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
  1 <= j & j < width GoB f & 1 <= i & i+1 < len GoB f implies not ( 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)
proof
  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;
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;
  per cases by A1,A2,A3,A4,A5,A6,Th56,Th57;
  suppose
A12: f/.(len f-'1) = (GoB f)*(i+2,j) & f/.(len f-'1) = (GoB f)*(i+1,j+ 1);
    [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_0:30;
    then j = j+1 by A12,GOBOARD1:5;
    hence contradiction;
  end;
  suppose
A13: f/.2 = (GoB f)*(i,j) & f/.2 = (GoB f)*(i+1,j+1);
    [i,j] in Indices GoB f & [i+1,j+1] in Indices GoB f by A1,A2,A3,A4,A6,A10
,A9,A8,MATRIX_0:30;
    then j = j+1 by A13,GOBOARD1:5;
    hence contradiction;
  end;
  suppose
A14: f/.2 = (GoB f)*(i+2,j) & f/.2 = (GoB f)*(i,j);
    [i+2,j] in Indices GoB f & [i,j] in Indices GoB f by A1,A2,A3,A9,A7,A11,
MATRIX_0:30;
    then i = i+2 by A14,GOBOARD1:5;
    hence contradiction;
  end;
  suppose
A15: f/.2 = (GoB f)*(i+2,j) & f/.2 = (GoB f)*(i+1,j+1);
    [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_0:30;
    then j = j+1 by A15,GOBOARD1:5;
    hence contradiction;
  end;
  suppose that
A16: f/.1 = (GoB f)*(i+1,j) and
A17: ex k 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));
    consider k 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;
  end;
  suppose that
A20: ex k 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);
    consider k 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;
  end;
  suppose that
A24: ex k 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 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));
    consider k1 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 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
      assume
A33:  k1 <> k2;
      per cases by A33,XXREAL_0:1;
      suppose
        k1 < k2;
        then k1+1 < k2+1 by XREAL_1:6;
        hence contradiction by A27,A29,A30,Th36,NAT_1:11;
      end;
      suppose
        k2 < k1;
        then k2+1 < k1+1 by XREAL_1:6;
        hence contradiction by A26,A27,A30,Th36,NAT_1:11;
      end;
    end;
    now
      per cases by A28,A31;
      suppose
A34:    f/.(k1+2) = (GoB f)*(i+2,j) & f/.(k2+2) = (GoB f)*(i+1,j+1);
        [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_0:30;
        then j = j+1 by A32,A34,GOBOARD1:5;
        hence contradiction;
      end;
      suppose
A35:    f/.k1 = (GoB f)*(i,j) & f/.k2 = (GoB f)*(i+1,j+1);
        [i,j] in Indices GoB f & [i+1,j+1] in Indices GoB f by A1,A2,A3,A4,A6
,A10,A9,A8,MATRIX_0:30;
        then j = j+1 by A32,A35,GOBOARD1:5;
        hence contradiction;
      end;
      suppose
A36:    f/.k1 = (GoB f)*(i+2,j) & f/.k2 = (GoB f)*(i,j);
        [i+2,j] in Indices GoB f & [i,j] in Indices GoB f by A1,A2,A3,A9,A7,A11
,MATRIX_0:30;
        then i = i+2 by A32,A36,GOBOARD1:5;
        hence contradiction;
      end;
      suppose
A37:    f/.k1 = (GoB f)*(i+2,j) & f/.k2 = (GoB f)*(i+1,j+1);
        [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_0:30;
        then j = j+1 by A32,A37,GOBOARD1:5;
        hence contradiction;
      end;
    end;
    hence contradiction;
  end;
end;
