reserve f for non empty FinSequence of TOP-REAL 2,
  i,j,k,k1,k2,n,i1,i2,j1,j2 for Nat,
  r,s,r1,r2 for Real,
  p,q,p1,q1 for Point of TOP-REAL 2,
  G for Go-board;
reserve f for non constant standard special_circular_sequence;

theorem Th54:
  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 implies 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 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
  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;
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 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
    assume k1 > 1;
    then k1 >= 1+1 by NAT_1:13;
    hence k1 = 2 or k1 > 2 by XXREAL_0:1;
  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 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
    assume k2 > 1;
    then k2 >= 1+1 by NAT_1:13;
    hence k2 = 2 or k2 > 2 by XXREAL_0:1;
  end;
A21: k1 = 1 or k1 > 1 by A10,XXREAL_0:1;
  now
    per cases by A16,A14,A20,A21,XXREAL_0:1;
    case that
A22:  k1 = 1 and
A23:  k2 = 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+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;
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;
      thus f/.1 = (GoB f)*(i,j) by A18,A23,A29,A24,A27,SPPOL_1:8;
      thus f/.(1+2) = (GoB f)*(i+1,j+1) by A12,A22,A28,A25,A27,SPPOL_1:8;
    end;
    case that
A30:  k1 = 1 and
A31:  k2 > 2;
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;
      thus f/.2 = (GoB f)*(i,j) by A12,A30,A32,A36,A37,A35,SPPOL_1:8;
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;
    end;
    case that
A40:  k2 = 1 and
A41:  k1 = 2;
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;
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;
      thus f/.1 = (GoB f)*(i+1,j+1) by A12,A41,A47,A42,A45,SPPOL_1:8;
      thus f/.(1+2) = (GoB f)*(i,j) by A18,A40,A46,A43,A45,SPPOL_1:8;
    end;
    case that
A48:  k2 = 1 and
A49:  k1 > 2;
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;
      thus f/.2 = (GoB f)*(i+1,j+1) by A18,A48,A50,A54,A55,A53,SPPOL_1:8;
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;
    end;
    case
      k1 = k2;
      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_0:30;
      [i,j] in Indices GoB f & [i,j+1] in Indices GoB f by A1,A3,A4,A15,A8,A7,
MATRIX_0:30;
      then j = j+1 by A58,A59,GOBOARD1:5;
      hence contradiction;
    end;
    case that
A60:  k1 > 1 and
A61:  k2 > k1;
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;
      thus f/.(k1+1) = (GoB f)*(i,j+1) by A12,A69,A66,A64,A68,SPPOL_1:8;
      thus f/.k1 = (GoB f)*(i,j) by A12,A69,A66,A64,A68,SPPOL_1:8;
      thus f/.(k1+2) = (GoB f)*(i+1,j+1) by A12,A69,A66,A64,A70,SPPOL_1:8;
    end;
    case that
A71:  k2 > 1 and
A72:  k1 > k2;
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;
      thus f/.(k2+1) = (GoB f)*(i,j+1) by A18,A80,A77,A75,A79,SPPOL_1:8;
      thus f/.k2 = (GoB f)*(i+1,j+1) by A18,A80,A77,A75,A79,SPPOL_1:8;
      thus f/.(k2+2) = (GoB f)*(i,j) by A18,A80,A77,A75,A81,SPPOL_1:8;
    end;
  end;
  hence thesis;
end;
