis: contradiction then f /. j in LSeg ((f /. (j + 1)),(f /. ((j + 1) + 1))) by A15, A20, A18, A22, Th8; then A25: f /. j in LSeg (f,(j + 1)) by A14, A16, TOPREAL1:def_3; (j + 1) + 1 = j + (1 + 1) ; then A26: (LSeg (f,j)) /\ (LSeg (f,(j + 1))) = {(f /. (j + 1))} by A14, A13, TOPREAL1:def_6; f /. j in LSeg (f,j) by A12, A13, TOPREAL1:21; then f /. j in (LSeg (f,j)) /\ (LSeg (f,(j + 1))) by A25, XBOOLE_0:def_4; then f /. j = f /. (j + 1) by A26, TARSKI:def_1; hence contradiction by A19, A17, Th29; ::_thesis: verum end; suppose (f /. j) `1 >= (f /. ((j + 1) + 1)) `1 ; ::_thesis: contradiction then f /. ((j + 1) + 1) in LSeg ((f /. j),(f /. (j + 1))) by A20, A18, A22, A24, Th8; then A27: f /. ((j + 1) + 1) in LSeg (f,j) by A12, A13, TOPREAL1:def_3; (j + 1) + 1 = j + (1 + 1) ; then A28: (LSeg (f,j)) /\ (LSeg (f,(j + 1))) = {(f /. (j + 1))} by A14, A13, TOPREAL1:def_6; f /. ((j + 1) + 1) in LSeg (f,(j + 1)) by A14, A16, TOPREAL1:21; then f /. ((j + 1) + 1) in (LSeg (f,j)) /\ (LSeg (f,(j + 1))) by A27, XBOOLE_0:def_4; then f /. ((j + 1) + 1) = f /. (j + 1) by A28, TARSKI:def_1; hence contradiction by A17, A21, Th29; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA29: (f /. ((j + 1) + 1)) `1 = (f /. (j + 1)) `1 ; ::_thesis: contradiction (f /. ((j + 1) + 1)) `2 = (f /. 1) `2 by A1, A21 .= (f /. (j + 1)) `2 by A1, A17 ; then f /. ((j + 1) + 1) = |[((f /. (j + 1)) `1),((f /. (j + 1)) `2)]| by A29, EUCLID:53 .= f /. (j + 1) by EUCLID:53 ; hence contradiction by A17, A21, Th29; ::_thesis: verum end; end; end; A30: ((len f) -' 1) + 1 = len f by A2, XREAL_1:235; then A31: ( 2 <= (len f) -' 1 & (len f) -' 1 < len f ) by A7, NAT_1:13; A32: S1[ 0 ] ; A33: for j being Element of NAT holds S1[j] from NAT_1:sch_1(A32, A9); then A34: (f /. ((len f) -' 1)) `1 <= (f /. 2) `1 by A31; (f /. (len f)) `1 < (f /. ((len f) -' 1)) `1 by A33, A30, A31; then (f /. (len f)) `1 < (f /. 2) `1 by A34, XXREAL_0:2; hence contradiction by A8, FINSEQ_6:def_1; ::_thesis: verum end; supposeA35: (f /. 2) `1 > (f /. 1) `1 ; ::_thesis: contradiction defpred S1[ Element of NAT ] means ( 2 <= $1 & $1 < len f implies ( (f /. $1) `1 >= (f /. 2) `1 & (f /. ($1 + 1)) `1 > (f /. $1) `1 ) ); A36: for j being Element of NAT st S1[j] holds S1[j + 1] proof let j be Element of NAT ; ::_thesis: ( S1[j] implies S1[j + 1] ) assume that A37: ( 2 <= j & j < len f implies ( (f /. j) `1 >= (f /. 2) `1 & (f /. (j + 1)) `1 > (f /. j) `1 ) ) and A38: 2 <= j + 1 and A39: j + 1 < len f ; ::_thesis: ( (f /. (j + 1)) `1 >= (f /. 2) `1 & (f /. ((j + 1) + 1)) `1 > (f /. (j + 1)) `1 ) 1 + 1 <= j + 1 by A38; then A40: 1 <= j by XREAL_1:6; thus (f /. (j + 1)) `1 >= (f /. 2) `1 ::_thesis: (f /. ((j + 1) + 1)) `1 > (f /. (j + 1)) `1 proof percases ( 2 = j + 1 or 2 < j + 1 ) by A38, XXREAL_0:1; suppose 2 = j + 1 ; ::_thesis: (f /. (j + 1)) `1 >= (f /. 2) `1 hence (f /. (j + 1)) `1 >= (f /. 2) `1 ; ::_thesis: verum end; suppose 2 < j + 1 ; ::_thesis: (f /. (j + 1)) `1 >= (f /. 2) `1 hence (f /. (j + 1)) `1 >= (f /. 2) `1 by A37, A39, NAT_1:13, XXREAL_0:2; ::_thesis: verum end; end; end; A41: (j + 1) + 1 <= len f by A39, NAT_1:13; A42: now__::_thesis:_(f_/._(j_+_1))_`1_>_(f_/._j)_`1 percases ( 1 + 1 = j + 1 or 2 < j + 1 ) by A38, XXREAL_0:1; suppose 1 + 1 = j + 1 ; ::_thesis: (f /. (j + 1)) `1 > (f /. j) `1 hence (f /. (j + 1)) `1 > (f /. j) `1 by A35; ::_thesis: verum end; suppose 2 < j + 1 ; ::_thesis: (f /. (j + 1)) `1 > (f /. j) `1 hence (f /. (j + 1)) `1 > (f /. j) `1 by A37, A39, NAT_1:13; ::_thesis: verum end; end; end; A43: 1 <= j + 1 by NAT_1:11; then A44: j + 1 in dom f by A39, FINSEQ_3:25; then A45: (f /. (j + 1)) `2 = (f /. 1) `2 by A1; j < len f by A39, NAT_1:13; then A46: j in dom f by A40, FINSEQ_3:25; then A47: (f /. j) `2 = (f /. 1) `2 by A1; 1 <= (j + 1) + 1 by NAT_1:11; then A48: (j + 1) + 1 in dom f by A41, FINSEQ_3:25; then A49: (f /. ((j + 1) + 1)) `2 = (f /. 1) `2 by A1; assume A50: (f /. ((j + 1) + 1)) `1 <= (f /. (j + 1)) `1 ; ::_thesis: contradiction percases ( (f /. ((j + 1) + 1)) `1 < (f /. (j + 1)) `1 or (f /. ((j + 1) + 1)) `1 = (f /. (j + 1)) `1 ) by A50, XXREAL_0:1; supposeA51: (f /. ((j + 1) + 1)) `1 < (f /. (j + 1)) `1 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( (f /. j) `1 >= (f /. ((j + 1) + 1)) `1 or (f /. j) `1 <= (f /. ((j + 1) + 1)) `1 ) ; suppose (f /. j) `1 >= (f /. ((j + 1) + 1)) `1 ; ::_thesis: contradiction then f /. j in LSeg ((f /. (j + 1)),(f /. ((j + 1) + 1))) by A42, A47, A45, A49, Th8; then A52: f /. j in LSeg (f,(j + 1)) by A41, A43, TOPREAL1:def_3; (j + 1) + 1 = j + (1 + 1) ; then A53: (LSeg (f,j)) /\ (LSeg (f,(j + 1))) = {(f /. (j + 1))} by A41, A40, TOPREAL1:def_6; f /. j in LSeg (f,j) by A39, A40, TOPREAL1:21; then f /. j in (LSeg (f,j)) /\ (LSeg (f,(j + 1))) by A52, XBOOLE_0:def_4; then f /. j = f /. (j + 1) by A53, TARSKI:def_1; hence contradiction by A46, A44, Th29; ::_thesis: verum end; suppose (f /. j) `1 <= (f /. ((j + 1) + 1)) `1 ; ::_thesis: contradiction then f /. ((j + 1) + 1) in LSeg ((f /. j),(f /. (j + 1))) by A47, A45, A49, A51, Th8; then A54: f /. ((j + 1) + 1) in LSeg (f,j) by A39, A40, TOPREAL1:def_3; (j + 1) + 1 = j + (1 + 1) ; then A55: (LSeg (f,j)) /\ (LSeg (f,(j + 1))) = {(f /. (j + 1))} by A41, A40, TOPREAL1:def_6; f /. ((j + 1) + 1) in LSeg (f,(j + 1)) by A41, A43, TOPREAL1:21; then f /. ((j + 1) + 1) in (LSeg (f,j)) /\ (LSeg (f,(j + 1))) by A54, XBOOLE_0:def_4; then f /. ((j + 1) + 1) = f /. (j + 1) by A55, TARSKI:def_1; hence contradiction by A44, A48, Th29; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA56: (f /. ((j + 1) + 1)) `1 = (f /. (j + 1)) `1 ; ::_thesis: contradiction (f /. ((j + 1) + 1)) `2 = (f /. 1) `2 by A1, A48 .= (f /. (j + 1)) `2 by A1, A44 ; then f /. ((j + 1) + 1) = |[((f /. (j + 1)) `1),((f /. (j + 1)) `2)]| by A56, EUCLID:53 .= f /. (j + 1) by EUCLID:53 ; hence contradiction by A44, A48, Th29; ::_thesis: verum end; end; end; A57: ((len f) -' 1) + 1 = len f by A2, XREAL_1:235; then A58: ( 2 <= (len f) -' 1 & (len f) -' 1 < len f ) by A7, NAT_1:13; A59: S1[ 0 ] ; A60: for j being Element of NAT holds S1[j] from NAT_1:sch_1(A59, A36); then A61: (f /. ((len f) -' 1)) `1 >= (f /. 2) `1 by A58; (f /. (len f)) `1 > (f /. ((len f) -' 1)) `1 by A60, A57, A58; then (f /. (len f)) `1 > (f /. 2) `1 by A61, XXREAL_0:2; hence contradiction by A35, FINSEQ_6:def_1; ::_thesis: verum end; end; end; theorem :: GOBOARD7:32 for f being non constant standard special_circular_sequence holds len (GoB f) > 1 proof let f be non constant standard special_circular_sequence; ::_thesis: len (GoB f) > 1 A1: len (GoB f) <> 0 by GOBOARD1:def_3; 1 in dom f by FINSEQ_5:6; then consider i2, j2 being Element of NAT such that A2: [i2,j2] in Indices (GoB f) and A3: f /. 1 = (GoB f) * (i2,j2) by GOBOARD2:14; A4: 1 <= i2 by A2, MATRIX_1:38; assume len (GoB f) <= 1 ; ::_thesis: contradiction then A5: len (GoB f) = 1 by A1, NAT_1:25; then i2 <= 1 by A2, MATRIX_1:38; then A6: i2 = 1 by A4, XXREAL_0:1; consider i being Element of NAT such that A7: i in dom f and A8: (f /. i) `1 <> (f /. 1) `1 by Th30; consider i1, j1 being Element of NAT such that A9: [i1,j1] in Indices (GoB f) and A10: f /. i = (GoB f) * (i1,j1) by A7, GOBOARD2:14; A11: ( 1 <= j1 & j1 <= width (GoB f) ) by A9, MATRIX_1:38; A12: 1 <= i1 by A9, MATRIX_1:38; i1 <= 1 by A5, A9, MATRIX_1:38; then i1 = 1 by A12, XXREAL_0:1; then A13: ((GoB f) * (i1,j1)) `1 = ((GoB f) * (1,1)) `1 by A5, A11, GOBOARD5:2; ( 1 <= j2 & j2 <= width (GoB f) ) by A2, MATRIX_1:38; hence contradiction by A5, A8, A10, A3, A13, A6, GOBOARD5:2; ::_thesis: verum end; theorem :: GOBOARD7:33 for f being non constant standard special_circular_sequence holds width (GoB f) > 1 proof let f be non constant standard special_circular_sequence; ::_thesis: width (GoB f) > 1 A1: width (GoB f) <> 0 by GOBOARD1:def_3; 1 in dom f by FINSEQ_5:6; then consider i2, j2 being Element of NAT such that A2: [i2,j2] in Indices (GoB f) and A3: f /. 1 = (GoB f) * (i2,j2) by GOBOARD2:14; A4: 1 <= j2 by A2, MATRIX_1:38; assume width (GoB f) <= 1 ; ::_thesis: contradiction then A5: width (GoB f) = 1 by A1, NAT_1:25; then j2 <= 1 by A2, MATRIX_1:38; then A6: j2 = 1 by A4, XXREAL_0:1; consider i being Element of NAT such that A7: i in dom f and A8: (f /. i) `2 <> (f /. 1) `2 by Th31; consider i1, j1 being Element of NAT such that A9: [i1,j1] in Indices (GoB f) and A10: f /. i = (GoB f) * (i1,j1) by A7, GOBOARD2:14; A11: ( 1 <= i1 & i1 <= len (GoB f) ) by A9, MATRIX_1:38; A12: 1 <= j1 by A9, MATRIX_1:38; j1 <= 1 by A5, A9, MATRIX_1:38; then j1 = 1 by A12, XXREAL_0:1; then A13: ((GoB f) * (i1,j1)) `2 = ((GoB f) * (1,1)) `2 by A5, A11, GOBOARD5:1; ( 1 <= i2 & i2 <= len (GoB f) ) by A2, MATRIX_1:38; hence contradiction by A5, A8, A10, A3, A13, A6, GOBOARD5:1; ::_thesis: verum end; theorem Th34: :: GOBOARD7:34 for f being non constant standard special_circular_sequence holds len f > 4 proof let f be non constant standard special_circular_sequence; ::_thesis: len f > 4 assume A1: len f <= 4 ; ::_thesis: contradiction A2: len f > 1 by Lm2; then A3: 1 in dom f by FINSEQ_3:25; A4: len f >= 1 + 1 by A2, NAT_1:13; then A5: 2 in dom f by FINSEQ_3:25; consider i2 being Element of NAT such that A6: i2 in dom f and A7: (f /. i2) `2 <> (f /. 1) `2 by Th31; consider i1 being Element of NAT such that A8: i1 in dom f and A9: (f /. i1) `1 <> (f /. 1) `1 by Th30; percases ( (f /. (1 + 1)) `1 = (f /. 1) `1 or (f /. (1 + 1)) `2 = (f /. 1) `2 ) by A4, TOPREAL1:def_5; supposeA10: (f /. (1 + 1)) `1 = (f /. 1) `1 ; ::_thesis: contradiction A11: i1 <= len f by A8, FINSEQ_3:25; A12: f /. (len f) = f /. 1 by FINSEQ_6:def_1; A13: i1 <> 0 by A8, FINSEQ_3:25; now__::_thesis:_contradiction percases ( i1 = 3 or i1 = 4 ) by A1, A9, A10, A11, A13, NAT_1:28, XXREAL_0:2; supposeA14: i1 = 3 ; ::_thesis: contradiction A15: now__::_thesis:_not_(f_/._(1_+_1))_`2_=_(f_/._1)_`2 assume (f /. (1 + 1)) `2 = (f /. 1) `2 ; ::_thesis: contradiction then f /. (1 + 1) = |[((f /. 1) `1),((f /. 1) `2)]| by A10, EUCLID:53 .= f /. 1 by EUCLID:53 ; hence contradiction by A3, A5, Th29; ::_thesis: verum end; A16: len f >= 3 by A8, A14, FINSEQ_3:25; then len f > 3 by A9, A12, A14, XXREAL_0:1; then A17: len f >= 3 + 1 by NAT_1:13; then A18: ( (f /. 3) `1 = (f /. (3 + 1)) `1 or (f /. 3) `2 = (f /. (3 + 1)) `2 ) by TOPREAL1:def_5; A19: len f = 4 by A1, A17, XXREAL_0:1; (f /. 2) `2 = (f /. (2 + 1)) `2 by A9, A10, A14, A16, TOPREAL1:def_5; hence contradiction by A9, A14, A19, A15, A18, FINSEQ_6:def_1; ::_thesis: verum end; suppose i1 = 4 ; ::_thesis: contradiction hence contradiction by A1, A9, A11, A12, XXREAL_0:1; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; supposeA20: (f /. (1 + 1)) `2 = (f /. 1) `2 ; ::_thesis: contradiction A21: i2 <= len f by A6, FINSEQ_3:25; A22: f /. (len f) = f /. 1 by FINSEQ_6:def_1; A23: i2 <> 0 by A6, FINSEQ_3:25; now__::_thesis:_contradiction percases ( i2 = 3 or i2 = 4 ) by A1, A7, A20, A21, A23, NAT_1:28, XXREAL_0:2; supposeA24: i2 = 3 ; ::_thesis: contradiction A25: now__::_thesis:_not_(f_/._(1_+_1))_`1_=_(f_/._1)_`1 assume (f /. (1 + 1)) `1 = (f /. 1) `1 ; ::_thesis: contradiction then f /. (1 + 1) = |[((f /. 1) `1),((f /. 1) `2)]| by A20, EUCLID:53 .= f /. 1 by EUCLID:53 ; hence contradiction by A3, A5, Th29; ::_thesis: verum end; A26: len f >= 3 by A6, A24, FINSEQ_3:25; then len f > 3 by A7, A22, A24, XXREAL_0:1; then A27: len f >= 3 + 1 by NAT_1:13; then A28: ( (f /. 3) `2 = (f /. (3 + 1)) `2 or (f /. 3) `1 = (f /. (3 + 1)) `1 ) by TOPREAL1:def_5; A29: len f = 4 by A1, A27, XXREAL_0:1; (f /. 2) `1 = (f /. (2 + 1)) `1 by A7, A20, A24, A26, TOPREAL1:def_5; hence contradiction by A7, A24, A29, A25, A28, FINSEQ_6:def_1; ::_thesis: verum end; suppose i2 = 4 ; ::_thesis: contradiction hence contradiction by A1, A7, A21, A22, XXREAL_0:1; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; theorem Th35: :: GOBOARD7:35 for f being circular s.c.c. FinSequence of (TOP-REAL 2) st len f > 4 holds for i, j being Element of NAT st 1 <= i & i < j & j < len f holds f /. i <> f /. j proof let f be circular s.c.c. FinSequence of (TOP-REAL 2); ::_thesis: ( len f > 4 implies for i, j being Element of NAT st 1 <= i & i < j & j < len f holds f /. i <> f /. j ) assume A1: len f > 4 ; ::_thesis: for i, j being Element of NAT st 1 <= i & i < j & j < len f holds f /. i <> f /. j let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i < j & j < len f implies f /. i <> f /. j ) assume that A2: 1 <= i and A3: i < j and A4: j < len f and A5: f /. i = f /. j ; ::_thesis: contradiction A6: j + 1 <= len f by A4, NAT_1:13; A7: ( i + 1 <= j & i <> 0 ) by A2, A3, NAT_1:13; 1 <= j by A2, A3, XXREAL_0:2; then A8: f /. j in LSeg (f,j) by A6, TOPREAL1:21; A9: i < len f by A3, A4, XXREAL_0:2; then i + 1 <= len f by NAT_1:13; then A10: f /. i in LSeg (f,i) by A2, TOPREAL1:21; percases ( ( i + 1 = j & i = 1 ) or ( i + 1 = j & i = 1 + 1 ) or i > 1 + 1 or ( i + 1 < j & i <> 1 ) or ( i + 1 < j & j + 1 <> len f ) or ( i + 1 < j & i = 1 & j + 1 = len f ) ) by A7, NAT_1:26, XXREAL_0:1; supposethat A11: i + 1 = j and A12: i = 1 ; ::_thesis: contradiction A13: ((len f) -' 1) + 1 = len f by A1, XREAL_1:235, XXREAL_0:2; (j + 1) + 1 < len f by A1, A11, A12; then A14: j + 1 < (len f) -' 1 by A13, XREAL_1:6; (len f) -' 1 < len f by A13, XREAL_1:29; then LSeg (f,j) misses LSeg (f,((len f) -' 1)) by A11, A12, A14, GOBOARD5:def_4; then A15: (LSeg (f,j)) /\ (LSeg (f,((len f) -' 1))) = {} by XBOOLE_0:def_7; A16: f /. i = f /. (len f) by A12, FINSEQ_6:def_1; 1 + 1 <= len f by A1, XXREAL_0:2; then 1 <= (len f) -' 1 by A13, XREAL_1:6; then f /. i in LSeg (f,((len f) -' 1)) by A13, A16, TOPREAL1:21; hence contradiction by A5, A8, A15, XBOOLE_0:def_4; ::_thesis: verum end; supposethat A17: i + 1 = j and A18: i = 1 + 1 ; ::_thesis: contradiction A19: (i -' 1) + 1 = i by A2, XREAL_1:235; j + 1 < len f by A1, A17, A18; then LSeg (f,(i -' 1)) misses LSeg (f,j) by A3, A19, GOBOARD5:def_4; then A20: (LSeg (f,(i -' 1))) /\ (LSeg (f,j)) = {} by XBOOLE_0:def_7; f /. i in LSeg (f,(i -' 1)) by A9, A18, A19, TOPREAL1:21; hence contradiction by A5, A8, A20, XBOOLE_0:def_4; ::_thesis: verum end; supposeA21: i > 1 + 1 ; ::_thesis: contradiction A22: (i -' 1) + 1 = i by A2, XREAL_1:235; then A23: 1 < i -' 1 by A21, XREAL_1:6; then LSeg (f,(i -' 1)) misses LSeg (f,j) by A3, A4, A22, GOBOARD5:def_4; then A24: (LSeg (f,(i -' 1))) /\ (LSeg (f,j)) = {} by XBOOLE_0:def_7; f /. i in LSeg (f,(i -' 1)) by A9, A22, A23, TOPREAL1:21; hence contradiction by A5, A8, A24, XBOOLE_0:def_4; ::_thesis: verum end; supposethat A25: i + 1 < j and A26: i <> 1 ; ::_thesis: contradiction 1 < i by A2, A26, XXREAL_0:1; then LSeg (f,i) misses LSeg (f,j) by A4, A25, GOBOARD5:def_4; then (LSeg (f,i)) /\ (LSeg (f,j)) = {} by XBOOLE_0:def_7; hence contradiction by A5, A8, A10, XBOOLE_0:def_4; ::_thesis: verum end; supposethat A27: i + 1 < j and A28: j + 1 <> len f ; ::_thesis: contradiction j + 1 < len f by A6, A28, XXREAL_0:1; then LSeg (f,i) misses LSeg (f,j) by A27, GOBOARD5:def_4; then (LSeg (f,i)) /\ (LSeg (f,j)) = {} by XBOOLE_0:def_7; hence contradiction by A5, A8, A10, XBOOLE_0:def_4; ::_thesis: verum end; supposethat A29: i + 1 < j and A30: i = 1 and A31: j + 1 = len f ; ::_thesis: contradiction A32: j < len f by A31, NAT_1:13; A33: (j -' 1) + 1 = j by A2, A3, XREAL_1:235, XXREAL_0:2; then A34: i + 1 <= j -' 1 by A29, NAT_1:13; i + 1 <> j -' 1 by A1, A30, A31, A33; then i + 1 < j -' 1 by A34, XXREAL_0:1; then LSeg (f,1) misses LSeg (f,(j -' 1)) by A30, A33, A32, GOBOARD5:def_4; then A35: (LSeg (f,1)) /\ (LSeg (f,(j -' 1))) = {} by XBOOLE_0:def_7; 1 <= j -' 1 by A30, A34, XXREAL_0:2; then f /. j in LSeg (f,(j -' 1)) by A4, A33, TOPREAL1:21; hence contradiction by A5, A10, A30, A35, XBOOLE_0:def_4; ::_thesis: verum end; end; end; theorem Th36: :: GOBOARD7:36 for f being non constant standard special_circular_sequence for i, j being Element of NAT st 1 <= i & i < j & j < len f holds f /. i <> f /. j proof let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st 1 <= i & i < j & j < len f holds f /. i <> f /. j len f > 4 by Th34; hence for i, j being Element of NAT st 1 <= i & i < j & j < len f holds f /. i <> f /. j by Th35; ::_thesis: verum end; theorem Th37: :: GOBOARD7:37 for f being non constant standard special_circular_sequence for i, j being Element of NAT st 1 < i & i < j & j <= len f holds f /. i <> f /. j proof let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st 1 < i & i < j & j <= len f holds f /. i <> f /. j let i, j be Element of NAT ; ::_thesis: ( 1 < i & i < j & j <= len f implies f /. i <> f /. j ) assume that A1: 1 < i and A2: i < j and A3: j <= len f ; ::_thesis: f /. i <> f /. j percases ( j < len f or j = len f ) by A3, XXREAL_0:1; suppose j < len f ; ::_thesis: f /. i <> f /. j hence f /. i <> f /. j by A1, A2, Th36; ::_thesis: verum end; suppose j = len f ; ::_thesis: f /. i <> f /. j then A4: f /. j = f /. 1 by FINSEQ_6:def_1; i < len f by A2, A3, XXREAL_0:2; hence f /. i <> f /. j by A1, A4, Th36; ::_thesis: verum end; end; end; theorem Th38: :: GOBOARD7:38 for f being non constant standard special_circular_sequence for i being Element of NAT st 1 < i & i <= len f & f /. i = f /. 1 holds i = len f proof let f be non constant standard special_circular_sequence; ::_thesis: for i being Element of NAT st 1 < i & i <= len f & f /. i = f /. 1 holds i = len f let i be Element of NAT ; ::_thesis: ( 1 < i & i <= len f & f /. i = f /. 1 implies i = len f ) assume that A1: 1 < i and A2: i <= len f and A3: f /. i = f /. 