reserve i,i1,i2,i9,i19,j,j1,j2,j9,j19,k,k1,k2,l,m,n for Nat;
reserve r,s,r9,s9 for Real;
reserve D for non empty set, f for FinSequence of D;
reserve f for FinSequence of TOP-REAL 2, G for Go-board;

theorem
  for G being Go-board, f being standard special_circular_sequence st 1
<= n & n+1 <= len f & f is_sequence_on G holds left_cell(f,n,G) c= left_cell(f,
  n) & right_cell(f,n,G) c= right_cell(f,n)
proof
  let G be Go-board,f be standard special_circular_sequence such that
A1: 1 <= n & n+1 <= len f and
A2: f is_sequence_on G;
  consider i1,j1,i2,j2 such that
A3: [i1,j1] in Indices G and
A4: f/.n = G*(i1,j1) and
A5: [i2,j2] in Indices G and
A6: f/.(n+1) = G*(i2,j2) and
A7: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2
  or i1 = i2 & j1 = j2+1 by A1,A2,JORDAN8:3;
A8: 1 <= j1 by A3,MATRIX_0:32;
A9: j1+1 > j1 & j2+1 > j2 by NAT_1:13;
A10: i1+1 > i1 & i2+1 > i2 by NAT_1:13;
A11: j2 <= width G by A5,MATRIX_0:32;
A12: j1 <= width G by A3,MATRIX_0:32;
A13: i2 <= len G by A5,MATRIX_0:32;
A14: 1 <= i2 by A5,MATRIX_0:32;
  then
A15: G*(i2,j1)`2 = G*(1,j1)`2 by A8,A12,A13,GOBOARD5:1;
A16: 1 <= j2 by A5,MATRIX_0:32;
  then
A17: G*(i2,j2)`1 = G*(i2,1)`1 by A14,A13,A11,GOBOARD5:2;
A18: i1 <= len G by A3,MATRIX_0:32;
  set F = GoB f;
A19: Values F c= Values G by A2,Th13;
  f is_sequence_on F by GOBOARD5:def 5;
  then consider m,k,i,j such that
A20: [m,k] in Indices F and
A21: f/.n = F*(m,k) and
A22: [i,j] in Indices F and
A23: f/.(n+1) = F*(i,j) and
  m = i & k+1 = j or m+1 = i & k = j or m = i+1 & k = j or m = i & k = j+1
  by A1,JORDAN8:3;
A24: 1 <= m by A20,MATRIX_0:32;
A25: 1 <= i1 by A3,MATRIX_0:32;
  then
A26: G*(i1,j1)`1 = G*(i1,1)`1 by A18,A8,A12,GOBOARD5:2;
A27: G*(i1,j1)`2 = G*(1,j1)`2 by A25,A18,A8,A12,GOBOARD5:1;
A28: m <= len F by A20,MATRIX_0:32;
A29: j+1 > j by NAT_1:13;
A30: k+1 > k by NAT_1:13;
A31: k+1 >= 1 by NAT_1:12;
A32: j+1 >= 1 by NAT_1:12;
A33: j <= width F by A22,MATRIX_0:32;
A34: i+1 > i by NAT_1:13;
A35: m+1 > m by NAT_1:13;
A36: i <= len F by A22,MATRIX_0:32;
A37: i+1 >= 1 by NAT_1:12;
A38: m+1 >= 1 by NAT_1:12;
A39: k <= width F by A20,MATRIX_0:32;
A40: 1 <= j by A22,MATRIX_0:32;
  then
A41: F*(m,j)`2 = F*(1,j)`2 by A24,A28,A33,GOBOARD5:1;
A42: 1 <= i by A22,MATRIX_0:32;
  then
A43: F*(i,j)`1 = F*(i,1)`1 by A36,A40,A33,GOBOARD5:2;
A44: F*(i,j)`2 = F*(1,j)`2 by A42,A36,A40,A33,GOBOARD5:1;
A45: 1 <= k by A20,MATRIX_0:32;
  then
A46: F*(i,k)`1 = F*(i,1)`1 by A39,A42,A36,GOBOARD5:2;
  per cases by A7;
  suppose
A47: i1 = i2 & j1+1 = j2;
A48: now
A49:  G*(i2,j2)`2 = G*(1,j2)`2 by A14,A13,A16,A11,GOBOARD5:1;
      assume
A50:  k+1 < j;
      then
A51:  k+1 < width F by A33,XXREAL_0:2;
      then consider l,i9 such that
A52:  l in dom f and
A53:  [i9,k+1] in Indices F and
A54:  f/.l = F*(i9,k+1) by JORDAN5D:8,NAT_1:12;
