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

theorem
  for f being non empty FinSequence of TOP-REAL 2 st f is_sequence_on G
  holds f is standard special
proof
  let f be non empty FinSequence of TOP-REAL 2 such that
A1: f is_sequence_on G;
  thus f is_sequence_on GoB f
  proof
    set F = GoB f;
    thus for n st n in dom f
    ex i,j st [i,j] in Indices GoB f & f/.n = (GoB f)*(i,j) by GOBOARD2:14;
    let n such that
A2: n in dom f and
A3: n+1 in dom f;
    let m,k,i,j such that
A4: [m,k] in Indices GoB f and
A5: [i,j] in Indices GoB f and
A6: f/.n = (GoB f)*(m,k) and
A7: f/.(n+1) = (GoB f)*(i,j);
A8: 1 <= m by A4,MATRIX_0:32;
A9: m <= len F by A4,MATRIX_0:32;
A10: 1 <= k by A4,MATRIX_0:32;
A11: k <= width F by A4,MATRIX_0:32;
A12: 1 <= i by A5,MATRIX_0:32;
A13: i <= len F by A5,MATRIX_0:32;
A14: 1 <= j by A5,MATRIX_0:32;
A15: j <= width F by A5,MATRIX_0:32;
    then
A16: F*(i,j)`1 = F*(i,1)`1 by A12,A13,A14,GOBOARD5:2;
A17: F*(i,k)`1 = F*(i,1)`1 by A10,A11,A12,A13,GOBOARD5:2;
A18: F*(i,j)`2 = F*(1,j)`2 by A12,A13,A14,A15,GOBOARD5:1;
A19: F*(m,j)`2 = F*(1,j)`2 by A8,A9,A14,A15,GOBOARD5:1;
A20: 1 <= n by A2,FINSEQ_3:25;
    n+1 <= len f by A3,FINSEQ_3:25;
    then consider i1,j1,i2,j2 such that
A21: [i1,j1] in Indices G and
A22: f/.n = G*(i1,j1) and
A23: [i2,j2] in Indices G and
A24: f/.(n+1) = G*(i2,j2) and
A25: 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,A20,Th3;
A26: 1 <= i1 by A21,MATRIX_0:32;
A27: i1 <= len G by A21,MATRIX_0:32;
A28: 1 <= j1 by A21,MATRIX_0:32;
A29: j1 <= width G by A21,MATRIX_0:32;
A30: 1 <= i2 by A23,MATRIX_0:32;
A31: i2 <= len G by A23,MATRIX_0:32;
A32: 1 <= j2 by A23,MATRIX_0:32;
A33: j2 <= width G by A23,MATRIX_0:32;
A34: G*(i1,j1)`1 = G*(i1,1)`1 by A26,A27,A28,A29,GOBOARD5:2;
A35: G*(i2,j2)`1 = G*(i2,1)`1 by A30,A31,A32,A33,GOBOARD5:2;
A36: G*(i1,j1)`2 = G*(1,j1)`2 by A26,A27,A28,A29,GOBOARD5:1;
A37: G*(i2,j1)`2 = G*(1,j1)`2 by A28,A29,A30,A31,GOBOARD5:1;
A38: k+1 >= 1 by NAT_1:12;
A39: j+1 >= 1 by NAT_1:12;
A40: m+1 >= 1 by NAT_1:12;
A41: i+1 >= 1 by NAT_1:12;
A42: k+1 > k by NAT_1:13;
A43: j+1 > j by NAT_1:13;
A44: m+1 > m by NAT_1:13;
A45: i+1 > i by NAT_1:13;
    per cases by A25;
    suppose
A46:  i1 = i2 & j1+1 = j2;
      now
        assume m <> i;
        then m < i or m > i by XXREAL_0:1;
        hence contradiction by A6,A7,A8,A9,A10,A11,A12,A13,A16,A17,A22,A24,A34
,A35,A46,GOBOARD5:3;
      end;
      then
A47:  |.m-i.| = 0 by ABSVALUE:2;
      now
        assume j <= k;
        then
A48:    F*(i,j)`2 <= F*(m,k)`2 by A8,A9,A11,A14,A18,A19,SPRECT_3:12;
        j1 < j2 by A46,NAT_1:13;
        hence contradiction by A6,A7,A22,A24,A28,A30,A31,A33,A36,A37,A48,
GOBOARD5:4;
      end;
      then
A49:  k+1 <= j by NAT_1:13;
      now
        assume
A50:    k+1 < j;
        then
A51:    k+1 < width F by A15,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:    1 <= i9 by A53,MATRIX_0:32;