1 ; ::_thesis: i = len f assume i <> len f ; ::_thesis: contradiction then i < len f by A2, XXREAL_0:1; hence contradiction by A1, A3, Th36; ::_thesis: verum end; theorem Th39: :: GOBOARD7:39 for i, j being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))) in L~ f holds ex k being Element of NAT st ( 1 <= k & k + 1 <= len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) ) proof let i, j be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))) in L~ f holds ex k being Element of NAT st ( 1 <= k & k + 1 <= len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))) in L~ f implies ex k being Element of NAT st ( 1 <= k & k + 1 <= len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) ) ) set mi = (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))); assume that A1: ( 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) ) and A2: (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))) in L~ f ; ::_thesis: ex k being Element of NAT st ( 1 <= k & k + 1 <= len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) ) L~ f = union { (LSeg (f,k)) where k is Element of NAT : ( 1 <= k & k + 1 <= len f ) } by TOPREAL1:def_4; then consider x being set such that A3: (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))) in x and A4: x in { (LSeg (f,k)) where k is Element of NAT : ( 1 <= k & k + 1 <= len f ) } by A2, TARSKI:def_4; consider k being Element of NAT such that A5: x = LSeg (f,k) and A6: 1 <= k and A7: k + 1 <= len f by A4; A8: f is_sequence_on GoB f by GOBOARD5:def_5; A9: (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))) in LSeg ((f /. k),(f /. (k + 1))) by A3, A5, A6, A7, TOPREAL1:def_3; k <= k + 1 by NAT_1:11; then k <= len f by A7, XXREAL_0:2; then A10: k in dom f by A6, FINSEQ_3:25; then consider i1, j1 being Element of NAT such that A11: [i1,j1] in Indices (GoB f) and A12: f /. k = (GoB f) * (i1,j1) by A8, GOBOARD1:def_9; A13: 1 <= i1 by A11, MATRIX_1:38; take k ; ::_thesis: ( 1 <= k & k + 1 <= len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) ) thus ( 1 <= k & k + 1 <= len f ) by A6, A7; ::_thesis: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) 1 <= k + 1 by NAT_1:11; then A14: k + 1 in dom f by A7, FINSEQ_3:25; then consider i2, j2 being Element of NAT such that A15: [i2,j2] in Indices (GoB f) and A16: f /. (k + 1) = (GoB f) * (i2,j2) by A8, GOBOARD1:def_9; A17: 1 <= i2 by A15, MATRIX_1:38; A18: j2 <= width (GoB f) by A15, MATRIX_1:38; (abs (i1 - i2)) + (abs (j1 - j2)) = 1 by A8, A10, A11, A12, A14, A15, A16, GOBOARD1:def_9; then A19: ( ( abs (i1 - i2) = 1 & j1 = j2 ) or ( abs (j1 - j2) = 1 & i1 = i2 ) ) by SEQM_3:42; A20: i1 <= len (GoB f) by A11, MATRIX_1:38; A21: j1 <= width (GoB f) by A11, MATRIX_1:38; A22: 1 <= j1 by A11, MATRIX_1:38; A23: i2 <= len (GoB f) by A15, MATRIX_1:38; A24: 1 <= j2 by A15, MATRIX_1:38; percases ( ( j1 = j2 & i1 = i2 + 1 ) or ( j1 = j2 & i1 + 1 = i2 ) or ( j1 = j2 + 1 & i1 = i2 ) or ( j1 + 1 = j2 & i1 = i2 ) ) by A19, SEQM_3:41; supposeA25: ( j1 = j2 & i1 = i2 + 1 ) ; ::_thesis: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) then (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))) in LSeg (((GoB f) * (i2,j2)),((GoB f) * ((i2 + 1),j2))) by A3, A5, A6, A7, A12, A16, TOPREAL1:def_3; hence LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) by A1, A20, A17, A24, A18, A25, Th28; ::_thesis: verum end; supposeA26: ( j1 = j2 & i1 + 1 = i2 ) ; ::_thesis: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) then (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))) in LSeg (((GoB f) * (i1,j1)),((GoB f) * ((i1 + 1),j1))) by A3, A5, A6, A7, A12, A16, TOPREAL1:def_3; hence LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) by A1, A13, A22, A21, A23, A26, Th28; ::_thesis: verum end; supposeA27: ( j1 = j2 + 1 & i1 = i2 ) ; ::_thesis: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) then ( i = i2 & j = j2 ) by A1, A12, A16, A13, A20, A21, A24, A9, Th25; hence LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) by A6, A7, A12, A16, A27, TOPREAL1:def_3; ::_thesis: verum end; supposeA28: ( j1 + 1 = j2 & i1 = i2 ) ; ::_thesis: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) then ( i = i1 & j = j1 ) by A1, A12, A16, A13, A20, A22, A18, A9, Th25; hence LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) by A6, A7, A12, A16, A28, TOPREAL1:def_3; ::_thesis: verum end; end; end; theorem Th40: :: GOBOARD7:40 for i, j being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j <= width (GoB f) & (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))) in L~ f holds ex k being Element of NAT st ( 1 <= k & k + 1 <= len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) ) proof let i, j be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j <= width (GoB f) & (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))) in L~ f holds ex k being Element of NAT st ( 1 <= k & k + 1 <= len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j <= width (GoB f) & (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))) in L~ f implies ex k being Element of NAT st ( 1 <= k & k + 1 <= len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) ) ) set mi = (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))); assume that A1: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j <= width (GoB f) ) and A2: (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))) in L~ f ; ::_thesis: ex k being Element of NAT st ( 1 <= k & k + 1 <= len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) ) L~ f = union { (LSeg (f,k)) where k is Element of NAT : ( 1 <= k & k + 1 <= len f ) } by TOPREAL1:def_4; then consider x being set such that A3: (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))) in x and A4: x in { (LSeg (f,k)) where k is Element of NAT : ( 1 <= k & k + 1 <= len f ) } by A2, TARSKI:def_4; consider k being Element of NAT such that A5: x = LSeg (f,k) and A6: 1 <= k and A7: k + 1 <= len f by A4; A8: f is_sequence_on GoB f by GOBOARD5:def_5; A9: (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))) in LSeg ((f /. k),(f /. (k + 1))) by A3, A5, A6, A7, TOPREAL1:def_3; k <= k + 1 by NAT_1:11; then k <= len f by A7, XXREAL_0:2; then A10: k in dom f by A6, FINSEQ_3:25; then consider i1, j1 being Element of NAT such that A11: [i1,j1] in Indices (GoB f) and A12: f /. k = (GoB f) * (i1,j1) by A8, GOBOARD1:def_9; A13: 1 <= j1 by A11, MATRIX_1:38; take k ; ::_thesis: ( 1 <= k & k + 1 <= len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) ) thus ( 1 <= k & k + 1 <= len f ) by A6, A7; ::_thesis: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) 1 <= k + 1 by NAT_1:11; then A14: k + 1 in dom f by A7, FINSEQ_3:25; then consider i2, j2 being Element of NAT such that A15: [i2,j2] in Indices (GoB f) and A16: f /. (k + 1) = (GoB f) * (i2,j2) by A8, GOBOARD1:def_9; A17: 1 <= j2 by A15, MATRIX_1:38; A18: i2 <= len (GoB f) by A15, MATRIX_1:38; (abs (j1 - j2)) + (abs (i1 - i2)) = 1 by A8, A10, A11, A12, A14, A15, A16, GOBOARD1:def_9; then A19: ( ( abs (j1 - j2) = 1 & i1 = i2 ) or ( abs (i1 - i2) = 1 & j1 = j2 ) ) by SEQM_3:42; A20: j1 <= width (GoB f) by A11, MATRIX_1:38; A21: i1 <= len (GoB f) by A11, MATRIX_1:38; A22: 1 <= i1 by A11, MATRIX_1:38; A23: j2 <= width (GoB f) by A15, MATRIX_1:38; A24: 1 <= i2 by A15, MATRIX_1:38; percases ( ( i1 = i2 & j1 = j2 + 1 ) or ( i1 = i2 & j1 + 1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) ) by A19, SEQM_3:41; supposeA25: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) then (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))) in LSeg (((GoB f) * (i2,j2)),((GoB f) * (i2,(j2 + 1)))) by A3, A5, A6, A7, A12, A16, TOPREAL1:def_3; hence LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) by A1, A20, A17, A24, A18, A25, Th27; ::_thesis: verum end; supposeA26: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) then (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))) in LSeg (((GoB f) * (i1,j1)),((GoB f) * (i1,(j1 + 1)))) by A3, A5, A6, A7, A12, A16, TOPREAL1:def_3; hence LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) by A1, A13, A22, A21, A23, A26, Th27; ::_thesis: verum end; supposeA27: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) then ( j = j2 & i = i2 ) by A1, A12, A16, A13, A20, A21, A24, A9, Th26; hence LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) by A6, A7, A12, A16, A27, TOPREAL1:def_3; ::_thesis: verum end; supposeA28: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) then ( j = j1 & i = i1 ) by A1, A12, A16, A13, A20, A22, A18, A9, Th26; hence LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) by A6, A7, A12, A16, A28, TOPREAL1:def_3; ::_thesis: verum end; end; end; theorem :: GOBOARD7:41 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: 1 <= k and A6: k + 1 < len f and A7: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) and A8: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) A9: 1 <= j + 1 by NAT_1:11; A10: i < i + 1 by NAT_1:13; A11: 1 <= i + 1 by NAT_1:11; j < width (GoB f) by A4, NAT_1:13; then ((GoB f) * ((i + 1),j)) `1 = ((GoB f) * ((i + 1),1)) `1 by A2, A3, A11, GOBOARD5:2 .= ((GoB f) * ((i + 1),(j + 1))) `1 by A2, A4, A9, A11, GOBOARD5:2 ; then A12: (GoB f) * (i,(j + 1)) <> (GoB f) * ((i + 1),j) by A1, A2, A4, A9, A10, GOBOARD5:3; A13: 1 <= k + 1 by NAT_1:11; A14: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A6, NAT_1:13; then A15: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A8, A14, A13, TOPREAL1:def_3; then A16: ( ( (GoB f) * ((i + 1),j) = f /. (k + 2) & (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. (k + 2) ) ) by SPPOL_1:8; A17: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. k),(f /. (k + 1))) by A5, A6, A7, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) & (GoB f) * (i,(j + 1)) = f /. k ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. k & (GoB f) * (i,(j + 1)) = f /. (k + 1) ) ) by SPPOL_1:8; hence f /. k = (GoB f) * (i,(j + 1)) by A15, A12, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) thus f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) by A17, A16, A12, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * ((i + 1),j) thus f /. (k + 2) = (GoB f) * ((i + 1),j) by A17, A16, A12, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:42 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 < width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 < width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 < width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) assume that A1: ( 1 <= i & i <= len (GoB f) & 1 <= j ) and A2: j + 1 < width (GoB f) and A3: 1 <= k and A4: k + 1 < len f and A5: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg (f,k) and A6: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) A7: 1 <= k + 1 by NAT_1:11; A8: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A4, NAT_1:13; then A9: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A6, A8, A7, TOPREAL1:def_3; then A10: ( ( (GoB f) * (i,j) = f /. (k + 2) & (GoB f) * (i,(j + 1)) = f /. (k + 1) ) or ( (GoB f) * (i,j) = f /. (k + 1) & (GoB f) * (i,(j + 1)) = f /. (k + 2) ) ) by SPPOL_1:8; A11: j < j + 2 by XREAL_1:29; j + (1 + 1) = (j + 1) + 1 ; then j + 2 <= width (GoB f) by A2, NAT_1:13; then A12: ((GoB f) * (i,j)) `2 < ((GoB f) * (i,(j + 2))) `2 by A1, A11, GOBOARD5:4; A13: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg ((f /. k),(f /. (k + 1))) by A3, A4, A5, TOPREAL1:def_3; then ( ( (GoB f) * (i,(j + 1)) = f /. (k + 1) & (GoB f) * (i,(j + 2)) = f /. k ) or ( (GoB f) * (i,(j + 1)) = f /. k & (GoB f) * (i,(j + 2)) = f /. (k + 1) ) ) by SPPOL_1:8; hence f /. k = (GoB f) * (i,(j + 2)) by A9, A12, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) thus f /. (k + 1) = (GoB f) * (i,(j + 1)) by A13, A10, A12, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * (i,j) thus f /. (k + 2) = (GoB f) * (i,j) by A13, A10, A12, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:43 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: 1 <= k and A6: k + 1 < len f and A7: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) and A8: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) A9: j < width (GoB f) by A4, NAT_1:13; A10: j < j + 1 by NAT_1:13; A11: 1 <= i + 1 by NAT_1:11; i < len (GoB f) by A2, NAT_1:13; then ((GoB f) * (i,j)) `2 = ((GoB f) * (1,j)) `2 by A1, A3, A9, GOBOARD5:1 .= ((GoB f) * ((i + 1),j)) `2 by A2, A3, A11, A9, GOBOARD5:1 ; then A12: (GoB f) * (i,j) <> (GoB f) * ((i + 1),(j + 1)) by A2, A3, A4, A11, A10, GOBOARD5:4; A13: 1 <= k + 1 by NAT_1:11; A14: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A6, NAT_1:13; then A15: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A8, A14, A13, TOPREAL1:def_3; then A16: ( ( (GoB f) * (i,j) = f /. (k + 2) & (GoB f) * (i,(j + 1)) = f /. (k + 1) ) or ( (GoB f) * (i,j) = f /. (k + 1) & (GoB f) * (i,(j + 1)) = f /. (k + 2) ) ) by SPPOL_1:8; A17: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. k),(f /. (k + 1))) by A5, A6, A7, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) & (GoB f) * (i,(j + 1)) = f /. k ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. k & (GoB f) * (i,(j + 1)) = f /. (k + 1) ) ) by SPPOL_1:8; hence f /. k = (GoB f) * ((i + 1),(j + 1)) by A15, A12, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) thus f /. (k + 1) = (GoB f) * (i,(j + 1)) by A17, A16, A12, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * (i,j) thus f /. (k + 2) = (GoB f) * (i,j) by A17, A16, A12, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:44 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: 1 <= k and A6: k + 1 < len f and A7: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) and A8: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) A9: 1 <= i + 1 by NAT_1:11; A10: j < j + 1 by NAT_1:13; A11: 1 <= j + 1 by NAT_1:11; i < len (GoB f) by A2, NAT_1:13; then ((GoB f) * (i,(j + 1))) `2 = ((GoB f) * (1,(j + 1))) `2 by A1, A4, A11, GOBOARD5:1 .= ((GoB f) * ((i + 1),(j + 1))) `2 by A2, A4, A9, A11, GOBOARD5:1 ; then A12: (GoB f) * ((i + 1),j) <> (GoB f) * (i,(j + 1)) by A2, A3, A4, A9, A10, GOBOARD5:4; A13: 1 <= k + 1 by NAT_1:11; A14: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A6, NAT_1:13; then A15: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A8, A14, A13, TOPREAL1:def_3; then A16: ( ( (GoB f) * (i,(j + 1)) = f /. (k + 2) & (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. (k + 2) ) ) by SPPOL_1:8; A17: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. k),(f /. (k + 1))) by A5, A6, A7, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) & (GoB f) * ((i + 1),j) = f /. k ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. k & (GoB f) * ((i + 1),j) = f /. (k + 1) ) ) by SPPOL_1:8; hence f /. k = (GoB f) * ((i + 1),j) by A15, A12, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) thus f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) by A17, A16, A12, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * (i,(j + 1)) thus f /. (k + 2) = (GoB f) * (i,(j + 1)) by A17, A16, A12, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:45 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 < len (GoB f) & 1 <= j & j <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,j) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 < len (GoB f) & 1 <= j & j <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,j) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 < len (GoB f) & 1 <= j & j <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) assume that A1: 1 <= i and A2: i + 1 < len (GoB f) and A3: ( 1 <= j & j <= width (GoB f) ) and A4: 1 <= k and A5: k + 1 < len f and A6: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg (f,k) and A7: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,j) ) A8: i < i + 2 by XREAL_1:29; i + (1 + 1) = (i + 1) + 1 ; then i + 2 <= len (GoB f) by A2, NAT_1:13; then A9: ((GoB f) * (i,j)) `1 < ((GoB f) * ((i + 2),j)) `1 by A1, A3, A8, GOBOARD5:3; A10: 1 <= k + 1 by NAT_1:11; A11: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A5, NAT_1:13; then A12: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A7, A11, A10, TOPREAL1:def_3; then A13: ( ( (GoB f) * (i,j) = f /. (k + 2) & (GoB f) * ((i + 1),j) = f /. (k + 1) ) or ( (GoB f) * (i,j) = f /. (k + 1) & (GoB f) * ((i + 1),j) = f /. (k + 2) ) ) by SPPOL_1:8; A14: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg ((f /. k),(f /. (k + 1))) by A4, A5, A6, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 1),j) = f /. (k + 1) & (GoB f) * ((i + 2),j) = f /. k ) or ( (GoB f) * ((i + 1),j) = f /. k & (GoB f) * ((i + 2),j) = f /. (k + 1) ) ) by SPPOL_1:8; hence f /. k = (GoB f) * ((i + 2),j) by A12, A9, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,j) ) thus f /. (k + 1) = (GoB f) * ((i + 1),j) by A14, A13, A9, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * (i,j) thus f /. (k + 2) = (GoB f) * (i,j) by A14, A13, A9, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:46 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,j) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,j) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: 1 <= k and A6: k + 1 < len f and A7: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) and A8: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,j) ) A9: i < len (GoB f) by A2, NAT_1:13; A10: i < i + 1 by NAT_1:13; A11: 1 <= j + 1 by NAT_1:11; j < width (GoB f) by A4, NAT_1:13; then ((GoB f) * (i,j)) `1 = ((GoB f) * (i,1)) `1 by A1, A3, A9, GOBOARD5:2 .= ((GoB f) * (i,(j + 1))) `1 by A1, A4, A11, A9, GOBOARD5:2 ; then A12: (GoB f) * (i,j) <> (GoB f) * ((i + 1),(j + 1)) by A1, A2, A4, A11, A10, GOBOARD5:3; A13: 1 <= k + 1 by NAT_1:11; A14: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A6, NAT_1:13; then A15: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A8, A14, A13, TOPREAL1:def_3; then A16: ( ( (GoB f) * (i,j) = f /. (k + 2) & (GoB f) * ((i + 1),j) = f /. (k + 1) ) or ( (GoB f) * (i,j) = f /. (k + 1) & (GoB f) * ((i + 1),j) = f /. (k + 2) ) ) by SPPOL_1:8; A17: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. k),(f /. (k + 1))) by A5, A6, A7, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) & (GoB f) * ((i + 1),j) = f /. k ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. k & (GoB f) * ((i + 1),j) = f /. (k + 1) ) ) by SPPOL_1:8; hence f /. k = (GoB f) * ((i + 1),(j + 1)) by A15, A12, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,j) ) thus f /. (k + 1) = (GoB f) * ((i + 1),j) by A17, A16, A12, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * (i,j) thus f /. (k + 2) = (GoB f) * (i,j) by A17, A16, A12, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:47 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: 1 <= k and A6: k + 1 < len f and A7: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) and A8: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) A9: 1 <= j + 1 by NAT_1:11; A10: i < i + 1 by NAT_1:13; A11: 1 <= i + 1 by NAT_1:11; j < width (GoB f) by A4, NAT_1:13; then ((GoB f) * ((i + 1),j)) `1 = ((GoB f) * ((i + 1),1)) `1 by A2, A3, A11, GOBOARD5:2 .= ((GoB f) * ((i + 1),(j + 1))) `1 by A2, A4, A9, A11, GOBOARD5:2 ; then A12: (GoB f) * (i,(j + 1)) <> (GoB f) * ((i + 1),j) by A1, A2, A4, A9, A10, GOBOARD5:3; A13: 1 <= k + 1 by NAT_1:11; A14: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A6, NAT_1:13; then A15: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A8, A14, A13, TOPREAL1:def_3; then A16: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) & (GoB f) * (i,(j + 1)) = f /. (k + 2) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 2) & (GoB f) * (i,(j + 1)) = f /. (k + 1) ) ) by SPPOL_1:8; A17: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. k),(f /. (k + 1))) by A5, A6, A7, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 1),j) = f /. k & (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. k ) ) by SPPOL_1:8; hence f /. k = (GoB f) * ((i + 1),j) by A15, A12, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) thus f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) by A17, A16, A12, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * (i,(j + 1)) thus f /. (k + 2) = (GoB f) * (i,(j + 1)) by A17, A16, A12, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:48 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 < width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 < width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 < width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) ) assume that A1: ( 1 <= i & i <= len (GoB f) & 1 <= j ) and A2: j + 1 < width (GoB f) and A3: 1 <= k and A4: k + 1 < len f and A5: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) and A6: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) A7: 1 <= k + 1 by NAT_1:11; A8: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A4, NAT_1:13; then A9: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A6, A8, A7, TOPREAL1:def_3; then A10: ( ( (GoB f) * (i,(j + 1)) = f /. (k + 1) & (GoB f) * (i,(j + 2)) = f /. (k + 2) ) or ( (GoB f) * (i,(j + 1)) = f /. (k + 2) & (GoB f) * (i,(j + 2)) = f /. (k + 1) ) ) by SPPOL_1:8; A11: j < j + 2 by XREAL_1:29; j + (1 + 1) = (j + 1) + 1 ; then j + 2 <= width (GoB f) by A2, NAT_1:13; then A12: ((GoB f) * (i,j)) `2 < ((GoB f) * (i,(j + 2))) `2 by A1, A11, GOBOARD5:4; A13: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg ((f /. k),(f /. (k + 1))) by A3, A4, A5, TOPREAL1:def_3; then ( ( (GoB f) * (i,j) = f /. k & (GoB f) * (i,(j + 1)) = f /. (k + 1) ) or ( (GoB f) * (i,j) = f /. (k + 1) & (GoB f) * (i,(j + 1)) = f /. k ) ) by SPPOL_1:8; hence f /. k = (GoB f) * (i,j) by A9, A12, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) thus f /. (k + 1) = (GoB f) * (i,(j + 1)) by A13, A10, A12, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * (i,(j + 2)) thus f /. (k + 2) = (GoB f) * (i,(j + 2)) by A13, A10, A12, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:49 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: 1 <= k and A6: k + 1 < len f and A7: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k) and A8: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) A9: j < width (GoB f) by A4, NAT_1:13; A10: j < j + 1 by NAT_1:13; A11: 1 <= i + 1 by NAT_1:11; i < len (GoB f) by A2, NAT_1:13; then ((GoB f) * (i,j)) `2 = ((GoB f) * (1,j)) `2 by A1, A3, A9, GOBOARD5:1 .