A55:  F*(m,k+1)`2 = F*(1,k+1)`2 by A24,A28,A31,A51,GOBOARD5:1;
      1 <= i9 & i9 <= len F by A53,MATRIX_0:32;
      then
A56:  F*(i9,k+1)`2 = F*(1,k+1)`2 by A31,A51,GOBOARD5:1;
      consider i19,j19 such that
A57:  [i19,j19] in Indices G and
A58:  f/.l = G*(i19,j19) by A2,A52,GOBOARD1:def 9;
A59:  1 <= j19 by A57,MATRIX_0:32;
A60:  1 <= i19 & i19 <= len G by A57,MATRIX_0:32;
      then
A61:  G*(i19,j1)`2 = G*(1,j1)`2 by A8,A12,GOBOARD5:1;
A62:  now
        assume j1 >= j19;
        then G*(i19,j19)`2 <= G*(i1,j1)`2 by A12,A27,A60,A59,A61,SPRECT_3:12;
        hence contradiction by A21,A24,A28,A45,A4,A30,A51,A54,A56,A55,A58,
GOBOARD5:4;
      end;
A63:  j19 <= width G by A57,MATRIX_0:32;
A64:  G*(i19,j2)`2 = G*(1,j2)`2 by A16,A11,A60,GOBOARD5:1;
      now
        assume j2 <= j19;
        then G*(i2,j2)`2 <= G*(i19,j19)`2 by A16,A60,A63,A49,A64,SPRECT_3:12;
        hence contradiction by A23,A24,A28,A33,A44,A41,A6,A31,A50,A54,A56,A55
,A58,GOBOARD5:4;
      end;
      hence contradiction by A47,A62,NAT_1:13;
    end;
    now
      assume j <= k;
      then
A65:  F*(i,j)`2 <= F*(m,k)`2 by A24,A28,A39,A40,A44,A41,SPRECT_3:12;
      j1 < j2 by A47,NAT_1:13;
      hence contradiction by A21,A23,A4,A6,A8,A14,A13,A11,A27,A15,A65,
GOBOARD5:4;
    end;
    then k+1 <= j by NAT_1:13;
    then k+1 = j by A48,XXREAL_0:1;
    then
A66: right_cell(f,n) = cell(F,m,k) & left_cell(f,n) = cell(F,m-'1,k) by A1,A20
,A21,A22,A23,A30,A29,GOBOARD5:def 6,def 7;
    right_cell(f,n,G) = cell(G,i1,j1) & left_cell(f,n,G) = cell(G,i1-'1,
    j1) by A1,A2,A3,A4,A5,A6,A9,A47,Def1,Def2;
    hence thesis by A19,A20,A21,A3,A4,A66,Th10,Th11;
  end;
  suppose
A67: i1+1 = i2 & j1 = j2;
A68: now
      assume
A69:  m+1 < i;
      then
A70:  m+1 < len F by A36,XXREAL_0:2;
      then consider l,j9 such that
A71:  l in dom f and
A72:  [m+1,j9] in Indices F and
A73:  f/.l = F*(m+1,j9) by JORDAN5D:7,NAT_1:12;
A74:  F*(m+1,k)`1 = F*(m+1,1)`1 by A45,A39,A38,A70,GOBOARD5:2;
      1 <= j9 & j9 <= width F by A72,MATRIX_0:32;
      then
A75:  F*(m+1,j9)`1 = F*(m+1,1)`1 by A38,A70,GOBOARD5:2;
A76:  1 <= m+1 & F*(m+1,j)`1 = F*(m+1,1)`1 by A40,A33,A38,A70,GOBOARD5:2;
A77:  G*(i2,j2)`1 = G*(i2,1)`1 by A14,A13,A16,A11,GOBOARD5:2;
      consider i19,j19 such that
A78:  [i19,j19] in Indices G and
A79:  f/.l = G*(i19,j19) by A2,A71,GOBOARD1:def 9;
A80:  i19 <= len G by A78,MATRIX_0:32;
A81:  1 <= j19 & j19 <= width G by A78,MATRIX_0:32;
A82:  1 <= i19 by A78,MATRIX_0:32;
      then
A83:  G*(i19,j19)`1 = G*(i19,1)`1 by A80,A81,GOBOARD5:2;
A84:  G*(i19,j1)`1 = G*(i19,1)`1 by A8,A12,A82,A80,GOBOARD5:2;
A85:  now
        assume i1 >= i19;
        then G*(i19,j19)`1 <= G*(i1,j1)`1 by A18,A8,A12,A82,A83,A84,SPRECT_3:13
;
        hence contradiction by A21,A24,A45,A39,A4,A35,A70,A73,A75,A74,A79,