        i9 <= len F by A53,MATRIX_0:32;
        then
A56:    F*(i9,k+1)`2 = F*(1,k+1)`2 by A38,A51,A55,GOBOARD5:1;
A57:    F*(m,k+1)`2 = F*(1,k+1)`2 by A8,A9,A38,A51,GOBOARD5:1;
        consider i19,j19 such that
A58:    [i19,j19] in Indices G and
A59:    f/.l = G*(i19,j19) by A1,A52;
A60:    1 <= i19 by A58,MATRIX_0:32;
A61:    i19 <= len G by A58,MATRIX_0:32;
A62:    1 <= j19 by A58,MATRIX_0:32;
A63:    j19 <= width G by A58,MATRIX_0:32;
A64:    G*(i19,j1)`2 = G*(1,j1)`2 by A28,A29,A60,A61,GOBOARD5:1;
A65:    G*(i2,j2)`2 = G*(1,j2)`2 by A30,A31,A32,A33,GOBOARD5:1;
A66:    G*(i19,j2)`2 = G*(1,j2)`2 by A32,A33,A60,A61,GOBOARD5:1;
A67:    now
          assume j1 >= j19;
          then G*(i19,j19)`2 <= G*(i1,j1)`2 by A29,A36,A60,A61,A62,A64,
SPRECT_3:12;
          hence contradiction by A6,A8,A9,A10,A22,A42,A51,A54,A56,A57,A59,
GOBOARD5:4;
        end;
        now
          assume j2 <= j19;
          then G*(i2,j2)`2 <= G*(i19,j19)`2 by A32,A60,A61,A63,A65,A66,
SPRECT_3:12;
hence contradiction by A7,A8,A9,A15,A18,A19,A24,A38,A50,A54,A56,A57,A59,
GOBOARD5:4;
        end;
        hence contradiction by A46,A67,NAT_1:13;
      end;
      then k+1 = j by A49,XXREAL_0:1;
      then |.j-k.| = 1 by ABSVALUE:def 1;
      hence thesis by A47,UNIFORM1:11;
    end;
    suppose
A68:  i1+1 = i2 & j1 = j2;
      now
        assume k <> j;
        then k < j or k > j by XXREAL_0:1;
        hence contradiction by A6,A7,A8,A9,A10,A11,A14,A15,A18,A19,A22,A24,A36
,A37,A68,GOBOARD5:4;
      end;
      then
A69:  |.k-j.| = 0 by ABSVALUE:2;
      now
        assume i <= m;
        then
A70:    F*(i,j)`1 <= F*(m,k)`1 by A9,A10,A11,A12,A16,A17,SPRECT_3:13;
        i1 < i2 by A68,NAT_1:13;
        hence contradiction by A6,A7,A22,A24,A26,A28,A29,A31,A68,A70,GOBOARD5:3
;
      end;
      then
A71:  m+1 <= i by NAT_1:13;
      now
        assume
A72:    m+1 < i;
        then
A73:    m+1 < len F by A13,XXREAL_0:2;
        then consider l,j9 such that
A74:    l in dom f and
A75:    [m+1,j9] in Indices F and
A76:    f/.l = F*(m+1,j9) by JORDAN5D:7,NAT_1:12;
A77:    1 <= j9 by A75,MATRIX_0:32;
        j9 <= width F by A75,MATRIX_0:32;
        then
A78:    F*(m+1,j9)`1 = F*(m+1,1)`1 by A40,A73,A77,GOBOARD5:2;
A79:    F*(m+1,k)`1 = F*(m+1,1)`1 by A10,A11,A40,A73,GOBOARD5:2;
        consider i19,j19 such that
A80:    [i19,j19] in Indices G and
A81:    f/.l = G*(i19,j19) by A1,A74;
A82:    1 <= i19 by A80,MATRIX_0:32;
A83:    i19 <= len G by A80,MATRIX_0:32;
A84:    1 <= j19 by A80,MATRIX_0:32;
A85:    j19 <= width G by A80,MATRIX_0:32;
        then
A86:    G*(i1,j19)`1 = G*(i1,1)`1 by A26,A27,A84,GOBOARD5:2;
A87:    G*(i2,j2)`1 = G*(i2,1)`1 by A30,A31,A32,A33,GOBOARD5:2;
A88:    G*(i2,j19)`1 = G*(i2,1)`1 by A30,A31,A84,A85,GOBOARD5:2;
A89:    now
          assume i1 >= i19;
          then G*(i19,j19)`1 <= G*(i1,j1)`1 by A27,A34,A82,A84,A85,A86,