= ((GoB f) * ((i + 1),j)) `2 by A2, A3, A11, A9, GOBOARD5:1 ; then A12: (GoB f) * (i,j) <> (GoB f) * ((i + 1),(j + 1)) by A2, A3, A4, A11, A10, GOBOARD5:4; A13: 1 <= k + 1 by NAT_1:11; A14: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A6, NAT_1:13; then A15: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A8, A14, A13, TOPREAL1:def_3; then A16: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) & (GoB f) * (i,(j + 1)) = f /. (k + 2) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 2) & (GoB f) * (i,(j + 1)) = f /. (k + 1) ) ) by SPPOL_1:8; A17: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg ((f /. k),(f /. (k + 1))) by A5, A6, A7, TOPREAL1:def_3; then ( ( (GoB f) * (i,j) = f /. k & (GoB f) * (i,(j + 1)) = f /. (k + 1) ) or ( (GoB f) * (i,j) = f /. (k + 1) & (GoB f) * (i,(j + 1)) = f /. k ) ) by SPPOL_1:8; hence f /. k = (GoB f) * (i,j) by A15, A12, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) thus f /. (k + 1) = (GoB f) * (i,(j + 1)) by A17, A16, A12, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) thus f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) by A17, A16, A12, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:50 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: 1 <= k and A6: k + 1 < len f and A7: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k) and A8: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) A9: 1 <= i + 1 by NAT_1:11; A10: j < j + 1 by NAT_1:13; A11: 1 <= j + 1 by NAT_1:11; i < len (GoB f) by A2, NAT_1:13; then ((GoB f) * (i,(j + 1))) `2 = ((GoB f) * (1,(j + 1))) `2 by A1, A4, A11, GOBOARD5:1 .= ((GoB f) * ((i + 1),(j + 1))) `2 by A2, A4, A9, A11, GOBOARD5:1 ; then A12: (GoB f) * ((i + 1),j) <> (GoB f) * (i,(j + 1)) by A2, A3, A4, A9, A10, GOBOARD5:4; A13: 1 <= k + 1 by NAT_1:11; A14: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A6, NAT_1:13; then A15: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A8, A14, A13, TOPREAL1:def_3; then A16: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) & (GoB f) * ((i + 1),j) = f /. (k + 2) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 2) & (GoB f) * ((i + 1),j) = f /. (k + 1) ) ) by SPPOL_1:8; A17: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. k),(f /. (k + 1))) by A5, A6, A7, TOPREAL1:def_3; then ( ( (GoB f) * (i,(j + 1)) = f /. k & (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. k ) ) by SPPOL_1:8; hence f /. k = (GoB f) * (i,(j + 1)) by A15, A12, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) thus f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) by A17, A16, A12, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * ((i + 1),j) thus f /. (k + 2) = (GoB f) * ((i + 1),j) by A17, A16, A12, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:51 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 < len (GoB f) & 1 <= j & j <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 < len (GoB f) & 1 <= j & j <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 < len (GoB f) & 1 <= j & j <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) ) assume that A1: 1 <= i and A2: i + 1 < len (GoB f) and A3: ( 1 <= j & j <= width (GoB f) ) and A4: 1 <= k and A5: k + 1 < len f and A6: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) and A7: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) A8: i < i + 2 by XREAL_1:29; i + (1 + 1) = (i + 1) + 1 ; then i + 2 <= len (GoB f) by A2, NAT_1:13; then A9: ((GoB f) * (i,j)) `1 < ((GoB f) * ((i + 2),j)) `1 by A1, A3, A8, GOBOARD5:3; A10: 1 <= k + 1 by NAT_1:11; A11: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A5, NAT_1:13; then A12: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A7, A11, A10, TOPREAL1:def_3; then A13: ( ( (GoB f) * ((i + 1),j) = f /. (k + 1) & (GoB f) * ((i + 2),j) = f /. (k + 2) ) or ( (GoB f) * ((i + 1),j) = f /. (k + 2) & (GoB f) * ((i + 2),j) = f /. (k + 1) ) ) by SPPOL_1:8; A14: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg ((f /. k),(f /. (k + 1))) by A4, A5, A6, TOPREAL1:def_3; then ( ( (GoB f) * (i,j) = f /. k & (GoB f) * ((i + 1),j) = f /. (k + 1) ) or ( (GoB f) * (i,j) = f /. (k + 1) & (GoB f) * ((i + 1),j) = f /. k ) ) by SPPOL_1:8; hence f /. k = (GoB f) * (i,j) by A12, A9, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) thus f /. (k + 1) = (GoB f) * ((i + 1),j) by A14, A13, A9, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * ((i + 2),j) thus f /. (k + 2) = (GoB f) * ((i + 2),j) by A14, A13, A9, SPPOL_1:8; ::_thesis: verum end; theorem :: GOBOARD7:52 for i, j, k being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) proof let i, j, k be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) holds ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & 1 <= k & k + 1 < len f & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) implies ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: 1 <= k and A6: k + 1 < len f and A7: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k) and A8: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,(k + 1)) ; ::_thesis: ( f /. k = (GoB f) * (i,j) & f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) A9: i < len (GoB f) by A2, NAT_1:13; A10: i < i + 1 by NAT_1:13; A11: 1 <= j + 1 by NAT_1:11; j < width (GoB f) by A4, NAT_1:13; then ((GoB f) * (i,j)) `1 = ((GoB f) * (i,1)) `1 by A1, A3, A9, GOBOARD5:2 .= ((GoB f) * (i,(j + 1))) `1 by A1, A4, A11, A9, GOBOARD5:2 ; then A12: (GoB f) * (i,j) <> (GoB f) * ((i + 1),(j + 1)) by A1, A2, A4, A11, A10, GOBOARD5:3; A13: 1 <= k + 1 by NAT_1:11; A14: k + (1 + 1) = (k + 1) + 1 ; then k + 2 <= len f by A6, NAT_1:13; then A15: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg ((f /. (k + 1)),(f /. (k + 2))) by A8, A14, A13, TOPREAL1:def_3; then A16: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 1) & (GoB f) * ((i + 1),j) = f /. (k + 2) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k + 2) & (GoB f) * ((i + 1),j) = f /. (k + 1) ) ) by SPPOL_1:8; A17: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg ((f /. k),(f /. (k + 1))) by A5, A6, A7, TOPREAL1:def_3; then ( ( (GoB f) * (i,j) = f /. k & (GoB f) * ((i + 1),j) = f /. (k + 1) ) or ( (GoB f) * (i,j) = f /. (k + 1) & (GoB f) * ((i + 1),j) = f /. k ) ) by SPPOL_1:8; hence f /. k = (GoB f) * (i,j) by A15, A12, SPPOL_1:8; ::_thesis: ( f /. (k + 1) = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) thus f /. (k + 1) = (GoB f) * ((i + 1),j) by A17, A16, A12, SPPOL_1:8; ::_thesis: f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) thus f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) by A17, A16, A12, SPPOL_1:8; ::_thesis: verum end; theorem Th53: :: GOBOARD7:53 for i, j being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 < width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) c= L~ f & not ( f /. 1 = (GoB f) * (i,(j + 1)) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) ) or ( f /. 2 = (GoB f) * (i,(j + 2)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) proof let i, j be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 < width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) c= L~ f & not ( f /. 1 = (GoB f) * (i,(j + 1)) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) ) or ( f /. 2 = (GoB f) * (i,(j + 2)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i <= len (GoB f) & 1 <= j & j + 1 < width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) c= L~ f & not ( f /. 1 = (GoB f) * (i,(j + 1)) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) ) or ( f /. 2 = (GoB f) * (i,(j + 2)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) implies ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) assume that A1: ( 1 <= i & i <= len (GoB f) ) and A2: 1 <= j and A3: j + 1 < width (GoB f) and A4: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f and A5: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) c= L~ f ; ::_thesis: ( ( f /. 1 = (GoB f) * (i,(j + 1)) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) ) or ( f /. 2 = (GoB f) * (i,(j + 2)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) A6: 1 <= j + 1 by NAT_1:11; (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))) in LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) by RLTOPSP1:69; then consider k1 being Element of NAT such that A7: 1 <= k1 and A8: k1 + 1 <= len f and A9: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k1) by A1, A2, A3, A4, Th39; A10: k1 < len f by A8, NAT_1:13; A11: now__::_thesis:_(_not_k1_>_1_or_k1_=_2_or_k1_>_2_) assume k1 > 1 ; ::_thesis: ( k1 = 2 or k1 > 2 ) then k1 >= 1 + 1 by NAT_1:13; hence ( k1 = 2 or k1 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A12: j < width (GoB f) by A3, NAT_1:13; A13: j + (1 + 1) = (j + 1) + 1 ; then A14: 1 <= j + 2 by NAT_1:11; A15: j + 2 <= width (GoB f) by A3, A13, NAT_1:13; (1 / 2) * (((GoB f) * (i,(j + 1))) + ((GoB f) * (i,(j + 2)))) in LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) by RLTOPSP1:69; then consider k2 being Element of NAT such that A16: 1 <= k2 and A17: k2 + 1 <= len f and A18: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) = LSeg (f,k2) by A1, A5, A6, A13, A15, Th39; A19: k2 < len f by A17, NAT_1:13; A20: now__::_thesis:_(_not_k2_>_1_or_k2_=_2_or_k2_>_2_) assume k2 > 1 ; ::_thesis: ( k2 = 2 or k2 > 2 ) then k2 >= 1 + 1 by NAT_1:13; hence ( k2 = 2 or k2 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A21: ( k1 = 1 or k1 > 1 ) by A7, XXREAL_0:1; now__::_thesis:_(_(_k1_=_1_&_k2_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._1_=_(GoB_f)_*_(i,j)_&_f_/._(1_+_2)_=_(GoB_f)_*_(i,(j_+_2))_)_or_(_k1_=_1_&_k2_>_2_&_f_/._1_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._2_=_(GoB_f)_*_(i,j)_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_(i,(j_+_2))_)_or_(_k2_=_1_&_k1_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._1_=_(GoB_f)_*_(i,(j_+_2))_&_f_/._(1_+_2)_=_(GoB_f)_*_(i,j)_)_or_(_k2_=_1_&_k1_>_2_&_f_/._1_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._2_=_(GoB_f)_*_(i,(j_+_2))_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_(i,j)_)_or_(_k1_=_k2_&_contradiction_)_or_(_k1_>_1_&_k2_>_k1_&_1_<=_k1_&_k1_+_1_<_len_f_&_f_/._(k1_+_1)_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._k1_=_(GoB_f)_*_(i,j)_&_f_/._(k1_+_2)_=_(GoB_f)_*_(i,(j_+_2))_)_or_(_k2_>_1_&_k1_>_k2_&_1_<=_k2_&_k2_+_1_<_len_f_&_f_/._(k2_+_1)_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._k2_=_(GoB_f)_*_(i,(j_+_2))_&_f_/._(k2_+_2)_=_(GoB_f)_*_(i,j)_)_) percases ( ( k1 = 1 & k2 = 2 ) or ( k1 = 1 & k2 > 2 ) or ( k2 = 1 & k1 = 2 ) or ( k2 = 1 & k1 > 2 ) or k1 = k2 or ( k1 > 1 & k2 > k1 ) or ( k2 > 1 & k1 > k2 ) ) by A16, A11, A20, A21, XXREAL_0:1; casethat A22: k1 = 1 and A23: k2 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * (i,(j + 1)) & f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * (i,(j + 2)) ) A24: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A17, A23, TOPREAL1:def_3; then A25: ( ( (GoB f) * (i,(j + 1)) = f /. 2 & (GoB f) * (i,(j + 2)) = f /. (2 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (2 + 1) & (GoB f) * (i,(j + 2)) = f /. 2 ) ) by A18, A23, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A17, A23, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * (i,(j + 1)) & f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * (i,(j + 2)) ) A26: 3 < len f by Th34, XXREAL_0:2; then A27: f /. 1 <> f /. 3 by Th36; A28: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A8, A22, TOPREAL1:def_3; then A29: ( ( (GoB f) * (i,j) = f /. 1 & (GoB f) * (i,(j + 1)) = f /. 2 ) or ( (GoB f) * (i,j) = f /. 2 & (GoB f) * (i,(j + 1)) = f /. 1 ) ) by A9, A22, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * (i,(j + 1)) by A25, A26, Th36; ::_thesis: ( f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * (i,(j + 2)) ) thus f /. 1 = (GoB f) * (i,j) by A18, A23, A29, A24, A27, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * (i,(j + 2)) thus f /. (1 + 2) = (GoB f) * (i,(j + 2)) by A9, A22, A28, A25, A27, SPPOL_1:8; ::_thesis: verum end; casethat A30: k1 = 1 and A31: k2 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * (i,(j + 1)) & f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) ) A32: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A8, A30, TOPREAL1:def_3; then A33: ( ( (GoB f) * (i,j) = f /. 1 & (GoB f) * (i,(j + 1)) = f /. 2 ) or ( (GoB f) * (i,j) = f /. 2 & (GoB f) * (i,(j + 1)) = f /. 1 ) ) by A9, A30, SPPOL_1:8; A34: 2 < k2 + 1 by A31, NAT_1:13; then A35: f /. (k2 + 1) <> f /. 2 by A17, Th37; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then A36: ( ( (GoB f) * (i,(j + 1)) = f /. k2 & (GoB f) * (i,(j + 2)) = f /. (k2 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k2 + 1) & (GoB f) * (i,(j + 2)) = f /. k2 ) ) by A18, SPPOL_1:8; A37: f /. k2 <> f /. 2 by A19, A31, Th36; hence f /. 1 = (GoB f) * (i,(j + 1)) by A9, A30, A32, A36, A35, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) ) thus f /. 2 = (GoB f) * (i,j) by A9, A30, A32, A36, A37, A35, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) A38: k2 > 1 by A31, XXREAL_0:2; then A39: k2 + 1 > 1 by NAT_1:13; then k2 + 1 = len f by A17, A19, A31, A33, A36, A38, A34, Th37, Th38; then k2 + 1 = ((len f) -' 1) + 1 by A39, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) by A19, A31, A33, A36, A38, Th36; ::_thesis: verum end; casethat A40: k2 = 1 and A41: k1 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * (i,(j + 1)) & f /. 1 = (GoB f) * (i,(j + 2)) & f /. (1 + 2) = (GoB f) * (i,j) ) A42: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A8, A41, TOPREAL1:def_3; then A43: ( ( (GoB f) * (i,(j + 1)) = f /. 2 & (GoB f) * (i,j) = f /. (2 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (2 + 1) & (GoB f) * (i,j) = f /. 2 ) ) by A9, A41, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A8, A41, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * (i,(j + 1)) & f /. 1 = (GoB f) * (i,(j + 2)) & f /. (1 + 2) = (GoB f) * (i,j) ) A44: 3 < len f by Th34, XXREAL_0:2; then A45: f /. 1 <> f /. 3 by Th36; A46: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A17, A40, TOPREAL1:def_3; then A47: ( ( (GoB f) * (i,(j + 2)) = f /. 1 & (GoB f) * (i,(j + 1)) = f /. 2 ) or ( (GoB f) * (i,(j + 2)) = f /. 2 & (GoB f) * (i,(j + 1)) = f /. 1 ) ) by A18, A40, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * (i,(j + 1)) by A43, A44, Th36; ::_thesis: ( f /. 1 = (GoB f) * (i,(j + 2)) & f /. (1 + 2) = (GoB f) * (i,j) ) thus f /. 1 = (GoB f) * (i,(j + 2)) by A9, A41, A47, A42, A45, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * (i,j) thus f /. (1 + 2) = (GoB f) * (i,j) by A18, A40, A46, A43, A45, SPPOL_1:8; ::_thesis: verum end; casethat A48: k2 = 1 and A49: k1 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * (i,(j + 1)) & f /. 2 = (GoB f) * (i,(j + 2)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) A50: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A17, A48, TOPREAL1:def_3; then A51: ( ( (GoB f) * (i,(j + 2)) = f /. 1 & (GoB f) * (i,(j + 1)) = f /. 2 ) or ( (GoB f) * (i,(j + 2)) = f /. 2 & (GoB f) * (i,(j + 1)) = f /. 1 ) ) by A18, A48, SPPOL_1:8; A52: 2 < k1 + 1 by A49, NAT_1:13; then A53: f /. (k1 + 1) <> f /. 2 by A8, Th37; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A7, A8, TOPREAL1:def_3; then A54: ( ( (GoB f) * (i,(j + 1)) = f /. k1 & (GoB f) * (i,j) = f /. (k1 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k1 + 1) & (GoB f) * (i,j) = f /. k1 ) ) by A9, SPPOL_1:8; A55: f /. k1 <> f /. 2 by A10, A49, Th36; hence f /. 1 = (GoB f) * (i,(j + 1)) by A18, A48, A50, A54, A53, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * (i,(j + 2)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) thus f /. 2 = (GoB f) * (i,(j + 2)) by A18, A48, A50, A54, A55, A53, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * (i,j) A56: k1 > 1 by A49, XXREAL_0:2; then A57: k1 + 1 > 1 by NAT_1:13; then k1 + 1 = len f by A8, A10, A49, A51, A54, A56, A52, Th37, Th38; then k1 + 1 = ((len f) -' 1) + 1 by A57, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * (i,j) by A10, A49, A51, A54, A56, Th36; ::_thesis: verum end; case k1 = k2 ; ::_thesis: contradiction then A58: ( (GoB f) * (i,j) = (GoB f) * (i,(j + 2)) or (GoB f) * (i,j) = (GoB f) * (i,(j + 1)) ) by A9, A18, SPPOL_1:8; A59: [i,(j + 2)] in Indices (GoB f) by A1, A15, A14, MATRIX_1:36; ( [i,j] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A1, A2, A3, A6, A12, MATRIX_1:36; then ( j = j + 1 or j = j + 2 ) by A58, A59, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; casethat A60: k1 > 1 and A61: k2 > k1 ; ::_thesis: ( 1 <= k1 & k1 + 1 < len f & f /. (k1 + 1) = (GoB f) * (i,(j + 1)) & f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * (i,(j + 2)) ) A62: ( 1 < k1 + 1 & k1 + 1 < k2 + 1 ) by A60, A61, NAT_1:13, XREAL_1:6; A63: k1 < k2 + 1 by A61, NAT_1:13; then A64: f /. k1 <> f /. (k2 + 1) by A17, A60, Th37; A65: k1 + 1 <= k2 by A61, NAT_1:13; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then A66: ( ( (GoB f) * (i,(j + 1)) = f /. k2 & (GoB f) * (i,(j + 2)) = f /. (k2 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k2 + 1) & (GoB f) * (i,(j + 2)) = f /. k2 ) ) by A18, SPPOL_1:8; A67: k2 < len f by A17, NAT_1:13; then A68: f /. k1 <> f /. k2 by A60, A61, Th37; A69: LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A7, A8, TOPREAL1:def_3; then ( ( (GoB f) * (i,j) = f /. k1 & (GoB f) * (i,(j + 1)) = f /. (k1 + 1) ) or ( (GoB f) * (i,j) = f /. (k1 + 1) & (GoB f) * (i,(j + 1)) = f /. k1 ) ) by A9, SPPOL_1:8; then k1 + 1 >= k2 by A17, A60, A61, A66, A63, A67, A62, Th37; then A70: k1 + 1 = k2 by A65, XXREAL_0:1; hence ( 1 <= k1 & k1 + 1 < len f ) by A17, A60, NAT_1:13; ::_thesis: ( f /. (k1 + 1) = (GoB f) * (i,(j + 1)) & f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * (i,(j + 2)) ) thus f /. (k1 + 1) = (GoB f) * (i,(j + 1)) by A9, A69, A66, A64, A68, SPPOL_1:8; ::_thesis: ( f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * (i,(j + 2)) ) thus f /. k1 = (GoB f) * (i,j) by A9, A69, A66, A64, A68, SPPOL_1:8; ::_thesis: f /. (k1 + 2) = (GoB f) * (i,(j + 2)) thus f /. (k1 + 2) = (GoB f) * (i,(j + 2)) by A9, A69, A66, A64, A70, SPPOL_1:8; ::_thesis: verum end; casethat A71: k2 > 1 and A72: k1 > k2 ; ::_thesis: ( 1 <= k2 & k2 + 1 < len f & f /. (k2 + 1) = (GoB f) * (i,(j + 1)) & f /. k2 = (GoB f) * (i,(j + 2)) & f /. (k2 + 2) = (GoB f) * (i,j) ) A73: ( 1 < k2 + 1 & k2 + 1 < k1 + 1 ) by A71, A72, NAT_1:13, XREAL_1:6; A74: k2 < k1 + 1 by A72, NAT_1:13; then A75: f /. k2 <> f /. (k1 + 1) by A8, A71, Th37; A76: k2 + 1 <= k1 by A72, NAT_1:13; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A7, A8, TOPREAL1:def_3; then A77: ( ( (GoB f) * (i,(j + 1)) = f /. k1 & (GoB f) * (i,j) = f /. (k1 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k1 + 1) & (GoB f) * (i,j) = f /. k1 ) ) by A9, SPPOL_1:8; A78: k1 < len f by A8, NAT_1:13; then A79: f /. k2 <> f /. k1 by A71, A72, Th37; A80: LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then ( ( (GoB f) * (i,(j + 2)) = f /. k2 & (GoB f) * (i,(j + 1)) = f /. (k2 + 1) ) or ( (GoB f) * (i,(j + 2)) = f /. (k2 + 1) & (GoB f) * (i,(j + 1)) = f /. k2 ) ) by A18, SPPOL_1:8; then k2 + 1 >= k1 by A8, A71, A72, A77, A74, A78, A73, Th37; then A81: k2 + 1 = k1 by A76, XXREAL_0:1; hence ( 1 <= k2 & k2 + 1 < len f ) by A8, A71, NAT_1:13; ::_thesis: ( f /. (k2 + 1) = (GoB f) * (i,(j + 1)) & f /. k2 = (GoB f) * (i,(j + 2)) & f /. (k2 + 2) = (GoB f) * (i,j) ) thus f /. (k2 + 1) = (GoB f) * (i,(j + 1)) by A18, A80, A77, A75, A79, SPPOL_1:8; ::_thesis: ( f /. k2 = (GoB f) * (i,(j + 2)) & f /. (k2 + 2) = (GoB f) * (i,j) ) thus f /. k2 = (GoB f) * (i,(j + 2)) by A18, A80, A77, A75, A79, SPPOL_1:8; ::_thesis: f /. (k2 + 2) = (GoB f) * (i,j) thus f /. (k2 + 2) = (GoB f) * (i,j) by A18, A80, A77, A75, A81, SPPOL_1:8; ::_thesis: verum end; end; end; hence ( ( f /. 