GOBOARD5:3;
      end;
A86:  G*(i2,j19)`1 = G*(i2,1)`1 by A14,A13,A81,GOBOARD5:2;
      now
        assume i2 <= i19;
        then G*(i2,j2)`1 <= G*(i19,j19)`1 by A14,A80,A81,A77,A86,SPRECT_3:13;
        hence contradiction by A23,A36,A40,A33,A6,A69,A73,A75,A76,A79,
GOBOARD5:3;
      end;
      hence contradiction by A67,A85,NAT_1:13;
    end;
    now
      assume i <= m;
      then
A87:  F*(i,j)`1 <= F*(m,k)`1 by A28,A45,A39,A42,A43,A46,SPRECT_3:13;
      i1 < i2 by A67,NAT_1:13;
      hence contradiction by A21,A23,A4,A6,A25,A8,A12,A13,A67,A87,GOBOARD5:3;
    end;
    then m+1 <= i by NAT_1:13;
    then m+1 = i by A68,XXREAL_0:1;
    then
A88: right_cell(f,n) = cell(F,m, k-'1) & left_cell(f,n) = cell(F,m,k) by A1,A20
,A21,A22,A23,A35,A34,GOBOARD5:def 6,def 7;
    right_cell(f,n,G) = cell(G,i1,j1-'1) & left_cell(f,n,G) = cell(G,i1,
    j1) by A1,A2,A3,A4,A5,A6,A10,A67,Def1,Def2;
    hence thesis by A19,A20,A21,A3,A4,A88,Th10,Th12;
  end;
  suppose
A89: i1 = i2+1 & j1 = j2;
A90: now
      assume
A91:  m > i+1;
      then
A92:  i+1 < len F by A28,XXREAL_0:2;
      then consider l,j9 such that
A93:  l in dom f and
A94:  [i+1,j9] in Indices F and
A95:  f/.l = F*(i+1,j9) by JORDAN5D:7,NAT_1:12;
A96:  1 <= i+1 & F*(i+1,k)`1 = F*(i+1,1)`1 by A45,A39,A37,A92,GOBOARD5:2;
      1 <= j9 & j9 <= width F by A94,MATRIX_0:32;
      then
A97:  F*(i+1,j9)`1 = F*(i+1,1)`1 by A37,A92,GOBOARD5:2;
A98:  F*(i+1,j)`1 = F*(i+1,1)`1 by A40,A33,A37,A92,GOBOARD5:2;
A99:  G*(i2,j2)`1 = G*(i2,1)`1 by A14,A13,A16,A11,GOBOARD5:2;
      consider i19,j19 such that
A100: [i19,j19] in Indices G and
A101: f/.l = G*(i19,j19) by A2,A93,GOBOARD1:def 9;
A102: 1 <= i19 by A100,MATRIX_0:32;
A103: 1 <= j19 & j19 <= width G by A100,MATRIX_0:32;
A104: i19 <= len G by A100,MATRIX_0:32;
      then
A105: G*(i19,j19)`1 = G*(i19,1)`1 by A102,A103,GOBOARD5:2;
A106: G*(i19,j1)`1 = G*(i19,1)`1 by A8,A12,A102,A104,GOBOARD5:2;
A107: now
        assume i1 <= i19;
        then G*(i19,j19)`1 >= G*(i1,j1)`1 by A25,A8,A12,A104,A105,A106,
SPRECT_3:13;
        hence contradiction by A21,A28,A45,A39,A4,A91,A95,A97,A96,A101,
GOBOARD5:3;
      end;
A108: G*(i2,j19)`1 = G*(i2,1)`1 by A14,A13,A103,GOBOARD5:2;
      now
        assume i2 >= i19;
        then G*(i2,j2)`1 >= G*(i19,j19)`1 by A13,A102,A103,A99,A108,SPRECT_3:13
;
        hence contradiction by A23,A42,A40,A33,A6,A34,A92,A95,A97,A98,A101,
GOBOARD5:3;
      end;
      hence contradiction by A89,A107,NAT_1:13;
    end;
    now
      assume m <= i;
      then
A109: F*(i,j)`1 >= F*(m,k)`1 by A24,A45,A39,A36,A43,A46,SPRECT_3:13;
      i1 > i2 by A89,NAT_1:13;
      hence contradiction by A21,A23,A4,A6,A18,A8,A12,A14,A89,A109,GOBOARD5:3;
    end;
    then i+1 <= m by NAT_1:13;