SPRECT_3:13;
          hence contradiction by A6,A8,A10,A11,A22,A44,A73,A76,A78,A79,A81,
GOBOARD5:3;
        end;
        now
          assume i2 <= i19;
          then G*(i2,j2)`1 <= G*(i19,j19)`1 by A30,A83,A84,A85,A87,A88,
SPRECT_3:13;
          hence contradiction by A7,A10,A11,A13,A16,A17,A24,A40,A72,A76,A78,A79
,A81,GOBOARD5:3;
        end;
        hence contradiction by A68,A89,NAT_1:13;
      end;
      then m+1 = i by A71,XXREAL_0:1;
      then |.i-m.| = 1 by ABSVALUE:def 1;
      hence thesis by A69,UNIFORM1:11;
    end;
    suppose
A90:  i1 = i2+1 & j1 = j2;
      now
        assume k <> j;
        then k < j or k > j by XXREAL_0:1;
        hence contradiction by A6,A7,A8,A9,A10,A11,A14,A15,A18,A19,A22,A24,A36
,A37,A90,GOBOARD5:4;
      end;
      then
A91:  |.k-j.| = 0 by ABSVALUE:2;
      now
        assume m <= i;
        then
A92:    F*(i,j)`1 >= F*(m,k)`1 by A8,A10,A11,A13,A16,A17,SPRECT_3:13;
        i1 > i2 by A90,NAT_1:13;
        hence contradiction by A6,A7,A22,A24,A27,A28,A29,A30,A90,A92,GOBOARD5:3
;
      end;
      then
A93:  i+1 <= m by NAT_1:13;
      now
        assume
A94:    i+1 < m;
        then
A95:    i+1 < len F by A9,XXREAL_0:2;
        then consider l,j9 such that
A96:    l in dom f and
A97:    [i+1,j9] in Indices F and
A98:    f/.l = F*(i+1,j9) by JORDAN5D:7,NAT_1:12;
A99:    1 <= j9 by A97,MATRIX_0:32;
        j9 <= width F by A97,MATRIX_0:32;
        then
A100:   F*(i+1,j9)`1 = F*(i+1,1)`1 by A41,A95,A99,GOBOARD5:2;
A101:   F*(i+1,k)`1 = F*(i+1,1)`1 by A10,A11,A41,A95,GOBOARD5:2;
A102:   F*(i+1,j)`1 = F*(i+1,1)`1 by A14,A15,A41,A95,GOBOARD5:2;
        consider i19,j19 such that
A103:   [i19,j19] in Indices G and
A104:   f/.l = G*(i19,j19) by A1,A96;
A105:   1 <= i19 by A103,MATRIX_0:32;
A106:   i19 <= len G by A103,MATRIX_0:32;
A107:   1 <= j19 by A103,MATRIX_0:32;
A108:   j19 <= width G by A103,MATRIX_0:32;
        then
A109:   G*(i1,j19)`1 = G*(i1,1)`1 by A26,A27,A107,GOBOARD5:2;
A110:   G*(i2,j2)`1 = G*(i2,1)`1 by A30,A31,A32,A33,GOBOARD5:2;
A111:   G*(i2,j19)`1 = G*(i2,1)`1 by A30,A31,A107,A108,GOBOARD5:2;
A112:   now
          assume i2 >= i19;
          then G*(i19,j19)`1 <= G*(i2,j2)`1 by A31,A105,A107,A108,A110,A111,
SPRECT_3:13;
          hence contradiction by A7,A12,A14,A15,A24,A45,A95,A98,A100,A102,A104,
GOBOARD5:3;
        end;
        now
          assume i1 <= i19;
          then G*(i1,j1)`1 <= G*(i19,j19)`1 by A26,A34,A106,A107,A108,A109,
SPRECT_3:13;
          hence contradiction by A6,A9,A10,A11,A22,A41,A94,A98,A100,A101,A104,
GOBOARD5:3;
        end;
        hence contradiction by A90,A112,NAT_1:13;
      end;
      then i+1 = m by A93,XXREAL_0:1;
      hence thesis by A91,ABSVALUE:def 1;
    end;
    suppose
A113: i1 = i2 & j1 = j2+1;
      now
        assume m <> i;
        then m < i or m > i by XXREAL_0:1;
        hence contradiction by A6,A7,A8,A9,A10,A11,A12,A13,A16,A17,A22,A24,A34