1 = (GoB f) * (i,(j + 1)) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) ) or ( f /. 2 = (GoB f) * (i,(j + 2)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) ; ::_thesis: verum end; theorem Th54: :: GOBOARD7:54 for i, j being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & not ( f /. 1 = (GoB f) * (i,(j + 1)) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) proof let i, j be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & not ( f /. 1 = (GoB f) * (i,(j + 1)) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & not ( f /. 1 = (GoB f) * (i,(j + 1)) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) implies ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f and A6: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f ; ::_thesis: ( ( f /. 1 = (GoB f) * (i,(j + 1)) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) A7: i < len (GoB f) by A2, NAT_1:13; A8: j < width (GoB f) by A4, NAT_1:13; A9: 1 <= i + 1 by NAT_1:11; (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * (i,(j + 1)))) in LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) by RLTOPSP1:69; then consider k1 being Element of NAT such that A10: 1 <= k1 and A11: k1 + 1 <= len f and A12: LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) = LSeg (f,k1) by A1, A3, A4, A5, A7, Th39; A13: k1 < len f by A11, NAT_1:13; A14: now__::_thesis:_(_not_k1_>_1_or_k1_=_2_or_k1_>_2_) assume k1 > 1 ; ::_thesis: ( k1 = 2 or k1 > 2 ) then k1 >= 1 + 1 by NAT_1:13; hence ( k1 = 2 or k1 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A15: 1 <= j + 1 by NAT_1:11; (1 / 2) * (((GoB f) * (i,(j + 1))) + ((GoB f) * ((i + 1),(j + 1)))) in LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) by RLTOPSP1:69; then consider k2 being Element of NAT such that A16: 1 <= k2 and A17: k2 + 1 <= len f and A18: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k2) by A1, A2, A4, A6, A15, Th40; A19: k2 < len f by A17, NAT_1:13; A20: now__::_thesis:_(_not_k2_>_1_or_k2_=_2_or_k2_>_2_) assume k2 > 1 ; ::_thesis: ( k2 = 2 or k2 > 2 ) then k2 >= 1 + 1 by NAT_1:13; hence ( k2 = 2 or k2 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A21: ( k1 = 1 or k1 > 1 ) by A10, XXREAL_0:1; now__::_thesis:_(_(_k1_=_1_&_k2_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._1_=_(GoB_f)_*_(i,j)_&_f_/._(1_+_2)_=_(GoB_f)_*_((i_+_1),(j_+_1))_)_or_(_k1_=_1_&_k2_>_2_&_f_/._1_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._2_=_(GoB_f)_*_(i,j)_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_((i_+_1),(j_+_1))_)_or_(_k2_=_1_&_k1_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._1_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._(1_+_2)_=_(GoB_f)_*_(i,j)_)_or_(_k2_=_1_&_k1_>_2_&_f_/._1_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._2_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_(i,j)_)_or_(_k1_=_k2_&_contradiction_)_or_(_k1_>_1_&_k2_>_k1_&_1_<=_k1_&_k1_+_1_<_len_f_&_f_/._(k1_+_1)_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._k1_=_(GoB_f)_*_(i,j)_&_f_/._(k1_+_2)_=_(GoB_f)_*_((i_+_1),(j_+_1))_)_or_(_k2_>_1_&_k1_>_k2_&_1_<=_k2_&_k2_+_1_<_len_f_&_f_/._(k2_+_1)_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._k2_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._(k2_+_2)_=_(GoB_f)_*_(i,j)_)_) percases ( ( k1 = 1 & k2 = 2 ) or ( k1 = 1 & k2 > 2 ) or ( k2 = 1 & k1 = 2 ) or ( k2 = 1 & k1 > 2 ) or k1 = k2 or ( k1 > 1 & k2 > k1 ) or ( k2 > 1 & k1 > k2 ) ) by A16, A14, A20, A21, XXREAL_0:1; casethat A22: k1 = 1 and A23: k2 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * (i,(j + 1)) & f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) A24: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A17, A23, TOPREAL1:def_3; then A25: ( ( (GoB f) * (i,(j + 1)) = f /. 2 & (GoB f) * ((i + 1),(j + 1)) = f /. (2 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (2 + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. 2 ) ) by A18, A23, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A17, A23, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * (i,(j + 1)) & f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) A26: 3 < len f by Th34, XXREAL_0:2; then A27: f /. 1 <> f /. 3 by Th36; A28: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A11, A22, TOPREAL1:def_3; then A29: ( ( (GoB f) * (i,j) = f /. 1 & (GoB f) * (i,(j + 1)) = f /. 2 ) or ( (GoB f) * (i,j) = f /. 2 & (GoB f) * (i,(j + 1)) = f /. 1 ) ) by A12, A22, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * (i,(j + 1)) by A25, A26, Th36; ::_thesis: ( f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) thus f /. 1 = (GoB f) * (i,j) by A18, A23, A29, A24, A27, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * ((i + 1),(j + 1)) thus f /. (1 + 2) = (GoB f) * ((i + 1),(j + 1)) by A12, A22, A28, A25, A27, SPPOL_1:8; ::_thesis: verum end; casethat A30: k1 = 1 and A31: k2 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * (i,(j + 1)) & f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) A32: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A11, A30, TOPREAL1:def_3; then A33: ( ( (GoB f) * (i,j) = f /. 1 & (GoB f) * (i,(j + 1)) = f /. 2 ) or ( (GoB f) * (i,j) = f /. 2 & (GoB f) * (i,(j + 1)) = f /. 1 ) ) by A12, A30, SPPOL_1:8; A34: 2 < k2 + 1 by A31, NAT_1:13; then A35: f /. (k2 + 1) <> f /. 2 by A17, Th37; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then A36: ( ( (GoB f) * (i,(j + 1)) = f /. k2 & (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k2 + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. k2 ) ) by A18, SPPOL_1:8; A37: f /. k2 <> f /. 2 by A19, A31, Th36; hence f /. 1 = (GoB f) * (i,(j + 1)) by A12, A30, A32, A36, A35, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) thus f /. 2 = (GoB f) * (i,j) by A12, A30, A32, A36, A37, A35, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) A38: k2 > 1 by A31, XXREAL_0:2; then A39: k2 + 1 > 1 by NAT_1:13; then k2 + 1 = len f by A17, A19, A31, A33, A36, A38, A34, Th37, Th38; then k2 + 1 = ((len f) -' 1) + 1 by A39, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) by A19, A31, A33, A36, A38, Th36; ::_thesis: verum end; casethat A40: k2 = 1 and A41: k1 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * (i,(j + 1)) & f /. 1 = (GoB f) * ((i + 1),(j + 1)) & f /. (1 + 2) = (GoB f) * (i,j) ) A42: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A11, A41, TOPREAL1:def_3; then A43: ( ( (GoB f) * (i,(j + 1)) = f /. 2 & (GoB f) * (i,j) = f /. (2 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (2 + 1) & (GoB f) * (i,j) = f /. 2 ) ) by A12, A41, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A11, A41, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * (i,(j + 1)) & f /. 1 = (GoB f) * ((i + 1),(j + 1)) & f /. (1 + 2) = (GoB f) * (i,j) ) A44: 3 < len f by Th34, XXREAL_0:2; then A45: f /. 1 <> f /. 3 by Th36; A46: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A17, A40, TOPREAL1:def_3; then A47: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. 1 & (GoB f) * (i,(j + 1)) = f /. 2 ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. 2 & (GoB f) * (i,(j + 1)) = f /. 1 ) ) by A18, A40, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * (i,(j + 1)) by A43, A44, Th36; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & f /. (1 + 2) = (GoB f) * (i,j) ) thus f /. 1 = (GoB f) * ((i + 1),(j + 1)) by A12, A41, A47, A42, A45, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * (i,j) thus f /. (1 + 2) = (GoB f) * (i,j) by A18, A40, A46, A43, A45, SPPOL_1:8; ::_thesis: verum end; casethat A48: k2 = 1 and A49: k1 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * (i,(j + 1)) & f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) A50: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A17, A48, TOPREAL1:def_3; then A51: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. 1 & (GoB f) * (i,(j + 1)) = f /. 2 ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. 2 & (GoB f) * (i,(j + 1)) = f /. 1 ) ) by A18, A48, SPPOL_1:8; A52: 2 < k1 + 1 by A49, NAT_1:13; then A53: f /. (k1 + 1) <> f /. 2 by A11, Th37; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A10, A11, TOPREAL1:def_3; then A54: ( ( (GoB f) * (i,(j + 1)) = f /. k1 & (GoB f) * (i,j) = f /. (k1 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k1 + 1) & (GoB f) * (i,j) = f /. k1 ) ) by A12, SPPOL_1:8; A55: f /. k1 <> f /. 2 by A13, A49, Th36; hence f /. 1 = (GoB f) * (i,(j + 1)) by A18, A48, A50, A54, A53, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) thus f /. 2 = (GoB f) * ((i + 1),(j + 1)) by A18, A48, A50, A54, A55, A53, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * (i,j) A56: k1 > 1 by A49, XXREAL_0:2; then A57: k1 + 1 > 1 by NAT_1:13; then k1 + 1 = len f by A11, A13, A49, A51, A54, A56, A52, Th37, Th38; then k1 + 1 = ((len f) -' 1) + 1 by A57, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * (i,j) by A13, A49, A51, A54, A56, Th36; ::_thesis: verum end; case k1 = k2 ; ::_thesis: contradiction then A58: ( (GoB f) * (i,j) = (GoB f) * ((i + 1),(j + 1)) or (GoB f) * (i,j) = (GoB f) * (i,(j + 1)) ) by A12, A18, SPPOL_1:8; A59: [(i + 1),(j + 1)] in Indices (GoB f) by A2, A4, A15, A9, MATRIX_1:36; ( [i,j] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A1, A3, A4, A15, A8, A7, MATRIX_1:36; then j = j + 1 by A58, A59, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; casethat A60: k1 > 1 and A61: k2 > k1 ; ::_thesis: ( 1 <= k1 & k1 + 1 < len f & f /. (k1 + 1) = (GoB f) * (i,(j + 1)) & f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) A62: ( 1 < k1 + 1 & k1 + 1 < k2 + 1 ) by A60, A61, NAT_1:13, XREAL_1:6; A63: k1 < k2 + 1 by A61, NAT_1:13; then A64: f /. k1 <> f /. (k2 + 1) by A17, A60, Th37; A65: k1 + 1 <= k2 by A61, NAT_1:13; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then A66: ( ( (GoB f) * (i,(j + 1)) = f /. k2 & (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k2 + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. k2 ) ) by A18, SPPOL_1:8; A67: k2 < len f by A17, NAT_1:13; then A68: f /. k1 <> f /. k2 by A60, A61, Th37; A69: LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A10, A11, TOPREAL1:def_3; then ( ( (GoB f) * (i,j) = f /. k1 & (GoB f) * (i,(j + 1)) = f /. (k1 + 1) ) or ( (GoB f) * (i,j) = f /. (k1 + 1) & (GoB f) * (i,(j + 1)) = f /. k1 ) ) by A12, SPPOL_1:8; then k1 + 1 >= k2 by A17, A60, A61, A66, A63, A67, A62, Th37; then A70: k1 + 1 = k2 by A65, XXREAL_0:1; hence ( 1 <= k1 & k1 + 1 < len f ) by A17, A60, NAT_1:13; ::_thesis: ( f /. (k1 + 1) = (GoB f) * (i,(j + 1)) & f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) thus f /. (k1 + 1) = (GoB f) * (i,(j + 1)) by A12, A69, A66, A64, A68, SPPOL_1:8; ::_thesis: ( f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) thus f /. k1 = (GoB f) * (i,j) by A12, A69, A66, A64, A68, SPPOL_1:8; ::_thesis: f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 1)) thus f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 1)) by A12, A69, A66, A64, A70, SPPOL_1:8; ::_thesis: verum end; casethat A71: k2 > 1 and A72: k1 > k2 ; ::_thesis: ( 1 <= k2 & k2 + 1 < len f & f /. (k2 + 1) = (GoB f) * (i,(j + 1)) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) & f /. (k2 + 2) = (GoB f) * (i,j) ) A73: ( 1 < k2 + 1 & k2 + 1 < k1 + 1 ) by A71, A72, NAT_1:13, XREAL_1:6; A74: k2 < k1 + 1 by A72, NAT_1:13; then A75: f /. k2 <> f /. (k1 + 1) by A11, A71, Th37; A76: k2 + 1 <= k1 by A72, NAT_1:13; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A10, A11, TOPREAL1:def_3; then A77: ( ( (GoB f) * (i,(j + 1)) = f /. k1 & (GoB f) * (i,j) = f /. (k1 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k1 + 1) & (GoB f) * (i,j) = f /. k1 ) ) by A12, SPPOL_1:8; A78: k1 < len f by A11, NAT_1:13; then A79: f /. k2 <> f /. k1 by A71, A72, Th37; A80: LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 1),(j + 1)) = f /. k2 & (GoB f) * (i,(j + 1)) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) & (GoB f) * (i,(j + 1)) = f /. k2 ) ) by A18, SPPOL_1:8; then k2 + 1 >= k1 by A11, A71, A72, A77, A74, A78, A73, Th37; then A81: k2 + 1 = k1 by A76, XXREAL_0:1; hence ( 1 <= k2 & k2 + 1 < len f ) by A11, A71, NAT_1:13; ::_thesis: ( f /. (k2 + 1) = (GoB f) * (i,(j + 1)) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) & f /. (k2 + 2) = (GoB f) * (i,j) ) thus f /. (k2 + 1) = (GoB f) * (i,(j + 1)) by A18, A80, A77, A75, A79, SPPOL_1:8; ::_thesis: ( f /. k2 = (GoB f) * ((i + 1),(j + 1)) & f /. (k2 + 2) = (GoB f) * (i,j) ) thus f /. k2 = (GoB f) * ((i + 1),(j + 1)) by A18, A80, A77, A75, A79, SPPOL_1:8; ::_thesis: f /. (k2 + 2) = (GoB f) * (i,j) thus f /. (k2 + 2) = (GoB f) * (i,j) by A18, A80, A77, A75, A81, SPPOL_1:8; ::_thesis: verum end; end; end; hence ( ( f /. 1 = (GoB f) * (i,(j + 1)) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) ; ::_thesis: verum end; theorem Th55: :: GOBOARD7:55 for i, j being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 1),j))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) or ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) proof let i, j be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 1),j))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) or ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 1),j))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) or ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) ) ) implies ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f and A6: LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 1),j))) c= L~ f ; ::_thesis: ( ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) or ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) ) A7: 1 <= j + 1 by NAT_1:11; (1 / 2) * (((GoB f) * (i,(j + 1))) + ((GoB f) * ((i + 1),(j + 1)))) in LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) by RLTOPSP1:69; then consider k1 being Element of NAT such that A8: 1 <= k1 and A9: k1 + 1 <= len f and A10: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k1) by A1, A2, A4, A5, A7, Th40; A11: k1 < len f by A9, NAT_1:13; A12: now__::_thesis:_(_not_k1_>_1_or_k1_=_2_or_k1_>_2_) assume k1 > 1 ; ::_thesis: ( k1 = 2 or k1 > 2 ) then k1 >= 1 + 1 by NAT_1:13; hence ( k1 = 2 or k1 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A13: ( j < width (GoB f) & i < len (GoB f) ) by A2, A4, NAT_1:13; A14: 1 <= i + 1 by NAT_1:11; (1 / 2) * (((GoB f) * ((i + 1),j)) + ((GoB f) * ((i + 1),(j + 1)))) in LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 1),j))) by RLTOPSP1:69; then consider k2 being Element of NAT such that A15: 1 <= k2 and A16: k2 + 1 <= len f and A17: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k2) by A2, A3, A4, A6, A14, Th39; A18: k2 < len f by A16, NAT_1:13; A19: now__::_thesis:_(_not_k2_>_1_or_k2_=_2_or_k2_>_2_) assume k2 > 1 ; ::_thesis: ( k2 = 2 or k2 > 2 ) then k2 >= 1 + 1 by NAT_1:13; hence ( k2 = 2 or k2 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A20: ( k1 = 1 or k1 > 1 ) by A8, XXREAL_0:1; now__::_thesis:_(_(_k1_=_1_&_k2_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._1_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._(1_+_2)_=_(GoB_f)_*_((i_+_1),j)_)_or_(_k1_=_1_&_k2_>_2_&_f_/._1_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._2_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_((i_+_1),j)_)_or_(_k2_=_1_&_k1_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._1_=_(GoB_f)_*_((i_+_1),j)_&_f_/._(1_+_2)_=_(GoB_f)_*_(i,(j_+_1))_)_or_(_k2_=_1_&_k1_>_2_&_f_/._1_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._2_=_(GoB_f)_*_((i_+_1),j)_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_(i,(j_+_1))_)_or_(_k1_=_k2_&_contradiction_)_or_(_k1_>_1_&_k2_>_k1_&_1_<=_k1_&_k1_+_1_<_len_f_&_f_/._(k1_+_1)_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._k1_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._(k1_+_2)_=_(GoB_f)_*_((i_+_1),j)_)_or_(_k2_>_1_&_k1_>_k2_&_1_<=_k2_&_k2_+_1_<_len_f_&_f_/._(k2_+_1)_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._k2_=_(GoB_f)_*_((i_+_1),j)_&_f_/._(k2_+_2)_=_(GoB_f)_*_(i,(j_+_1))_)_) percases ( ( k1 = 1 & k2 = 2 ) or ( k1 = 1 & k2 > 2 ) or ( k2 = 1 & k1 = 2 ) or ( k2 = 1 & k1 > 2 ) or k1 = k2 or ( k1 > 1 & k2 > k1 ) or ( k2 > 1 & k1 > k2 ) ) by A15, A12, A19, A20, XXREAL_0:1; casethat A21: k1 = 1 and A22: k2 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. 1 = (GoB f) * (i,(j + 1)) & f /. (1 + 2) = (GoB f) * ((i + 1),j) ) A23: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A16, A22, TOPREAL1:def_3; then A24: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. 2 & (GoB f) * ((i + 1),j) = f /. (2 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (2 + 1) & (GoB f) * ((i + 1),j) = f /. 2 ) ) by A17, A22, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A16, A22, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. 1 = (GoB f) * (i,(j + 1)) & f /. (1 + 2) = (GoB f) * ((i + 1),j) ) A25: 3 < len f by Th34, XXREAL_0:2; then A26: f /. 1 <> f /. 3 by Th36; A27: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A9, A21, TOPREAL1:def_3; then A28: ( ( (GoB f) * (i,(j + 1)) = f /. 1 & (GoB f) * ((i + 1),(j + 1)) = f /. 2 ) or ( (GoB f) * (i,(j + 1)) = f /. 2 & (GoB f) * ((i + 1),(j + 1)) = f /. 1 ) ) by A10, A21, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) by A24, A25, Th36; ::_thesis: ( f /. 1 = (GoB f) * (i,(j + 1)) & f /. (1 + 2) = (GoB f) * ((i + 1),j) ) thus f /. 1 = (GoB f) * (i,(j + 1)) by A17, A22, A28, A23, A26, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * ((i + 1),j) thus f /. (1 + 2) = (GoB f) * ((i + 1),j) by A10, A21, A27, A24, A26, SPPOL_1:8; ::_thesis: verum end; casethat A29: k1 = 1 and A30: k2 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) A31: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A9, A29, TOPREAL1:def_3; then A32: ( ( (GoB f) * (i,(j + 1)) = f /. 