    then i+1 = m by A90,XXREAL_0:1;
    then
A110: right_cell(f,n) = cell(F,i,j) & left_cell(f,n) = cell(F,i,j-'1) by A1,A20
,A21,A22,A23,A35,A34,GOBOARD5:def 6,def 7;
    right_cell(f,n,G) = cell(G,i2,j2) & left_cell(f,n,G) = cell(G,i2,j2
    -'1) by A1,A2,A3,A4,A5,A6,A10,A89,Def1,Def2;
    hence thesis by A19,A22,A23,A5,A6,A110,Th10,Th12;
  end;
  suppose
A111: i1 = i2 & j1 = j2+1;
A112: now
A113: G*(i2,j2)`2 = G*(1,j2)`2 by A14,A13,A16,A11,GOBOARD5:1;
      assume
A114: j+1 < k;
      then
A115: j+1 < width F by A39,XXREAL_0:2;
      then consider l,i9 such that
A116: l in dom f and
A117: [i9,j+1] in Indices F and
A118: f/.l = F*(i9,j+1) by JORDAN5D:8,NAT_1:12;
A119: F*(m,j+1)`2 = F*(1,j+1)`2 by A24,A28,A32,A115,GOBOARD5:1;
      1 <= i9 & i9 <= len F by A117,MATRIX_0:32;
      then
A120: F*(i9,j+1)`2 = F*(1,j+1)`2 by A32,A115,GOBOARD5:1;
      consider i19,j19 such that
A121: [i19,j19] in Indices G and
A122: f/.l = G*(i19,j19) by A2,A116,GOBOARD1:def 9;
A123: j19 <= width G by A121,MATRIX_0:32;
A124: 1 <= i19 & i19 <= len G by A121,MATRIX_0:32;
      then
A125: G*(i19,j1)`2 = G*(1,j1)`2 by A8,A12,GOBOARD5:1;
A126: now
        assume j1 <= j19;
        then G*(i19,j19)`2 >= G*(i1,j1)`2 by A8,A27,A124,A123,A125,SPRECT_3:12;
        hence
        contradiction by A21,A24,A28,A39,A4,A32,A114,A118,A120,A119,A122,
GOBOARD5:4;
      end;
A127: F*(i,j+1)`2 = F*(1,j+1)`2 by A42,A36,A32,A115,GOBOARD5:1;
A128: 1 <= j19 by A121,MATRIX_0:32;
A129: G*(i19,j2)`2 = G*(1,j2)`2 by A16,A11,A124,GOBOARD5:1;
      now
        assume j2 >= j19;
        then G*(i2,j2)`2 >= G*(i19,j19)`2 by A11,A124,A128,A113,A129,
SPRECT_3:12;
        hence contradiction by A23,A42,A36,A40,A6,A29,A115,A118,A120,A127,A122,
GOBOARD5:4;
      end;
      hence contradiction by A111,A126,NAT_1:13;
    end;
    now
      assume j >= k;
      then
A130: F*(i,j)`2 >= F*(m,k)`2 by A24,A28,A45,A33,A44,A41,SPRECT_3:12;
      j1 > j2 by A111,NAT_1:13;
      hence contradiction by A21,A23,A4,A6,A12,A14,A13,A16,A27,A15,A130,
GOBOARD5:4;
    end;
    then j+1 <= k by NAT_1:13;
    then j+1 = k by A112,XXREAL_0:1;
    then
A131: right_cell(f,n) = cell(F,m -'1,j) & left_cell(f,n) = cell(F,m,j) by A1
,A20,A21,A22,A23,A30,A29,GOBOARD5:def 6,def 7;
A132: now
      assume
A133: m <> i;
      per cases by A133,XXREAL_0:1;
      suppose
        m < i;
        hence contradiction by A21,A23,A24,A45,A39,A36,A43,A46,A4,A6,A26,A17
,A111,GOBOARD5:3;
      end;
      suppose
        m > i;
        hence contradiction by A21,A23,A28,A45,A39,A42,A43,A46,A4,A6,A26,A17
,A111,GOBOARD5:3;
      end;
    end;
    right_cell(f,n,G) = cell(G,i1-'1,j2) & left_cell(f,n,G) = cell(G,i1,
    j2) by A1,A2,A3,A4,A5,A6,A9,A111,Def1,Def2;
    hence thesis by A19,A22,A23,A5,A6,A111,A132,A131,Th10,Th11;
  end;
end;