,A35,A113,GOBOARD5:3;
      end;
      then
A114: |.m-i.| = 0 by ABSVALUE:2;
      now
        assume j >= k;
        then
A115:   F*(i,j)`2 >= F*(m,k)`2 by A8,A9,A10,A15,A18,A19,SPRECT_3:12;
        j1 > j2 by A113,NAT_1:13;
        hence contradiction by A6,A7,A22,A24,A29,A30,A31,A32,A36,A37,A115,
GOBOARD5:4;
      end;
      then
A116: j+1 <= k by NAT_1:13;
      now
        assume
A117:   j+1 < k;
        then
A118:   j+1 < width F by A11,XXREAL_0:2;
        then consider l,i9 such that
A119:   l in dom f and
A120:   [i9,j+1] in Indices F and
A121:   f/.l = F*(i9,j+1) by JORDAN5D:8,NAT_1:12;
A122:   1 <= i9 by A120,MATRIX_0:32;
        i9 <= len F by A120,MATRIX_0:32;
        then
A123:   F*(i9,j+1)`2 = F*(1,j+1)`2 by A39,A118,A122,GOBOARD5:1;
A124:   F*(i,j+1)`2 = F*(1,j+1)`2 by A12,A13,A39,A118,GOBOARD5:1;
A125:   F*(m,j+1)`2 = F*(1,j+1)`2 by A8,A9,A39,A118,GOBOARD5:1;
        consider i19,j19 such that
A126:   [i19,j19] in Indices G and
A127:   f/.l = G*(i19,j19) by A1,A119;
A128:   1 <= i19 by A126,MATRIX_0:32;
A129:   i19 <= len G by A126,MATRIX_0:32;
A130:   1 <= j19 by A126,MATRIX_0:32;
A131:   j19 <= width G by A126,MATRIX_0:32;
A132:   G*(i19,j1)`2 = G*(1,j1)`2 by A28,A29,A128,A129,GOBOARD5:1;
A133:   G*(i2,j2)`2 = G*(1,j2)`2 by A30,A31,A32,A33,GOBOARD5:1;
A134:   G*(i19,j2)`2 = G*(1,j2)`2 by A32,A33,A128,A129,GOBOARD5:1;
A135:   now
          assume j2 >= j19;
          then G*(i19,j19)`2 <= G*(i2,j2)`2 by A33,A128,A129,A130,A133,A134,
SPRECT_3:12;
          hence contradiction by A7,A12,A13,A14,A24,A43,A118,A121,A123,A124
,A127,GOBOARD5:4;
        end;
        now
          assume j1 <= j19;
          then G*(i1,j1)`2 <= G*(i19,j19)`2 by A28,A36,A128,A129,A131,A132,
SPRECT_3:12;
          hence contradiction by A6,A8,A9,A11,A22,A39,A117,A121,A123,A125,A127,
GOBOARD5:4;
        end;
        hence contradiction by A113,A135,NAT_1:13;
      end;
      then j+1 = k by A116,XXREAL_0:1;
      hence thesis by A114,ABSVALUE:def 1;
    end;
  end;
  thus f is special
  proof
    let i be Nat;
    assume that
A136: 1 <= i and
A137: i+1 <= len f;
    consider i1,j1,i2,j2 such that
A138: [i1,j1] in Indices G and
A139: f/.i = G*(i1,j1) and
A140: [i2,j2] in Indices G and
A141: f/.(i+1) = G*(i2,j2) and
A142: 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,A136,A137,Th3;
A143: 1 <= i1 by A138,MATRIX_0:32;
A144: i1 <= len G by A138,MATRIX_0:32;
A145: 1 <= j1 by A138,MATRIX_0:32;
A146: j1 <= width G by A138,MATRIX_0:32;
A147: 1 <= i2 by A140,MATRIX_0:32;
A148: i2 <= len G by A140,MATRIX_0:32;
A149: 1 <= j2 by A140,MATRIX_0:32;
A150: j2 <= width G by A140,MATRIX_0:32;
A151: G*(i1,j1)`2 = G*(1,j1)`2 by A143,A144,A145,A146,GOBOARD5:1;
    G*(i1,j1)`1 = G*(i1,1)`1 by A143,A144,A145,A146,GOBOARD5:2;
    hence thesis by A139,A141,A142,A147,A148,A149,A150,A151,GOBOARD5:1,2;
  end;
end;