1 & (GoB f) * ((i + 1),(j + 1)) = f /. 2 ) or ( (GoB f) * (i,(j + 1)) = f /. 2 & (GoB f) * ((i + 1),(j + 1)) = f /. 1 ) ) by A10, A29, SPPOL_1:8; A33: 2 < k2 + 1 by A30, NAT_1:13; then A34: f /. (k2 + 1) <> f /. 2 by A16, Th37; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A15, A16, TOPREAL1:def_3; then A35: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. k2 & (GoB f) * ((i + 1),j) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) & (GoB f) * ((i + 1),j) = f /. k2 ) ) by A17, SPPOL_1:8; A36: f /. k2 <> f /. 2 by A18, A30, Th36; hence f /. 1 = (GoB f) * ((i + 1),(j + 1)) by A10, A29, A31, A35, A34, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) thus f /. 2 = (GoB f) * (i,(j + 1)) by A10, A29, A31, A35, A36, A34, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) A37: k2 > 1 by A30, XXREAL_0:2; then A38: k2 + 1 > 1 by NAT_1:13; then k2 + 1 = len f by A16, A18, A30, A32, A35, A37, A33, Th37, Th38; then k2 + 1 = ((len f) -' 1) + 1 by A38, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) by A18, A30, A32, A35, A37, Th36; ::_thesis: verum end; casethat A39: k2 = 1 and A40: k1 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. 1 = (GoB f) * ((i + 1),j) & f /. (1 + 2) = (GoB f) * (i,(j + 1)) ) A41: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A9, A40, TOPREAL1:def_3; then A42: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. 2 & (GoB f) * (i,(j + 1)) = f /. (2 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (2 + 1) & (GoB f) * (i,(j + 1)) = f /. 2 ) ) by A10, A40, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A9, A40, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. 1 = (GoB f) * ((i + 1),j) & f /. (1 + 2) = (GoB f) * (i,(j + 1)) ) A43: 3 < len f by Th34, XXREAL_0:2; then A44: f /. 1 <> f /. 3 by Th36; A45: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A16, A39, TOPREAL1:def_3; then A46: ( ( (GoB f) * ((i + 1),j) = f /. 1 & (GoB f) * ((i + 1),(j + 1)) = f /. 2 ) or ( (GoB f) * ((i + 1),j) = f /. 2 & (GoB f) * ((i + 1),(j + 1)) = f /. 1 ) ) by A17, A39, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) by A42, A43, Th36; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),j) & f /. (1 + 2) = (GoB f) * (i,(j + 1)) ) thus f /. 1 = (GoB f) * ((i + 1),j) by A10, A40, A46, A41, A44, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * (i,(j + 1)) thus f /. (1 + 2) = (GoB f) * (i,(j + 1)) by A17, A39, A45, A42, A44, SPPOL_1:8; ::_thesis: verum end; casethat A47: k2 = 1 and A48: k1 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) A49: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A16, A47, TOPREAL1:def_3; then A50: ( ( (GoB f) * ((i + 1),j) = f /. 1 & (GoB f) * ((i + 1),(j + 1)) = f /. 2 ) or ( (GoB f) * ((i + 1),j) = f /. 2 & (GoB f) * ((i + 1),(j + 1)) = f /. 1 ) ) by A17, A47, SPPOL_1:8; A51: 2 < k1 + 1 by A48, NAT_1:13; then A52: f /. (k1 + 1) <> f /. 2 by A9, Th37; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A8, A9, TOPREAL1:def_3; then A53: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. k1 & (GoB f) * (i,(j + 1)) = f /. (k1 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k1 + 1) & (GoB f) * (i,(j + 1)) = f /. k1 ) ) by A10, SPPOL_1:8; A54: f /. k1 <> f /. 2 by A11, A48, Th36; hence f /. 1 = (GoB f) * ((i + 1),(j + 1)) by A17, A47, A49, A53, A52, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) thus f /. 2 = (GoB f) * ((i + 1),j) by A17, A47, A49, A53, A54, A52, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) A55: k1 > 1 by A48, XXREAL_0:2; then A56: k1 + 1 > 1 by NAT_1:13; then k1 + 1 = len f by A9, A11, A48, A50, A53, A55, A51, Th37, Th38; then k1 + 1 = ((len f) -' 1) + 1 by A56, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) by A11, A48, A50, A53, A55, Th36; ::_thesis: verum end; case k1 = k2 ; ::_thesis: contradiction then A57: ( (GoB f) * (i,(j + 1)) = (GoB f) * ((i + 1),j) or (GoB f) * (i,(j + 1)) = (GoB f) * ((i + 1),(j + 1)) ) by A10, A17, SPPOL_1:8; A58: [(i + 1),(j + 1)] in Indices (GoB f) by A2, A4, A7, A14, MATRIX_1:36; ( [(i + 1),j] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A1, A2, A3, A4, A7, A14, A13, MATRIX_1:36; then ( i = i + 1 or j = j + 1 ) by A57, A58, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; casethat A59: k1 > 1 and A60: k2 > k1 ; ::_thesis: ( 1 <= k1 & k1 + 1 < len f & f /. (k1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. k1 = (GoB f) * (i,(j + 1)) & f /. (k1 + 2) = (GoB f) * ((i + 1),j) ) A61: ( 1 < k1 + 1 & k1 + 1 < k2 + 1 ) by A59, A60, NAT_1:13, XREAL_1:6; A62: k1 < k2 + 1 by A60, NAT_1:13; then A63: f /. k1 <> f /. (k2 + 1) by A16, A59, Th37; A64: k1 + 1 <= k2 by A60, NAT_1:13; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A15, A16, TOPREAL1:def_3; then A65: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. k2 & (GoB f) * ((i + 1),j) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) & (GoB f) * ((i + 1),j) = f /. k2 ) ) by A17, SPPOL_1:8; A66: k2 < len f by A16, NAT_1:13; then A67: f /. k1 <> f /. k2 by A59, A60, Th37; A68: LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A8, A9, TOPREAL1:def_3; then ( ( (GoB f) * (i,(j + 1)) = f /. k1 & (GoB f) * ((i + 1),(j + 1)) = f /. (k1 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k1 + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. k1 ) ) by A10, SPPOL_1:8; then k1 + 1 >= k2 by A16, A59, A60, A65, A62, A66, A61, Th37; then A69: k1 + 1 = k2 by A64, XXREAL_0:1; hence ( 1 <= k1 & k1 + 1 < len f ) by A16, A59, NAT_1:13; ::_thesis: ( f /. (k1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. k1 = (GoB f) * (i,(j + 1)) & f /. (k1 + 2) = (GoB f) * ((i + 1),j) ) thus f /. (k1 + 1) = (GoB f) * ((i + 1),(j + 1)) by A10, A68, A65, A63, A67, SPPOL_1:8; ::_thesis: ( f /. k1 = (GoB f) * (i,(j + 1)) & f /. (k1 + 2) = (GoB f) * ((i + 1),j) ) thus f /. k1 = (GoB f) * (i,(j + 1)) by A10, A68, A65, A63, A67, SPPOL_1:8; ::_thesis: f /. (k1 + 2) = (GoB f) * ((i + 1),j) thus f /. (k1 + 2) = (GoB f) * ((i + 1),j) by A10, A68, A65, A63, A69, SPPOL_1:8; ::_thesis: verum end; casethat A70: k2 > 1 and A71: k1 > k2 ; ::_thesis: ( 1 <= k2 & k2 + 1 < len f & f /. (k2 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. k2 = (GoB f) * ((i + 1),j) & f /. (k2 + 2) = (GoB f) * (i,(j + 1)) ) A72: ( 1 < k2 + 1 & k2 + 1 < k1 + 1 ) by A70, A71, NAT_1:13, XREAL_1:6; A73: k2 < k1 + 1 by A71, NAT_1:13; then A74: f /. k2 <> f /. (k1 + 1) by A9, A70, Th37; A75: k2 + 1 <= k1 by A71, NAT_1:13; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A8, A9, TOPREAL1:def_3; then A76: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. k1 & (GoB f) * (i,(j + 1)) = f /. (k1 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k1 + 1) & (GoB f) * (i,(j + 1)) = f /. k1 ) ) by A10, SPPOL_1:8; A77: k1 < len f by A9, NAT_1:13; then A78: f /. k2 <> f /. k1 by A70, A71, Th37; A79: LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A15, A16, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 1),j) = f /. k2 & (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k2 + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. k2 ) ) by A17, SPPOL_1:8; then k2 + 1 >= k1 by A9, A70, A71, A76, A73, A77, A72, Th37; then A80: k2 + 1 = k1 by A75, XXREAL_0:1; hence ( 1 <= k2 & k2 + 1 < len f ) by A9, A70, NAT_1:13; ::_thesis: ( f /. (k2 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. k2 = (GoB f) * ((i + 1),j) & f /. (k2 + 2) = (GoB f) * (i,(j + 1)) ) thus f /. (k2 + 1) = (GoB f) * ((i + 1),(j + 1)) by A17, A79, A76, A74, A78, SPPOL_1:8; ::_thesis: ( f /. k2 = (GoB f) * ((i + 1),j) & f /. (k2 + 2) = (GoB f) * (i,(j + 1)) ) thus f /. k2 = (GoB f) * ((i + 1),j) by A17, A79, A76, A74, A78, SPPOL_1:8; ::_thesis: f /. (k2 + 2) = (GoB f) * (i,(j + 1)) thus f /. (k2 + 2) = (GoB f) * (i,(j + 1)) by A17, A79, A76, A74, A80, SPPOL_1:8; ::_thesis: verum end; end; end; hence ( ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) or ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) ) ; ::_thesis: verum end; theorem Th56: :: GOBOARD7:56 for i, j being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 < len (GoB f) & 1 <= j & j <= width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),j) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) ) or ( f /. 2 = (GoB f) * ((i + 2),j) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) proof let i, j be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 < len (GoB f) & 1 <= j & j <= width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),j) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) ) or ( f /. 2 = (GoB f) * ((i + 2),j) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 < len (GoB f) & 1 <= j & j <= width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),j) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) ) or ( f /. 2 = (GoB f) * ((i + 2),j) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) implies ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) assume that A1: 1 <= i and A2: i + 1 < len (GoB f) and A3: ( 1 <= j & j <= width (GoB f) ) and A4: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f and A5: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) c= L~ f ; ::_thesis: ( ( f /. 1 = (GoB f) * ((i + 1),j) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) ) or ( f /. 2 = (GoB f) * ((i + 2),j) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) A6: 1 <= i + 1 by NAT_1:11; (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))) in LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) by RLTOPSP1:69; then consider k1 being Element of NAT such that A7: 1 <= k1 and A8: k1 + 1 <= len f and A9: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k1) by A1, A2, A3, A4, Th40; A10: k1 < len f by A8, NAT_1:13; A11: now__::_thesis:_(_not_k1_>_1_or_k1_=_2_or_k1_>_2_) assume k1 > 1 ; ::_thesis: ( k1 = 2 or k1 > 2 ) then k1 >= 1 + 1 by NAT_1:13; hence ( k1 = 2 or k1 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A12: i < len (GoB f) by A2, NAT_1:13; A13: i + (1 + 1) = (i + 1) + 1 ; then A14: 1 <= i + 2 by NAT_1:11; A15: i + 2 <= len (GoB f) by A2, A13, NAT_1:13; (1 / 2) * (((GoB f) * ((i + 1),j)) + ((GoB f) * ((i + 2),j))) in LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) by RLTOPSP1:69; then consider k2 being Element of NAT such that A16: 1 <= k2 and A17: k2 + 1 <= len f and A18: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) = LSeg (f,k2) by A3, A5, A6, A13, A15, Th40; A19: k2 < len f by A17, NAT_1:13; A20: now__::_thesis:_(_not_k2_>_1_or_k2_=_2_or_k2_>_2_) assume k2 > 1 ; ::_thesis: ( k2 = 2 or k2 > 2 ) then k2 >= 1 + 1 by NAT_1:13; hence ( k2 = 2 or k2 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A21: ( k1 = 1 or k1 > 1 ) by A7, XXREAL_0:1; now__::_thesis:_(_(_k1_=_1_&_k2_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_((i_+_1),j)_&_f_/._1_=_(GoB_f)_*_(i,j)_&_f_/._(1_+_2)_=_(GoB_f)_*_((i_+_2),j)_)_or_(_k1_=_1_&_k2_>_2_&_f_/._1_=_(GoB_f)_*_((i_+_1),j)_&_f_/._2_=_(GoB_f)_*_(i,j)_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_((i_+_2),j)_)_or_(_k2_=_1_&_k1_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_((i_+_1),j)_&_f_/._1_=_(GoB_f)_*_((i_+_2),j)_&_f_/._(1_+_2)_=_(GoB_f)_*_(i,j)_)_or_(_k2_=_1_&_k1_>_2_&_f_/._1_=_(GoB_f)_*_((i_+_1),j)_&_f_/._2_=_(GoB_f)_*_((i_+_2),j)_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_(i,j)_)_or_(_k1_=_k2_&_contradiction_)_or_(_k1_>_1_&_k2_>_k1_&_1_<=_k1_&_k1_+_1_<_len_f_&_f_/._(k1_+_1)_=_(GoB_f)_*_((i_+_1),j)_&_f_/._k1_=_(GoB_f)_*_(i,j)_&_f_/._(k1_+_2)_=_(GoB_f)_*_((i_+_2),j)_)_or_(_k2_>_1_&_k1_>_k2_&_1_<=_k2_&_k2_+_1_<_len_f_&_f_/._(k2_+_1)_=_(GoB_f)_*_((i_+_1),j)_&_f_/._k2_=_(GoB_f)_*_((i_+_2),j)_&_f_/._(k2_+_2)_=_(GoB_f)_*_(i,j)_)_) percases ( ( k1 = 1 & k2 = 2 ) or ( k1 = 1 & k2 > 2 ) or ( k2 = 1 & k1 = 2 ) or ( k2 = 1 & k1 > 2 ) or k1 = k2 or ( k1 > 1 & k2 > k1 ) or ( k2 > 1 & k1 > k2 ) ) by A16, A11, A20, A21, XXREAL_0:1; casethat A22: k1 = 1 and A23: k2 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * ((i + 1),j) & f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * ((i + 2),j) ) A24: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A17, A23, TOPREAL1:def_3; then A25: ( ( (GoB f) * ((i + 1),j) = f /. 2 & (GoB f) * ((i + 2),j) = f /. (2 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (2 + 1) & (GoB f) * ((i + 2),j) = f /. 2 ) ) by A18, A23, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A17, A23, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * ((i + 1),j) & f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * ((i + 2),j) ) A26: 3 < len f by Th34, XXREAL_0:2; then A27: f /. 1 <> f /. 3 by Th36; A28: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A8, A22, TOPREAL1:def_3; then A29: ( ( (GoB f) * (i,j) = f /. 1 & (GoB f) * ((i + 1),j) = f /. 2 ) or ( (GoB f) * (i,j) = f /. 2 & (GoB f) * ((i + 1),j) = f /. 1 ) ) by A9, A22, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * ((i + 1),j) by A25, A26, Th36; ::_thesis: ( f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * ((i + 2),j) ) thus f /. 1 = (GoB f) * (i,j) by A18, A23, A29, A24, A27, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * ((i + 2),j) thus f /. (1 + 2) = (GoB f) * ((i + 2),j) by A9, A22, A28, A25, A27, SPPOL_1:8; ::_thesis: verum end; casethat A30: k1 = 1 and A31: k2 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),j) & f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) ) A32: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A8, A30, TOPREAL1:def_3; then A33: ( ( (GoB f) * (i,j) = f /. 1 & (GoB f) * ((i + 1),j) = f /. 2 ) or ( (GoB f) * (i,j) = f /. 2 & (GoB f) * ((i + 1),j) = f /. 1 ) ) by A9, A30, SPPOL_1:8; A34: 2 < k2 + 1 by A31, NAT_1:13; then A35: f /. (k2 + 1) <> f /. 2 by A17, Th37; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then A36: ( ( (GoB f) * ((i + 1),j) = f /. k2 & (GoB f) * ((i + 2),j) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k2 + 1) & (GoB f) * ((i + 2),j) = f /. k2 ) ) by A18, SPPOL_1:8; A37: f /. k2 <> f /. 2 by A19, A31, Th36; hence f /. 1 = (GoB f) * ((i + 1),j) by A9, A30, A32, A36, A35, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) ) thus f /. 2 = (GoB f) * (i,j) by A9, A30, A32, A36, A37, A35, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) A38: k2 > 1 by A31, XXREAL_0:2; then A39: k2 + 1 > 1 by NAT_1:13; then k2 + 1 = len f by A17, A19, A31, A33, A36, A38, A34, Th37, Th38; then k2 + 1 = ((len f) -' 1) + 1 by A39, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) by A19, A31, A33, A36, A38, Th36; ::_thesis: verum end; casethat A40: k2 = 1 and A41: k1 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * ((i + 1),j) & f /. 1 = (GoB f) * ((i + 2),j) & f /. (1 + 2) = (GoB f) * (i,j) ) A42: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A8, A41, TOPREAL1:def_3; then A43: ( ( (GoB f) * ((i + 1),j) = f /. 2 & (GoB f) * (i,j) = f /. (2 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (2 + 1) & (GoB f) * (i,j) = f /. 2 ) ) by A9, A41, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A8, A41, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * ((i + 1),j) & f /. 1 = (GoB f) * ((i + 2),j) & f /. (1 + 2) = (GoB f) * (i,j) ) A44: 3 < len f by Th34, XXREAL_0:2; then A45: f /. 1 <> f /. 3 by Th36; A46: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A17, A40, TOPREAL1:def_3; then A47: ( ( (GoB f) * ((i + 2),j) = f /. 1 & (GoB f) * ((i + 1),j) = f /. 2 ) or ( (GoB f) * ((i + 2),j) = f /. 2 & (GoB f) * ((i + 1),j) = f /. 1 ) ) by A18, A40, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * ((i + 1),j) by A43, A44, Th36; ::_thesis: ( f /. 1 = (GoB f) * ((i + 2),j) & f /. (1 + 2) = (GoB f) * (i,j) ) thus f /. 1 = (GoB f) * ((i + 2),j) by A9, A41, A47, A42, A45, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * (i,j) thus f /. (1 + 2) = (GoB f) * (i,j) by A18, A40, A46, A43, A45, SPPOL_1:8; ::_thesis: verum end; casethat A48: k2 = 1 and A49: k1 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),j) & f /. 2 = (GoB f) * ((i + 2),j) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) A50: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A17, A48, TOPREAL1:def_3; then A51: ( ( (GoB f) * ((i + 2),j) = f /. 1 & (GoB f) * ((i + 1),j) = f /. 2 ) or ( (GoB f) * ((i + 2),j) = f /. 2 & (GoB f) * ((i + 1),j) = f /. 1 ) ) by A18, A48, SPPOL_1:8; A52: 2 < k1 + 1 by A49, NAT_1:13; then A53: f /. (k1 + 1) <> f /. 2 by A8, Th37; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A7, A8, TOPREAL1:def_3; then A54: ( ( (GoB f) * ((i + 1),j) = f /. k1 & (GoB f) * (i,j) = f /. (k1 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k1 + 1) & (GoB f) * (i,j) = f /. k1 ) ) by A9, SPPOL_1:8; A55: f /. k1 <> f /. 2 by A10, A49, Th36; hence f /. 1 = (GoB f) * ((i + 1),j) by A18, A48, A50, A54, A53, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * ((i + 2),j) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) thus f /. 2 = (GoB f) * ((i + 2),j) by A18, A48, A50, A54, A55, A53, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * (i,j) A56: k1 > 1 by A49, XXREAL_0:2; then A57: k1 + 1 > 1 by NAT_1:13; then k1 + 1 = len f by A8, A10, A49, A51, A54, A56, A52, Th37, Th38; then k1 + 1 = ((len f) -' 1) + 1 by A57, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * (i,j) by A10, A49, A51, A54, A56, Th36; ::_thesis: verum end; case k1 = k2 ; ::_thesis: contradiction then A58: ( (GoB f) * (i,j) = (GoB f) * ((i + 2),j) or (GoB f) * (i,j) = (GoB f) * ((i + 1),j) ) by A9, A18, SPPOL_1:8; A59: [(i + 2),j] in Indices (GoB f) by A3, A15, A14, MATRIX_1:36; ( [i,j] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A1, A2, A3, A6, A12, MATRIX_1:36; then ( i = i + 1 or i = i + 2 ) by A58, A59, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; casethat A60: k1 > 1 and A61: k2 > k1 ; ::_thesis: ( 1 <= k1 & k1 + 1 < len f & f /. (k1 + 1) = (GoB f) * ((i + 1),j) & f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * ((i + 2),j) ) A62: ( 1 < k1 + 1 & k1 + 1 < k2 + 1 ) by A60, A61, NAT_1:13, XREAL_1:6; A63: k1 < k2 + 1 by A61, NAT_1:13; then A64: f /. k1 <> f /. (k2 + 1) by A17, A60, Th37; A65: k1 + 1 <= k2 by A61, NAT_1:13; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then A66: ( ( (GoB f) * ((i + 1),j) = f /. k2 & (GoB f) * ((i + 2),j) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k2 + 1) & (GoB f) * ((i + 2),j) = f /. k2 ) ) by A18, SPPOL_1:8; A67: k2 < len f by A17, NAT_1:13; then A68: f /. k1 <> f /. k2 by A60, A61, Th37; A69: LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A7, A8, TOPREAL1:def_3; then ( ( (GoB f) * (i,j) = f /. k1 & (GoB f) * ((i + 1),j) = f /. (k1 + 1) ) or ( (GoB f) * (i,j) = f /. (k1 + 1) & (GoB f) * ((i + 1),j) = f /. k1 ) ) by A9, SPPOL_1:8; then k1 + 1 >= k2 by A17, A60, A61, A66, A63, A67, A62, Th37; then A70: k1 + 1 = k2 by A65, XXREAL_0:1; hence ( 1 <= k1 & k1 + 1 < len f ) by A17, A60, NAT_1:13; ::_thesis: ( f /. (k1 + 1) = (GoB f) * ((i + 1),j) & f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * ((i + 2),j) ) thus f /. (k1 + 1) = (GoB f) * ((i + 1),j) by A9, A69, A66, A64, A68, SPPOL_1:8; ::_thesis: ( f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * ((i + 2),j) ) thus f /. k1 = (GoB f) * (i,j) by A9, A69, A66, A64, A68, SPPOL_1:8; ::_thesis: f /. (k1 + 2) = (GoB f) * ((i + 2),j) thus f /. (k1 + 2) = (GoB f) * ((i + 2),j) by A9, A69, A66, A64, A70, SPPOL_1:8; ::_thesis: verum end; casethat A71: k2 > 1 and A72: k1 > k2 ; ::_thesis: ( 1 <= k2 & k2 + 1 < len f & f /. (k2 + 1) = (GoB f) * ((i + 1),j) & f /. k2 = (GoB f) * ((i + 2),j) & f /. (k2 + 2) = (GoB f) * (i,j) ) A73: ( 1 < k2 + 1 & k2 + 1 < k1 + 1 ) by A71, A72, NAT_1:13, XREAL_1:6; A74: k2 < k1 + 1 by A72, NAT_1:13; then A75: f /. k2 <> f /. (k1 + 1) by A8, A71, Th37; A76: k2 + 1 <= k1 by A72, NAT_1:13; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A7, A8, TOPREAL1:def_3; then A77: ( ( (GoB f) * ((i + 1),j) = f /. k1 & (GoB f) * (i,j) = f /. (k1 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k1 + 1) & (GoB f) * (i,j) = f /. k1 ) ) by A9, SPPOL_1:8; A78: k1 < len f by A8, NAT_1:13; then A79: f /. k2 <> f /. k1 by A71, A72, Th37; A80: LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 2),j) = f /. k2 & (GoB f) * ((i + 1),j) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 2),j) = f /. (k2 + 1) & (GoB f) * ((i + 1),j) = f /. k2 ) ) by A18, SPPOL_1:8; then k2 + 1 >= k1 by A8, A71, A72, A77, A74, A78, A73, Th37; then A81: k2 + 1 = k1 by A76, XXREAL_0:1; hence ( 1 <= k2 & k2 + 1 < len f ) by A8, A71, NAT_1:13; ::_thesis: ( f /. (k2 + 1) = (GoB f) * ((i + 1),j) & f /. k2 = (GoB f) * ((i + 2),j) & f /. (k2 + 2) = (GoB f) * (i,j) ) thus f /. (k2 + 1) = (GoB f) * ((i + 1),j) by A18, A80, A77, A75, A79, SPPOL_1:8; ::_thesis: ( f /. k2 = (GoB f) * ((i + 2),j) & f /. (k2 + 2) = (GoB f) * (i,j) ) thus f /. k2 = (GoB f) * ((i + 2),j) by A18, A80, A77, A75, A79, SPPOL_1:8; ::_thesis: f /. (k2 + 2) = (GoB f) * (i,j) thus f /. (k2 + 2) = (GoB f) * (i,j) by A18, A80, A77, A75, A81, SPPOL_1:8; ::_thesis: verum end; end; end; hence ( ( f /. 1 = (GoB f) * ((i + 1),j) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) ) or ( f /. 2 = (GoB f) * ((i + 2),j) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) ; ::_thesis: verum end; theorem Th57: :: GOBOARD7:57 for i, j being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),j) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) proof let i, j be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),j) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),j) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) implies ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f and A6: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f ; ::_thesis: ( ( f /. 1 = (GoB f) * ((i + 1),j) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) A7: j < width (GoB f) by A4, NAT_1:13; A8: i < len (GoB f) by A2, NAT_1:13; A9: 1 <= j + 1 by NAT_1:11; (1 / 2) * (((GoB f) * (i,j)) + ((GoB f) * ((i + 1),j))) in LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) by RLTOPSP1:69; then consider k1 being Element of NAT such that A10: 1 <= k1 and A11: k1 + 1 <= len f and A12: LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) = LSeg (f,k1) by A1, A2, A3, A5, A7, Th40; A13: k1 < len f by A11, NAT_1:13; A14: now__::_thesis:_(_not_k1_>_1_or_k1_=_2_or_k1_>_2_) assume k1 > 1 ; ::_thesis: ( k1 = 2 or k1 > 2 ) then k1 >= 1 + 1 by NAT_1:13; hence ( k1 = 2 or k1 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A15: 1 <= i + 1 by NAT_1:11; (1 / 2) * (((GoB f) * ((i + 1),j)) + ((GoB f) * ((i + 1),(j + 1)))) in LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) by RLTOPSP1:69; then consider k2 being Element of NAT such that A16: 1 <= k2 and A17: k2 + 1 <= len f and A18: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k2) by A2, A3, A4, A6, A15, Th39; A19: k2 < len f by A17, NAT_1:13; A20: now__::_thesis:_(_not_k2_>_1_or_k2_=_2_or_k2_>_2_) assume k2 > 1 ; ::_thesis: ( k2 = 2 or k2 > 2 ) then k2 >= 1 + 1 by NAT_1:13; hence ( k2 = 2 or k2 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A21: ( k1 = 1 or k1 > 1 ) by A10, XXREAL_0:1; now__::_thesis:_(_(_k1_=_1_&_k2_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_((i_+_1),j)_&_f_/._1_=_(GoB_f)_*_(i,j)_&_f_/._(1_+_2)_=_(GoB_f)_*_((i_+_1),(j_+_1))_)_or_(_k1_=_1_&_k2_>_2_&_f_/._1_=_(GoB_f)_*_((i_+_1),j)_&_f_/._2_=_(GoB_f)_*_(i,j)_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_((i_+_1),(j_+_1))_)_or_(_k2_=_1_&_k1_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_((i_+_1),j)_&_f_/._1_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._(1_+_2)_=_(GoB_f)_*_(i,j)_)_or_(_k2_=_1_&_k1_>_2_&_f_/._1_=_(GoB_f)_*_((i_+_1),j)_&_f_/._2_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_(i,j)_)_or_(_k1_=_k2_&_contradiction_)_or_(_k1_>_1_&_k2_>_k1_&_1_<=_k1_&_k1_+_1_<_len_f_&_f_/._(k1_+_1)_=_(GoB_f)_*_((i_+_1),j)_&_f_/._k1_=_(GoB_f)_*_(i,j)_&_f_/._(k1_+_2)_=_(GoB_f)_*_((i_+_1),(j_+_1))_)_or_(_k2_>_1_&_k1_>_k2_&_1_<=_k2_&_k2_+_1_<_len_f_&_f_/._(k2_+_1)_=_(GoB_f)_*_((i_+_1),j)_&_f_/._k2_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._(k2_+_2)_=_(GoB_f)_*_(i,j)_)_) percases ( ( k1 = 1 & k2 = 2 ) or ( k1 = 1 & k2 > 2 ) or ( k2 = 1 & k1 = 2 ) or ( k2 = 1 & k1 > 2 ) or k1 = k2 or ( k1 > 1 & k2 > k1 ) or ( k2 > 1 & k1 > k2 ) ) by A16, A14, A20, A21, XXREAL_0:1; casethat A22: k1 = 1 and A23: k2 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * ((i + 1),j) & f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) A24: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A17, A23, TOPREAL1:def_3; then A25: ( ( (GoB f) * ((i + 1),j) = f /. 2 & (GoB f) * ((i + 1),(j + 1)) = f /. (2 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (2 + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. 2 ) ) by A18, A23, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A17, A23, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * ((i + 1),j) & f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) A26: 3 < len f by Th34, XXREAL_0:2; then A27: f /. 1 <> f /. 3 by Th36; A28: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A11, A22, TOPREAL1:def_3; then A29: ( ( (GoB f) * (i,j) = f /. 1 & (GoB f) * ((i + 1),j) = f /. 2 ) or ( (GoB f) * (i,j) = f /. 2 & (GoB f) * ((i + 1),j) = f /. 1 ) ) by A12, A22, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * ((i + 1),j) by A25, A26, Th36; ::_thesis: ( f /. 1 = (GoB f) * (i,j) & f /. (1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) thus f /. 1 = (GoB f) * (i,j) by A18, A23, A29, A24, A27, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * ((i + 1),(j + 1)) thus f /. (1 + 2) = (GoB f) * ((i + 1),(j + 1)) by A12, A22, A28, A25, A27, SPPOL_1:8; ::_thesis: verum end; casethat A30: k1 = 1 and A31: k2 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),j) & f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) A32: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A11, A30, TOPREAL1:def_3; then A33: ( ( (GoB f) * (i,j) = f /. 1 & (GoB f) * ((i + 1),j) = f /. 2 ) or ( (GoB f) * (i,j) = f /. 2 & (GoB f) * ((i + 1),j) = f /. 1 ) ) by A12, A30, SPPOL_1:8; A34: 2 < k2 + 1 by A31, NAT_1:13; then A35: f /. (k2 + 1) <> f /. 2 by A17, Th37; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then A36: ( ( (GoB f) * ((i + 1),j) = f /. k2 & (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k2 + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. k2 ) ) by A18, SPPOL_1:8; A37: f /. k2 <> f /. 2 by A19, A31, Th36; hence f /. 1 = (GoB f) * ((i + 1),j) by A12, A30, A32, A36, A35, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) thus f /. 2 = (GoB f) * (i,j) by A12, A30, A32, A36, A37, A35, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) A38: k2 > 1 by A31, XXREAL_0:2; then A39: k2 + 1 > 1 by NAT_1:13; then k2 + 1 = len f by A17, A19, A31, A33, A36, A38, A34, Th37, Th38; then k2 + 1 = ((len f) -' 1) + 1 by A39, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) by A19, A31, A33, A36, A38, Th36; ::_thesis: verum end; casethat A40: k2 = 1 and A41: k1 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * ((i + 1),j) & f /. 1 = (GoB f) * ((i + 1),(j + 1)) & f /. (1 + 2) = (GoB f) * (i,j) ) A42: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A11, A41, TOPREAL1:def_3; then A43: ( ( (GoB f) * ((i + 1),j) = f /. 2 & (GoB f) * (i,j) = f /. (2 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (2 + 1) & (GoB f) * (i,j) = f /. 2 ) ) by A12, A41, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A11, A41, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * ((i + 1),j) & f /. 1 = (GoB f) * ((i + 1),(j + 1)) & f /. (1 + 2) = (GoB f) * (i,j) ) A44: 3 < len f by Th34, XXREAL_0:2; then A45: f /. 1 <> f /. 3 by Th36; A46: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A17, A40, TOPREAL1:def_3; then A47: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. 1 & (GoB f) * ((i + 1),j) = f /. 2 ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. 2 & (GoB f) * ((i + 1),j) = f /. 1 ) ) by A18, A40, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * ((i + 1),j) by A43, A44, Th36; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & f /. (1 + 2) = (GoB f) * (i,j) ) thus f /. 1 = (GoB f) * ((i + 1),(j + 1)) by A12, A41, A47, A42, A45, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * (i,j) thus f /. (1 + 2) = (GoB f) * (i,j) by A18, A40, A46, A43, A45, SPPOL_1:8; ::_thesis: verum end; casethat A48: k2 = 1 and A49: k1 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),j) & f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) A50: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A17, A48, TOPREAL1:def_3; then A51: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. 1 & (GoB f) * ((i + 1),j) = f /. 2 ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. 2 & (GoB f) * ((i + 1),j) = f /. 1 ) ) by A18, A48, SPPOL_1:8; A52: 2 < k1 + 1 by A49, NAT_1:13; then A53: f /. (k1 + 1) <> f /. 2 by A11, Th37; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A10, A11, TOPREAL1:def_3; then A54: ( ( (GoB f) * ((i + 1),j) = f /. k1 & (GoB f) * (i,j) = f /. (k1 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k1 + 1) & (GoB f) * (i,j) = f /. k1 ) ) by A12, SPPOL_1:8; A55: f /. k1 <> f /. 2 by A13, A49, Th36; hence f /. 1 = (GoB f) * ((i + 1),j) by A18, A48, A50, A54, A53, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) thus f /. 2 = (GoB f) * ((i + 1),(j + 1)) by A18, A48, A50, A54, A55, A53, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * (i,j) A56: k1 > 1 by A49, XXREAL_0:2; then A57: k1 + 1 > 1 by NAT_1:13; then k1 + 1 = len f by A11, A13, A49, A51, A54, A56, A52, Th37, Th38; then k1 + 1 = ((len f) -' 1) + 1 by A57, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * (i,j) by A13, A49, A51, A54, A56, Th36; ::_thesis: verum end; case k1 = k2 ; ::_thesis: contradiction then A58: ( (GoB f) * (i,j) = (GoB f) * ((i + 1),(j + 1)) or (GoB f) * (i,j) = (GoB f) * ((i + 1),j) ) by A12, A18, SPPOL_1:8; A59: [(i + 1),(j + 1)] in Indices (GoB f) by A2, A4, A15, A9, MATRIX_1:36; ( [i,j] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A1, A2, A3, A15, A8, A7, MATRIX_1:36; then i = i + 1 by A58, A59, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; casethat A60: k1 > 1 and A61: k2 > k1 ; ::_thesis: ( 1 <= k1 & k1 + 1 < len f & f /. (k1 + 1) = (GoB f) * ((i + 1),j) & f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) A62: ( 1 < k1 + 1 & k1 + 1 < k2 + 1 ) by A60, A61, NAT_1:13, XREAL_1:6; A63: k1 < k2 + 1 by A61, NAT_1:13; then A64: f /. k1 <> f /. (k2 + 1) by A17, A60, Th37; A65: k1 + 1 <= k2 by A61, NAT_1:13; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then A66: ( ( (GoB f) * ((i + 1),j) = f /. k2 & (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k2 + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. k2 ) ) by A18, SPPOL_1:8; A67: k2 < len f by A17, NAT_1:13; then A68: f /. k1 <> f /. k2 by A60, A61, Th37; A69: LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A10, A11, TOPREAL1:def_3; then ( ( (GoB f) * (i,j) = f /. k1 & (GoB f) * ((i + 1),j) = f /. (k1 + 1) ) or ( (GoB f) * (i,j) = f /. (k1 + 1) & (GoB f) * ((i + 1),j) = f /. k1 ) ) by A12, SPPOL_1:8; then k1 + 1 >= k2 by A17, A60, A61, A66, A63, A67, A62, Th37; then A70: k1 + 1 = k2 by A65, XXREAL_0:1; hence ( 1 <= k1 & k1 + 1 < len f ) by A17, A60, NAT_1:13; ::_thesis: ( f /. (k1 + 1) = (GoB f) * ((i + 1),j) & f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) thus f /. (k1 + 1) = (GoB f) * ((i + 1),j) by A12, A69, A66, A64, A68, SPPOL_1:8; ::_thesis: ( f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 1)) ) thus f /. k1 = (GoB f) * (i,j) by A12, A69, A66, A64, A68, SPPOL_1:8; ::_thesis: f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 1)) thus f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 1)) by A12, A69, A66, A64, A70, SPPOL_1:8; ::_thesis: verum end; casethat A71: k2 > 1 and A72: k1 > k2 ; ::_thesis: ( 1 <= k2 & k2 + 1 < len f & f /. (k2 + 1) = (GoB f) * ((i + 1),j) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) & f /. (k2 + 2) = (GoB f) * (i,j) ) A73: ( 1 < k2 + 1 & k2 + 1 < k1 + 1 ) by A71, A72, NAT_1:13, XREAL_1:6; A74: k2 < k1 + 1 by A72, NAT_1:13; then A75: f /. k2 <> f /. (k1 + 1) by A11, A71, Th37; A76: k2 + 1 <= k1 by A72, NAT_1:13; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A10, A11, TOPREAL1:def_3; then A77: ( ( (GoB f) * ((i + 1),j) = f /. k1 & (GoB f) * (i,j) = f /. (k1 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k1 + 1) & (GoB f) * (i,j) = f /. k1 ) ) by A12, SPPOL_1:8; A78: k1 < len f by A11, NAT_1:13; then A79: f /. k2 <> f /. k1 by A71, A72, Th37; A80: LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A16, A17, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 1),(j + 1)) = f /. k2 & (GoB f) * ((i + 1),j) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) & (GoB f) * ((i + 1),j) = f /. k2 ) ) by A18, SPPOL_1:8; then k2 + 1 >= k1 by A11, A71, A72, A77, A74, A78, A73, Th37; then A81: k2 + 1 = k1 by A76, XXREAL_0:1; hence ( 1 <= k2 & k2 + 1 < len f ) by A11, A71, NAT_1:13; ::_thesis: ( f /. (k2 + 1) = (GoB f) * ((i + 1),j) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) & f /. (k2 + 2) = (GoB f) * (i,j) ) thus f /. (k2 + 1) = (GoB f) * ((i + 1),j) by A18, A80, A77, A75, A79, SPPOL_1:8; ::_thesis: ( f /. k2 = (GoB f) * ((i + 1),(j + 1)) & f /. (k2 + 2) = (GoB f) * (i,j) ) thus f /. k2 = (GoB f) * ((i + 1),(j + 1)) by A18, A80, A77, A75, A79, SPPOL_1:8; ::_thesis: f /. (k2 + 2) = (GoB f) * (i,j) thus f /. (k2 + 2) = (GoB f) * (i,j) by A18, A80, A77, A75, A81, SPPOL_1:8; ::_thesis: verum end; end; end; hence ( ( f /. 1 = (GoB f) * ((i + 1),j) & ( ( f /. 2 = (GoB f) * (i,j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * (i,j) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) ; ::_thesis: verum end; theorem Th58: :: GOBOARD7:58 for i, j being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * (i,(j + 1)))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) or ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) proof let i, j be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * (i,(j + 1)))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) or ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) ) ) holds ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i + 1 <= len (GoB f) & 1 <= j & j + 1 <= width (GoB f) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * (i,(j + 1)))) c= L~ f & not ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) or ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) ) ) implies ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) ) assume that A1: 1 <= i and A2: i + 1 <= len (GoB f) and A3: 1 <= j and A4: j + 1 <= width (GoB f) and A5: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f and A6: LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * (i,(j + 1)))) c= L~ f ; ::_thesis: ( ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) or ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) ) A7: 1 <= i + 1 by NAT_1:11; (1 / 2) * (((GoB f) * ((i + 1),j)) + ((GoB f) * ((i + 1),(j + 1)))) in LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) by RLTOPSP1:69; then consider k1 being Element of NAT such that A8: 1 <= k1 and A9: k1 + 1 <= len f and A10: LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k1) by A2, A3, A4, A5, A7, Th39; A11: k1 < len f by A9, NAT_1:13; A12: now__::_thesis:_(_not_k1_>_1_or_k1_=_2_or_k1_>_2_) assume k1 > 1 ; ::_thesis: ( k1 = 2 or k1 > 2 ) then k1 >= 1 + 1 by NAT_1:13; hence ( k1 = 2 or k1 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A13: ( i < len (GoB f) & j < width (GoB f) ) by A2, A4, NAT_1:13; A14: 1 <= j + 1 by NAT_1:11; (1 / 2) * (((GoB f) * (i,(j + 1))) + ((GoB f) * ((i + 1),(j + 1)))) in LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * (i,(j + 1)))) by RLTOPSP1:69; then consider k2 being Element of NAT such that A15: 1 <= k2 and A16: k2 + 1 <= len f and A17: LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) = LSeg (f,k2) by A1, A2, A4, A6, A14, Th40; A18: k2 < len f by A16, NAT_1:13; A19: now__::_thesis:_(_not_k2_>_1_or_k2_=_2_or_k2_>_2_) assume k2 > 1 ; ::_thesis: ( k2 = 2 or k2 > 2 ) then k2 >= 1 + 1 by NAT_1:13; hence ( k2 = 2 or k2 > 2 ) by XXREAL_0:1; ::_thesis: verum end; A20: ( k1 = 1 or k1 > 1 ) by A8, XXREAL_0:1; now__::_thesis:_(_(_k1_=_1_&_k2_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._1_=_(GoB_f)_*_((i_+_1),j)_&_f_/._(1_+_2)_=_(GoB_f)_*_(i,(j_+_1))_)_or_(_k1_=_1_&_k2_>_2_&_f_/._1_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._2_=_(GoB_f)_*_((i_+_1),j)_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_(i,(j_+_1))_)_or_(_k2_=_1_&_k1_=_2_&_1_<=_1_&_1_+_1_<_len_f_&_f_/._(1_+_1)_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._1_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._(1_+_2)_=_(GoB_f)_*_((i_+_1),j)_)_or_(_k2_=_1_&_k1_>_2_&_f_/._1_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._2_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._((len_f)_-'_1)_=_(GoB_f)_*_((i_+_1),j)_)_or_(_k1_=_k2_&_contradiction_)_or_(_k1_>_1_&_k2_>_k1_&_1_<=_k1_&_k1_+_1_<_len_f_&_f_/._(k1_+_1)_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._k1_=_(GoB_f)_*_((i_+_1),j)_&_f_/._(k1_+_2)_=_(GoB_f)_*_(i,(j_+_1))_)_or_(_k2_>_1_&_k1_>_k2_&_1_<=_k2_&_k2_+_1_<_len_f_&_f_/._(k2_+_1)_=_(GoB_f)_*_((i_+_1),(j_+_1))_&_f_/._k2_=_(GoB_f)_*_(i,(j_+_1))_&_f_/._(k2_+_2)_=_(GoB_f)_*_((i_+_1),j)_)_) percases ( ( k1 = 1 & k2 = 2 ) or ( k1 = 1 & k2 > 2 ) or ( k2 = 1 & k1 = 2 ) or ( k2 = 1 & k1 > 2 ) or k1 = k2 or ( k1 > 1 & k2 > k1 ) or ( k2 > 1 & k1 > k2 ) ) by A15, A12, A19, A20, XXREAL_0:1; casethat A21: k1 = 1 and A22: k2 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. 1 = (GoB f) * ((i + 1),j) & f /. (1 + 2) = (GoB f) * (i,(j + 1)) ) A23: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A16, A22, TOPREAL1:def_3; then A24: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. 2 & (GoB f) * (i,(j + 1)) = f /. (2 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (2 + 1) & (GoB f) * (i,(j + 1)) = f /. 2 ) ) by A17, A22, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A16, A22, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. 1 = (GoB f) * ((i + 1),j) & f /. (1 + 2) = (GoB f) * (i,(j + 1)) ) A25: 3 < len f by Th34, XXREAL_0:2; then A26: f /. 1 <> f /. 3 by Th36; A27: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A9, A21, TOPREAL1:def_3; then A28: ( ( (GoB f) * ((i + 1),j) = f /. 1 & (GoB f) * ((i + 1),(j + 1)) = f /. 2 ) or ( (GoB f) * ((i + 1),j) = f /. 2 & (GoB f) * ((i + 1),(j + 1)) = f /. 1 ) ) by A10, A21, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) by A24, A25, Th36; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),j) & f /. (1 + 2) = (GoB f) * (i,(j + 1)) ) thus f /. 1 = (GoB f) * ((i + 1),j) by A17, A22, A28, A23, A26, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * (i,(j + 1)) thus f /. (1 + 2) = (GoB f) * (i,(j + 1)) by A10, A21, A27, A24, A26, SPPOL_1:8; ::_thesis: verum end; casethat A29: k1 = 1 and A30: k2 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) A31: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A9, A29, TOPREAL1:def_3; then A32: ( ( (GoB f) * ((i + 1),j) = f /. 1 & (GoB f) * ((i + 1),(j + 1)) = f /. 2 ) or ( (GoB f) * ((i + 1),j) = f /. 2 & (GoB f) * ((i + 1),(j + 1)) = f /. 1 ) ) by A10, A29, SPPOL_1:8; A33: 2 < k2 + 1 by A30, NAT_1:13; then A34: f /. (k2 + 1) <> f /. 2 by A16, Th37; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A15, A16, TOPREAL1:def_3; then A35: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. k2 & (GoB f) * (i,(j + 1)) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) & (GoB f) * (i,(j + 1)) = f /. k2 ) ) by A17, SPPOL_1:8; A36: f /. k2 <> f /. 2 by A18, A30, Th36; hence f /. 1 = (GoB f) * ((i + 1),(j + 1)) by A10, A29, A31, A35, A34, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) thus f /. 2 = (GoB f) * ((i + 1),j) by A10, A29, A31, A35, A36, A34, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) A37: k2 > 1 by A30, XXREAL_0:2; then A38: k2 + 1 > 1 by NAT_1:13; then k2 + 1 = len f by A16, A18, A30, A32, A35, A37, A33, Th37, Th38; then k2 + 1 = ((len f) -' 1) + 1 by A38, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) by A18, A30, A32, A35, A37, Th36; ::_thesis: verum end; casethat A39: k2 = 1 and A40: k1 = 2 ; ::_thesis: ( 1 <= 1 & 1 + 1 < len f & f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. 1 = (GoB f) * (i,(j + 1)) & f /. (1 + 2) = (GoB f) * ((i + 1),j) ) A41: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A9, A40, TOPREAL1:def_3; then A42: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. 2 & (GoB f) * ((i + 1),j) = f /. (2 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (2 + 1) & (GoB f) * ((i + 1),j) = f /. 2 ) ) by A10, A40, SPPOL_1:8; thus ( 1 <= 1 & 1 + 1 < len f ) by A9, A40, NAT_1:13; ::_thesis: ( f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. 1 = (GoB f) * (i,(j + 1)) & f /. (1 + 2) = (GoB f) * ((i + 1),j) ) A43: 3 < len f by Th34, XXREAL_0:2; then A44: f /. 1 <> f /. 3 by Th36; A45: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A16, A39, TOPREAL1:def_3; then A46: ( ( (GoB f) * (i,(j + 1)) = f /. 1 & (GoB f) * ((i + 1),(j + 1)) = f /. 2 ) or ( (GoB f) * (i,(j + 1)) = f /. 2 & (GoB f) * ((i + 1),(j + 1)) = f /. 1 ) ) by A17, A39, SPPOL_1:8; hence f /. (1 + 1) = (GoB f) * ((i + 1),(j + 1)) by A42, A43, Th36; ::_thesis: ( f /. 1 = (GoB f) * (i,(j + 1)) & f /. (1 + 2) = (GoB f) * ((i + 1),j) ) thus f /. 1 = (GoB f) * (i,(j + 1)) by A10, A40, A46, A41, A44, SPPOL_1:8; ::_thesis: f /. (1 + 2) = (GoB f) * ((i + 1),j) thus f /. (1 + 2) = (GoB f) * ((i + 1),j) by A17, A39, A45, A42, A44, SPPOL_1:8; ::_thesis: verum end; casethat A47: k2 = 1 and A48: k1 > 2 ; ::_thesis: ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) A49: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A16, A47, TOPREAL1:def_3; then A50: ( ( (GoB f) * (i,(j + 1)) = f /. 1 & (GoB f) * ((i + 1),(j + 1)) = f /. 2 ) or ( (GoB f) * (i,(j + 1)) = f /. 2 & (GoB f) * ((i + 1),(j + 1)) = f /. 1 ) ) by A17, A47, SPPOL_1:8; A51: 2 < k1 + 1 by A48, NAT_1:13; then A52: f /. (k1 + 1) <> f /. 2 by A9, Th37; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A8, A9, TOPREAL1:def_3; then A53: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. k1 & (GoB f) * ((i + 1),j) = f /. (k1 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k1 + 1) & (GoB f) * ((i + 1),j) = f /. k1 ) ) by A10, SPPOL_1:8; A54: f /. k1 <> f /. 2 by A11, A48, Th36; hence f /. 1 = (GoB f) * ((i + 1),(j + 1)) by A17, A47, A49, A53, A52, SPPOL_1:8; ::_thesis: ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) thus f /. 2 = (GoB f) * (i,(j + 1)) by A17, A47, A49, A53, A54, A52, SPPOL_1:8; ::_thesis: f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) A55: k1 > 1 by A48, XXREAL_0:2; then A56: k1 + 1 > 1 by NAT_1:13; then k1 + 1 = len f by A9, A11, A48, A50, A53, A55, A51, Th37, Th38; then k1 + 1 = ((len f) -' 1) + 1 by A56, XREAL_1:235; hence f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) by A11, A48, A50, A53, A55, Th36; ::_thesis: verum end; case k1 = k2 ; ::_thesis: contradiction then A57: ( (GoB f) * ((i + 1),j) = (GoB f) * (i,(j + 1)) or (GoB f) * ((i + 1),j) = (GoB f) * ((i + 1),(j + 1)) ) by A10, A17, SPPOL_1:8; A58: [(i + 1),(j + 1)] in Indices (GoB f) by A2, A4, A7, A14, MATRIX_1:36; ( [i,(j + 1)] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A1, A2, A3, A4, A7, A14, A13, MATRIX_1:36; then ( j = j + 1 or i = i + 1 ) by A57, A58, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; casethat A59: k1 > 1 and A60: k2 > k1 ; ::_thesis: ( 1 <= k1 & k1 + 1 < len f & f /. (k1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. k1 = (GoB f) * ((i + 1),j) & f /. (k1 + 2) = (GoB f) * (i,(j + 1)) ) A61: ( 1 < k1 + 1 & k1 + 1 < k2 + 1 ) by A59, A60, NAT_1:13, XREAL_1:6; A62: k1 < k2 + 1 by A60, NAT_1:13; then A63: f /. k1 <> f /. (k2 + 1) by A16, A59, Th37; A64: k1 + 1 <= k2 by A60, NAT_1:13; LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A15, A16, TOPREAL1:def_3; then A65: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. k2 & (GoB f) * (i,(j + 1)) = f /. (k2 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) & (GoB f) * (i,(j + 1)) = f /. k2 ) ) by A17, SPPOL_1:8; A66: k2 < len f by A16, NAT_1:13; then A67: f /. k1 <> f /. k2 by A59, A60, Th37; A68: LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A8, A9, TOPREAL1:def_3; then ( ( (GoB f) * ((i + 1),j) = f /. k1 & (GoB f) * ((i + 1),(j + 1)) = f /. (k1 + 1) ) or ( (GoB f) * ((i + 1),j) = f /. (k1 + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. k1 ) ) by A10, SPPOL_1:8; then k1 + 1 >= k2 by A16, A59, A60, A65, A62, A66, A61, Th37; then A69: k1 + 1 = k2 by A64, XXREAL_0:1; hence ( 1 <= k1 & k1 + 1 < len f ) by A16, A59, NAT_1:13; ::_thesis: ( f /. (k1 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. k1 = (GoB f) * ((i + 1),j) & f /. (k1 + 2) = (GoB f) * (i,(j + 1)) ) thus f /. (k1 + 1) = (GoB f) * ((i + 1),(j + 1)) by A10, A68, A65, A63, A67, SPPOL_1:8; ::_thesis: ( f /. k1 = (GoB f) * ((i + 1),j) & f /. (k1 + 2) = (GoB f) * (i,(j + 1)) ) thus f /. k1 = (GoB f) * ((i + 1),j) by A10, A68, A65, A63, A67, SPPOL_1:8; ::_thesis: f /. (k1 + 2) = (GoB f) * (i,(j + 1)) thus f /. (k1 + 2) = (GoB f) * (i,(j + 1)) by A10, A68, A65, A63, A69, SPPOL_1:8; ::_thesis: verum end; casethat A70: k2 > 1 and A71: k1 > k2 ; ::_thesis: ( 1 <= k2 & k2 + 1 < len f & f /. (k2 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. k2 = (GoB f) * (i,(j + 1)) & f /. (k2 + 2) = (GoB f) * ((i + 1),j) ) A72: ( 1 < k2 + 1 & k2 + 1 < k1 + 1 ) by A70, A71, NAT_1:13, XREAL_1:6; A73: k2 < k1 + 1 by A71, NAT_1:13; then A74: f /. k2 <> f /. (k1 + 1) by A9, A70, Th37; A75: k2 + 1 <= k1 by A71, NAT_1:13; LSeg (f,k1) = LSeg ((f /. k1),(f /. (k1 + 1))) by A8, A9, TOPREAL1:def_3; then A76: ( ( (GoB f) * ((i + 1),(j + 1)) = f /. k1 & (GoB f) * ((i + 1),j) = f /. (k1 + 1) ) or ( (GoB f) * ((i + 1),(j + 1)) = f /. (k1 + 1) & (GoB f) * ((i + 1),j) = f /. k1 ) ) by A10, SPPOL_1:8; A77: k1 < len f by A9, NAT_1:13; then A78: f /. k2 <> f /. k1 by A70, A71, Th37; A79: LSeg (f,k2) = LSeg ((f /. k2),(f /. (k2 + 1))) by A15, A16, TOPREAL1:def_3; then ( ( (GoB f) * (i,(j + 1)) = f /. k2 & (GoB f) * ((i + 1),(j + 1)) = f /. (k2 + 1) ) or ( (GoB f) * (i,(j + 1)) = f /. (k2 + 1) & (GoB f) * ((i + 1),(j + 1)) = f /. k2 ) ) by A17, SPPOL_1:8; then k2 + 1 >= k1 by A9, A70, A71, A76, A73, A77, A72, Th37; then A80: k2 + 1 = k1 by A75, XXREAL_0:1; hence ( 1 <= k2 & k2 + 1 < len f ) by A9, A70, NAT_1:13; ::_thesis: ( f /. (k2 + 1) = (GoB f) * ((i + 1),(j + 1)) & f /. k2 = (GoB f) * (i,(j + 1)) & f /. (k2 + 2) = (GoB f) * ((i + 1),j) ) thus f /. (k2 + 1) = (GoB f) * ((i + 1),(j + 1)) by A17, A79, A76, A74, A78, SPPOL_1:8; ::_thesis: ( f /. k2 = (GoB f) * (i,(j + 1)) & f /. (k2 + 2) = (GoB f) * ((i + 1),j) ) thus f /. k2 = (GoB f) * (i,(j + 1)) by A17, A79, A76, A74, A78, SPPOL_1:8; ::_thesis: f /. (k2 + 2) = (GoB f) * ((i + 1),j) thus f /. (k2 + 2) = (GoB f) * ((i + 1),j) by A17, A79, A76, A74, A80, SPPOL_1:8; ::_thesis: verum end; end; end; hence ( ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. 2 = (GoB f) * ((i + 1),j) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) or ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) ) ) or ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) ) ; ::_thesis: verum end; theorem :: GOBOARD7:59 for i, j being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i < len (GoB f) & 1 <= j & j + 1 < width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) c= L~ f holds not LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f proof let i, j be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i < len (GoB f) & 1 <= j & j + 1 < width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) c= L~ f holds not LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i < len (GoB f) & 1 <= j & j + 1 < width (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) c= L~ f implies not LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f ) assume that A1: 1 <= i and A2: i < len (GoB f) and A3: 1 <= j and A4: j + 1 < width (GoB f) and A5: ( LSeg (((GoB f) * (i,j)),((GoB f) * (i,(j + 1)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * (i,(j + 2)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f ) ; ::_thesis: contradiction A6: i + 1 <= len (GoB f) by A2, NAT_1:13; j + (1 + 1) = (j + 1) + 1 ; then A7: j + 2 <= width (GoB f) by A4, NAT_1:13; A8: 1 <= j + 1 by NAT_1:11; A9: j < width (GoB f) by A4, NAT_1:13; A10: 1 <= i + 1 by NAT_1:11; j + 1 <= j + 2 by XREAL_1:6; then A11: 1 <= j + 2 by A8, XXREAL_0:2; percases ( ( f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * (i,j) & f /. 2 = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * (i,(j + 2)) & f /. 2 = (GoB f) * (i,j) ) or ( f /. 2 = (GoB f) * (i,(j + 2)) & f /. 2 = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 1 = (GoB f) * (i,(j + 1)) & ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) or ( ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) & f /. 1 = (GoB f) * (i,(j + 1)) ) or ( ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) & ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) ) by A1, A2, A3, A4, A5, A6, Th53, Th54; supposeA12: ( f /. ((len f) -' 1) = (GoB f) * (i,(j + 2)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [i,(j + 2)] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then i = i + 1 by A12, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA13: ( f /. 2 = (GoB f) * (i,j) & f /. 2 = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [i,j] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_1:36; then i = i + 1 by A13, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA14: ( f /. 2 = (GoB f) * (i,(j + 2)) & f /. 2 = (GoB f) * (i,j) ) ; ::_thesis: contradiction ( [i,(j + 2)] in Indices (GoB f) & [i,j] in Indices (GoB f) ) by A1, A2, A3, A9, A7, A11, MATRIX_1:36; then j = j + 2 by A14, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA15: ( f /. 2 = (GoB f) * (i,(j + 2)) & f /. 2 = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [i,(j + 2)] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then i = i + 1 by A15, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposethat A16: f /. 1 = (GoB f) * (i,(j + 1)) and A17: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ; ::_thesis: contradiction consider k being Element of NAT such that A18: 1 <= k and A19: ( k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) ) and ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) by A17; 1 < k + 1 by A18, NAT_1:13; hence contradiction by A16, A19, Th36; ::_thesis: verum end; supposethat A20: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) and A21: f /. 1 = (GoB f) * (i,(j + 1)) ; ::_thesis: contradiction consider k being Element of NAT such that A22: 1 <= k and A23: ( k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) ) and ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 2) = (GoB f) * (i,j) ) ) by A20; 1 < k + 1 by A22, NAT_1:13; hence contradiction by A21, A23, Th36; ::_thesis: verum end; supposethat A24: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k = (GoB f) * (i,(j + 2)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) and A25: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * (i,(j + 1)) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ; ::_thesis: contradiction consider k1 being Element of NAT such that 1 <= k1 and A26: k1 + 1 < len f and A27: f /. (k1 + 1) = (GoB f) * (i,(j + 1)) and A28: ( ( f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * (i,(j + 2)) ) or ( f /. k1 = (GoB f) * (i,(j + 2)) & f /. (k1 + 2) = (GoB f) * (i,j) ) ) by A24; consider k2 being Element of NAT such that 1 <= k2 and A29: k2 + 1 < len f and A30: f /. (k2 + 1) = (GoB f) * (i,(j + 1)) and A31: ( ( f /. k2 = (GoB f) * (i,j) & f /. (k2 + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k2 = (GoB f) * ((i + 1),(j + 1)) & f /. (k2 + 2) = (GoB f) * (i,j) ) ) by A25; A32: now__::_thesis:_not_k1_<>_k2 assume A33: k1 <> k2 ; ::_thesis: contradiction percases ( k1 < k2 or k2 < k1 ) by A33, XXREAL_0:1; suppose k1 < k2 ; ::_thesis: contradiction then k1 + 1 < k2 + 1 by XREAL_1:6; hence contradiction by A27, A29, A30, Th36, NAT_1:11; ::_thesis: verum end; suppose k2 < k1 ; ::_thesis: contradiction then k2 + 1 < k1 + 1 by XREAL_1:6; hence contradiction by A26, A27, A30, Th36, NAT_1:11; ::_thesis: verum end; end; end; now__::_thesis:_contradiction percases ( ( f /. (k1 + 2) = (GoB f) * (i,(j + 2)) & f /. (k2 + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k1 = (GoB f) * (i,j) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k1 = (GoB f) * (i,(j + 2)) & f /. k2 = (GoB f) * (i,j) ) or ( f /. k1 = (GoB f) * (i,(j + 2)) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) ) ) by A28, A31; supposeA34: ( f /. (k1 + 2) = (GoB f) * (i,(j + 2)) & f /. (k2 + 2) = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [i,(j + 2)] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then i = i + 1 by A32, A34, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA35: ( f /. k1 = (GoB f) * (i,j) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [i,j] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_1:36; then i = i + 1 by A32, A35, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA36: ( f /. k1 = (GoB f) * (i,(j + 2)) & f /. k2 = (GoB f) * (i,j) ) ; ::_thesis: contradiction ( [i,(j + 2)] in Indices (GoB f) & [i,j] in Indices (GoB f) ) by A1, A2, A3, A9, A7, A11, MATRIX_1:36; then j = j + 2 by A32, A36, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA37: ( f /. k1 = (GoB f) * (i,(j + 2)) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [i,(j + 2)] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then i = i + 1 by A32, A37, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; theorem :: GOBOARD7:60 for i, j being Element of NAT for f being non constant standard special_circular_sequence st 1 <= i & i < len (GoB f) & 1 <= j & j + 1 < width (GoB f) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 1),(j + 2)))) c= L~ f holds not LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f proof let i, j be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i < len (GoB f) & 1 <= j & j + 1 < width (GoB f) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 1),(j + 2)))) c= L~ f holds not LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i < len (GoB f) & 1 <= j & j + 1 < width (GoB f) & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 1),(j + 2)))) c= L~ f implies not LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f ) assume that A1: 1 <= i and A2: i < len (GoB f) and A3: 1 <= j and A4: j + 1 < width (GoB f) and A5: ( LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 1),(j + 2)))) c= L~ f & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f ) ; ::_thesis: contradiction A6: i + 1 <= len (GoB f) by A2, NAT_1:13; j + (1 + 1) = (j + 1) + 1 ; then A7: j + 2 <= width (GoB f) by A4, NAT_1:13; A8: 1 <= j + 1 by NAT_1:11; A9: j < width (GoB f) by A4, NAT_1:13; A10: 1 <= i + 1 by NAT_1:11; j + 1 <= j + 2 by XREAL_1:6; then A11: 1 <= j + 2 by A8, XXREAL_0:2; percases ( ( f /. 2 = (GoB f) * ((i + 1),j) & f /. 2 = (GoB f) * (i,(j + 1)) ) or ( f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 2)) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 2)) & f /. 2 = (GoB f) * (i,(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 1),(j + 2)) & f /. 2 = (GoB f) * ((i + 1),j) ) or ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) ) or ( ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 2)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 2)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) & f /. 1 = (GoB f) * ((i + 1),(j + 1)) ) or ( ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 2)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 2)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) & ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) ) ) by A1, A3, A4, A5, A6, A10, Th53, Th55; supposeA12: ( f /. 2 = (GoB f) * ((i + 1),j) & f /. 2 = (GoB f) * (i,(j + 1)) ) ; ::_thesis: contradiction ( [(i + 1),j] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_1:36; then i = i + 1 by A12, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA13: ( f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 2)) & f /. ((len f) -' 1) = (GoB f) * (i,(j + 1)) ) ; ::_thesis: contradiction ( [(i + 1),(j + 2)] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then i = i + 1 by A13, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA14: ( f /. 2 = (GoB f) * ((i + 1),(j + 2)) & f /. 2 = (GoB f) * (i,(j + 1)) ) ; ::_thesis: contradiction ( [(i + 1),(j + 2)] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then i = i + 1 by A14, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA15: ( f /. 2 = (GoB f) * ((i + 1),(j + 2)) & f /. 2 = (GoB f) * ((i + 1),j) ) ; ::_thesis: contradiction ( [(i + 1),(j + 2)] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A3, A6, A10, A9, A7, A11, MATRIX_1:36; then j = j + 2 by A15, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposethat A16: f /. 1 = (GoB f) * ((i + 1),(j + 1)) and A17: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) ; ::_thesis: contradiction consider k being Element of NAT such that A18: 1 <= k and A19: ( k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) ) and ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) by A17; 1 < k + 1 by A18, NAT_1:13; hence contradiction by A16, A19, Th36; ::_thesis: verum end; supposethat A20: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 2)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 2)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) and A21: f /. 1 = (GoB f) * ((i + 1),(j + 1)) ; ::_thesis: contradiction consider k being Element of NAT such that A22: 1 <= k and A23: ( k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) ) and ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 2)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 2)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) by A20; 1 < k + 1 by A22, NAT_1:13; hence contradiction by A21, A23, Th36; ::_thesis: verum end; supposethat A24: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 2)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 2)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) and A25: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) ; ::_thesis: contradiction consider k1 being Element of NAT such that 1 <= k1 and A26: k1 + 1 < len f and A27: f /. (k1 + 1) = (GoB f) * ((i + 1),(j + 1)) and A28: ( ( f /. k1 = (GoB f) * ((i + 1),j) & f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 2)) ) or ( f /. k1 = (GoB f) * ((i + 1),(j + 2)) & f /. (k1 + 2) = (GoB f) * ((i + 1),j) ) ) by A24; consider k2 being Element of NAT such that 1 <= k2 and A29: k2 + 1 < len f and A30: f /. (k2 + 1) = (GoB f) * ((i + 1),(j + 1)) and A31: ( ( f /. k2 = (GoB f) * (i,(j + 1)) & f /. (k2 + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k2 = (GoB f) * ((i + 1),j) & f /. (k2 + 2) = (GoB f) * (i,(j + 1)) ) ) by A25; A32: now__::_thesis:_not_k1_<>_k2 assume A33: k1 <> k2 ; ::_thesis: contradiction percases ( k1 < k2 or k2 < k1 ) by A33, XXREAL_0:1; suppose k1 < k2 ; ::_thesis: contradiction then k1 + 1 < k2 + 1 by XREAL_1:6; hence contradiction by A27, A29, A30, Th36, NAT_1:11; ::_thesis: verum end; suppose k2 < k1 ; ::_thesis: contradiction then k2 + 1 < k1 + 1 by XREAL_1:6; hence contradiction by A26, A27, A30, Th36, NAT_1:11; ::_thesis: verum end; end; end; now__::_thesis:_contradiction percases ( ( f /. k1 = (GoB f) * ((i + 1),j) & f /. k2 = (GoB f) * (i,(j + 1)) ) or ( f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 2)) & f /. (k2 + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k1 = (GoB f) * ((i + 1),(j + 2)) & f /. k2 = (GoB f) * (i,(j + 1)) ) or ( f /. k1 = (GoB f) * ((i + 1),(j + 2)) & f /. k2 = (GoB f) * ((i + 1),j) ) ) by A28, A31; supposeA34: ( f /. k1 = (GoB f) * ((i + 1),j) & f /. k2 = (GoB f) * (i,(j + 1)) ) ; ::_thesis: contradiction ( [(i + 1),j] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_1:36; then i = i + 1 by A32, A34, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA35: ( f /. (k1 + 2) = (GoB f) * ((i + 1),(j + 2)) & f /. (k2 + 2) = (GoB f) * (i,(j + 1)) ) ; ::_thesis: contradiction ( [(i + 1),(j + 2)] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then i = i + 1 by A32, A35, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA36: ( f /. k1 = (GoB f) * ((i + 1),(j + 2)) & f /. k2 = (GoB f) * (i,(j + 1)) ) ; ::_thesis: contradiction ( [(i + 1),(j + 2)] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then i = i + 1 by A32, A36, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA37: ( f /. k1 = (GoB f) * ((i + 1),(j + 2)) & f /. k2 = (GoB f) * ((i + 1),j) ) ; ::_thesis: contradiction ( [(i + 1),(j + 2)] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A3, A6, A10, A9, A7, A11, MATRIX_1:36; then j = j + 2 by A32, A37, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; theorem :: GOBOARD7:61 for j, i being Element of NAT for f being non constant standard special_circular_sequence st 1 <= j & j < width (GoB f) & 1 <= i & i + 1 < len (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) c= L~ f holds not LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f proof let j, i be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= j & j < width (GoB f) & 1 <= i & i + 1 < len (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) c= L~ f holds not LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= j & j < width (GoB f) & 1 <= i & i + 1 < len (GoB f) & LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) c= L~ f implies not LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f ) assume that A1: 1 <= j and A2: j < width (GoB f) and A3: 1 <= i and A4: i + 1 < len (GoB f) and A5: ( LSeg (((GoB f) * (i,j)),((GoB f) * ((i + 1),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 2),j))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f ) ; ::_thesis: contradiction A6: j + 1 <= width (GoB f) by A2, NAT_1:13; i + (1 + 1) = (i + 1) + 1 ; then A7: i + 2 <= len (GoB f) by A4, NAT_1:13; A8: 1 <= i + 1 by NAT_1:11; A9: i < len (GoB f) by A4, NAT_1:13; A10: 1 <= j + 1 by NAT_1:11; i + 1 <= i + 2 by XREAL_1:6; then A11: 1 <= i + 2 by A8, XXREAL_0:2; percases ( ( f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * (i,j) & f /. 2 = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 2 = (GoB f) * ((i + 2),j) & f /. 2 = (GoB f) * (i,j) ) or ( f /. 2 = (GoB f) * ((i + 2),j) & f /. 2 = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. 1 = (GoB f) * ((i + 1),j) & ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) or ( ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) & f /. 1 = (GoB f) * ((i + 1),j) ) or ( ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) & ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ) ) by A1, A2, A3, A4, A5, A6, Th56, Th57; supposeA12: ( f /. ((len f) -' 1) = (GoB f) * ((i + 2),j) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [(i + 2),j] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then j = j + 1 by A12, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA13: ( f /. 2 = (GoB f) * (i,j) & f /. 2 = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [i,j] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_1:36; then j = j + 1 by A13, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA14: ( f /. 2 = (GoB f) * ((i + 2),j) & f /. 2 = (GoB f) * (i,j) ) ; ::_thesis: contradiction ( [(i + 2),j] in Indices (GoB f) & [i,j] in Indices (GoB f) ) by A1, A2, A3, A9, A7, A11, MATRIX_1:36; then i = i + 2 by A14, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA15: ( f /. 2 = (GoB f) * ((i + 2),j) & f /. 2 = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [(i + 2),j] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then j = j + 1 by A15, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposethat A16: f /. 1 = (GoB f) * ((i + 1),j) and A17: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ; ::_thesis: contradiction consider k being Element of NAT such that A18: 1 <= k and A19: ( k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) ) and ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) by A17; 1 < k + 1 by A18, NAT_1:13; hence contradiction by A16, A19, Th36; ::_thesis: verum end; supposethat A20: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) and A21: f /. 1 = (GoB f) * ((i + 1),j) ; ::_thesis: contradiction consider k being Element of NAT such that A22: 1 <= k and A23: ( k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) ) and ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) by A20; 1 < k + 1 by A22, NAT_1:13; hence contradiction by A21, A23, Th36; ::_thesis: verum end; supposethat A24: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k = (GoB f) * ((i + 2),j) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) and A25: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),j) & ( ( f /. k = (GoB f) * (i,j) & f /. (k + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 1),(j + 1)) & f /. (k + 2) = (GoB f) * (i,j) ) ) ) ; ::_thesis: contradiction consider k1 being Element of NAT such that 1 <= k1 and A26: k1 + 1 < len f and A27: f /. (k1 + 1) = (GoB f) * ((i + 1),j) and A28: ( ( f /. k1 = (GoB f) * (i,j) & f /. (k1 + 2) = (GoB f) * ((i + 2),j) ) or ( f /. k1 = (GoB f) * ((i + 2),j) & f /. (k1 + 2) = (GoB f) * (i,j) ) ) by A24; consider k2 being Element of NAT such that 1 <= k2 and A29: k2 + 1 < len f and A30: f /. (k2 + 1) = (GoB f) * ((i + 1),j) and A31: ( ( f /. k2 = (GoB f) * (i,j) & f /. (k2 + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k2 = (GoB f) * ((i + 1),(j + 1)) & f /. (k2 + 2) = (GoB f) * (i,j) ) ) by A25; A32: now__::_thesis:_not_k1_<>_k2 assume A33: k1 <> k2 ; ::_thesis: contradiction percases ( k1 < k2 or k2 < k1 ) by A33, XXREAL_0:1; suppose k1 < k2 ; ::_thesis: contradiction then k1 + 1 < k2 + 1 by XREAL_1:6; hence contradiction by A27, A29, A30, Th36, NAT_1:11; ::_thesis: verum end; suppose k2 < k1 ; ::_thesis: contradiction then k2 + 1 < k1 + 1 by XREAL_1:6; hence contradiction by A26, A27, A30, Th36, NAT_1:11; ::_thesis: verum end; end; end; now__::_thesis:_contradiction percases ( ( f /. (k1 + 2) = (GoB f) * ((i + 2),j) & f /. (k2 + 2) = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k1 = (GoB f) * (i,j) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) ) or ( f /. k1 = (GoB f) * ((i + 2),j) & f /. k2 = (GoB f) * (i,j) ) or ( f /. k1 = (GoB f) * ((i + 2),j) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) ) ) by A28, A31; supposeA34: ( f /. (k1 + 2) = (GoB f) * ((i + 2),j) & f /. (k2 + 2) = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [(i + 2),j] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then j = j + 1 by A32, A34, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA35: ( f /. k1 = (GoB f) * (i,j) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [i,j] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_1:36; then j = j + 1 by A32, A35, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA36: ( f /. k1 = (GoB f) * ((i + 2),j) & f /. k2 = (GoB f) * (i,j) ) ; ::_thesis: contradiction ( [(i + 2),j] in Indices (GoB f) & [i,j] in Indices (GoB f) ) by A1, A2, A3, A9, A7, A11, MATRIX_1:36; then i = i + 2 by A32, A36, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA37: ( f /. k1 = (GoB f) * ((i + 2),j) & f /. k2 = (GoB f) * ((i + 1),(j + 1)) ) ; ::_thesis: contradiction ( [(i + 2),j] in Indices (GoB f) & [(i + 1),(j + 1)] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then j = j + 1 by A32, A37, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; theorem :: GOBOARD7:62 for j, i being Element of NAT for f being non constant standard special_circular_sequence st 1 <= j & j < width (GoB f) & 1 <= i & i + 1 < len (GoB f) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 2),(j + 1)))) c= L~ f holds not LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f proof let j, i be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= j & j < width (GoB f) & 1 <= i & i + 1 < len (GoB f) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 2),(j + 1)))) c= L~ f holds not LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= j & j < width (GoB f) & 1 <= i & i + 1 < len (GoB f) & LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 2),(j + 1)))) c= L~ f implies not LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f ) assume that A1: 1 <= j and A2: j < width (GoB f) and A3: 1 <= i and A4: i + 1 < len (GoB f) and A5: ( LSeg (((GoB f) * (i,(j + 1))),((GoB f) * ((i + 1),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),(j + 1))),((GoB f) * ((i + 2),(j + 1)))) c= L~ f & LSeg (((GoB f) * ((i + 1),j)),((GoB f) * ((i + 1),(j + 1)))) c= L~ f ) ; ::_thesis: contradiction A6: j + 1 <= width (GoB f) by A2, NAT_1:13; i + (1 + 1) = (i + 1) + 1 ; then A7: i + 2 <= len (GoB f) by A4, NAT_1:13; A8: 1 <= i + 1 by NAT_1:11; A9: i < len (GoB f) by A4, NAT_1:13; A10: 1 <= j + 1 by NAT_1:11; i + 1 <= i + 2 by XREAL_1:6; then A11: 1 <= i + 2 by A8, XXREAL_0:2; percases ( ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. 2 = (GoB f) * ((i + 1),j) ) or ( f /. ((len f) -' 1) = (GoB f) * ((i + 2),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) or ( f /. 2 = (GoB f) * ((i + 2),(j + 1)) & f /. 2 = (GoB f) * ((i + 1),j) ) or ( f /. 2 = (GoB f) * ((i + 2),(j + 1)) & f /. 2 = (GoB f) * (i,(j + 1)) ) or ( f /. 1 = (GoB f) * ((i + 1),(j + 1)) & ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) ) or ( ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 2),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) & f /. 1 = (GoB f) * ((i + 1),(j + 1)) ) or ( ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 2),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) & ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) ) ) by A1, A3, A4, A5, A6, A10, Th56, Th58; supposeA12: ( f /. 2 = (GoB f) * (i,(j + 1)) & f /. 2 = (GoB f) * ((i + 1),j) ) ; ::_thesis: contradiction ( [i,(j + 1)] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_1:36; then j = j + 1 by A12, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA13: ( f /. ((len f) -' 1) = (GoB f) * ((i + 2),(j + 1)) & f /. ((len f) -' 1) = (GoB f) * ((i + 1),j) ) ; ::_thesis: contradiction ( [(i + 2),(j + 1)] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then j = j + 1 by A13, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA14: ( f /. 2 = (GoB f) * ((i + 2),(j + 1)) & f /. 2 = (GoB f) * ((i + 1),j) ) ; ::_thesis: contradiction ( [(i + 2),(j + 1)] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then j = j + 1 by A14, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA15: ( f /. 2 = (GoB f) * ((i + 2),(j + 1)) & f /. 2 = (GoB f) * (i,(j + 1)) ) ; ::_thesis: contradiction ( [(i + 2),(j + 1)] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A3, A6, A10, A9, A7, A11, MATRIX_1:36; then i = i + 2 by A15, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposethat A16: f /. 1 = (GoB f) * ((i + 1),(j + 1)) and A17: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) ; ::_thesis: contradiction consider k being Element of NAT such that A18: 1 <= k and A19: ( k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) ) and ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) by A17; 1 < k + 1 by A18, NAT_1:13; hence contradiction by A16, A19, Th36; ::_thesis: verum end; supposethat A20: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 2),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) and A21: f /. 1 = (GoB f) * ((i + 1),(j + 1)) ; ::_thesis: contradiction consider k being Element of NAT such that A22: 1 <= k and A23: ( k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) ) and ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 2),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) by A20; 1 < k + 1 by A22, NAT_1:13; hence contradiction by A21, A23, Th36; ::_thesis: verum end; supposethat A24: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. k = (GoB f) * ((i + 2),(j + 1)) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) ) ) and A25: ex k being Element of NAT st ( 1 <= k & k + 1 < len f & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),j) & f /. (k + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k = (GoB f) * (i,(j + 1)) & f /. (k + 2) = (GoB f) * ((i + 1),j) ) ) ) ; ::_thesis: contradiction consider k1 being Element of NAT such that 1 <= k1 and A26: k1 + 1 < len f and A27: f /. (k1 + 1) = (GoB f) * ((i + 1),(j + 1)) and A28: ( ( f /. k1 = (GoB f) * (i,(j + 1)) & f /. (k1 + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. k1 = (GoB f) * ((i + 2),(j + 1)) & f /. (k1 + 2) = (GoB f) * (i,(j + 1)) ) ) by A24; consider k2 being Element of NAT such that 1 <= k2 and A29: k2 + 1 < len f and A30: f /. (k2 + 1) = (GoB f) * ((i + 1),(j + 1)) and A31: ( ( f /. k2 = (GoB f) * ((i + 1),j) & f /. (k2 + 2) = (GoB f) * (i,(j + 1)) ) or ( f /. k2 = (GoB f) * (i,(j + 1)) & f /. (k2 + 2) = (GoB f) * ((i + 1),j) ) ) by A25; A32: now__::_thesis:_not_k1_<>_k2 assume A33: k1 <> k2 ; ::_thesis: contradiction percases ( k1 < k2 or k2 < k1 ) by A33, XXREAL_0:1; suppose k1 < k2 ; ::_thesis: contradiction then k1 + 1 < k2 + 1 by XREAL_1:6; hence contradiction by A27, A29, A30, Th36, NAT_1:11; ::_thesis: verum end; suppose k2 < k1 ; ::_thesis: contradiction then k2 + 1 < k1 + 1 by XREAL_1:6; hence contradiction by A26, A27, A30, Th36, NAT_1:11; ::_thesis: verum end; end; end; now__::_thesis:_contradiction percases ( ( f /. k1 = (GoB f) * (i,(j + 1)) & f /. k2 = (GoB f) * ((i + 1),j) ) or ( f /. (k1 + 2) = (GoB f) * ((i + 2),(j + 1)) & f /. (k2 + 2) = (GoB f) * ((i + 1),j) ) or ( f /. k1 = (GoB f) * ((i + 2),(j + 1)) & f /. k2 = (GoB f) * ((i + 1),j) ) or ( f /. k1 = (GoB f) * ((i + 2),(j + 1)) & f /. k2 = (GoB f) * (i,(j + 1)) ) ) by A28, A31; supposeA34: ( f /. k1 = (GoB f) * (i,(j + 1)) & f /. k2 = (GoB f) * ((i + 1),j) ) ; ::_thesis: contradiction ( [i,(j + 1)] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_1:36; then j = j + 1 by A32, A34, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA35: ( f /. (k1 + 2) = (GoB f) * ((i + 2),(j + 1)) & f /. (k2 + 2) = (GoB f) * ((i + 1),j) ) ; ::_thesis: contradiction ( [(i + 2),(j + 1)] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then j = j + 1 by A32, A35, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA36: ( f /. k1 = (GoB f) * ((i + 2),(j + 1)) & f /. k2 = (GoB f) * ((i + 1),j) ) ; ::_thesis: contradiction ( [(i + 2),(j + 1)] in Indices (GoB f) & [(i + 1),j] in Indices (GoB f) ) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_1:36; then j = j + 1 by A32, A36, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; supposeA37: ( f /. k1 = (GoB f) * ((i + 2),(j + 1)) & f /. k2 = (GoB f) * (i,(j + 1)) ) ; ::_thesis: contradiction ( [(i + 2),(j + 1)] in Indices (GoB f) & [i,(j + 1)] in Indices (GoB f) ) by A3, A6, A10, A9, A7, A11, MATRIX_1:36; then i = i + 2 by A32, A37, GOBOARD1:5; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; theorem :: GOBOARD7:63 for p, q, p1, q1 being Point of (TOP-REAL 2) st LSeg (p,q) is vertical & LSeg (p1,q1) is vertical & p `1 = p1 `1 & p `2 <= p1 `2 & p1 `2 <= q1 `2 & q1 `2 <= q `2 holds LSeg (p1,q1) c= LSeg (p,q) proof let p, q, p1, q1 be Point of (TOP-REAL 2); ::_thesis: ( LSeg (p,q) is vertical & LSeg (p1,q1) is vertical & p `1 = p1 `1 & p `2 <= p1 `2 & p1 `2 <= q1 `2 & q1 `2 <= q `2 implies LSeg (p1,q1) c= LSeg (p,q) ) assume that A1: LSeg (p,q) is vertical and A2: LSeg (p1,q1) is vertical and A3: p `1 = p1 `1 and A4: p `2 <= p1 `2 and A5: p1 `2 <= q1 `2 and A6: q1 `2 <= q `2 ; ::_thesis: LSeg (p1,q1) c= LSeg (p,q) A7: p `1 = q `1 by A1, SPPOL_1:16; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (p1,q1) or x in LSeg (p,q) ) assume A8: x in LSeg (p1,q1) ; ::_thesis: x in LSeg (p,q) then reconsider r = x as Point of (TOP-REAL 2) ; p1 `2 <= r `2 by A5, A8, TOPREAL1:4; then A9: p `2 <= r `2 by A4, XXREAL_0:2; r `2 <= q1 `2 by A5, A8, TOPREAL1:4; then A10: r `2 <= q `2 by A6, XXREAL_0:2; p1 `1 = r `1 by A2, A8, SPPOL_1:41; hence x in LSeg (p,q) by A3, A7, A9, A10, Th7; ::_thesis: verum end; theorem :: GOBOARD7:64 for p, q, p1, q1 being Point of (TOP-REAL 2) st LSeg (p,q) is horizontal & LSeg (p1,q1) is horizontal & p `2 = p1 `2 & p `1 <= p1 `1 & p1 `1 <= q1 `1 & q1 `1 <= q `1 holds LSeg (p1,q1) c= LSeg (p,q) proof let p, q, p1, q1 be Point of (TOP-REAL 2); ::_thesis: ( LSeg (p,q) is horizontal & LSeg (p1,q1) is horizontal & p `2 = p1 `2 & p `1 <= p1 `1 & p1 `1 <= q1 `1 & q1 `1 <= q `1 implies LSeg (p1,q1) c= LSeg (p,q) ) assume that A1: LSeg (p,q) is horizontal and A2: LSeg (p1,q1) is horizontal and A3: p `2 = p1 `2 and A4: p `1 <= p1 `1 and A5: p1 `1 <= q1 `1 and A6: q1 `1 <= q `1 ; ::_thesis: LSeg (p1,q1) c= LSeg (p,q) A7: p `2 = q `2 by A1, SPPOL_1:15; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (p1,q1) or x in LSeg (p,q) ) assume A8: x in LSeg (p1,q1) ; ::_thesis: x in LSeg (p,q) then reconsider r = x as Point of (TOP-REAL 2) ; p1 `1 <= r `1 by A5, A8, TOPREAL1:3; then A9: p `1 <= r `1 by A4, XXREAL_0:2; r `1 <= q1 `1 by A5, A8, TOPREAL1:3; then A10: r `1 <= q `1 by A6, XXREAL_0:2; p1 `2 = r `2 by A2, A8, SPPOL_1:40; hence x in LSeg (p,q) by A3, A7, A9, A10, Th8; ::_thesis: verum end;