reserve m,k,j,j1,i,i1,i2,n for Nat,
  r,s for Real,
  C for compact non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p for Point of TOP-REAL 2;

theorem Th26:
  for i1,j1,i2,j2 being Nat, G being Go-board st [i1,j1
  ] in Indices G & [i2,j2] in Indices G & G*(i1,j1) = G*(i2,j2) holds i1 = i2 &
  j1 = j2
proof
  let i1,j1,i2,j2 be Nat, G be Go-board such that
A1: [i1,j1] in Indices G and
A2: [i2,j2] in Indices G and
A3: G*(i1,j1) = G*(i2,j2);
A4: 1 <= i1 by A1,MATRIX_0:32;
A5: j1 <= width G by A1,MATRIX_0:32;
A6: 1 <= j1 by A1,MATRIX_0:32;
A7: 1 <= i2 by A2,MATRIX_0:32;
A8: i1 <= len G by A1,MATRIX_0:32;
A9: i2 <= len G by A2,MATRIX_0:32;
A10: j2 <= width G by A2,MATRIX_0:32;
A11: 1 <= j2 by A2,MATRIX_0:32;
  then
A12: G*(i1,j2)`2 = G*(1,j2)`2 by A4,A8,A10,GOBOARD5:1
    .= G*(i1,j1)`2 by A3,A7,A9,A11,A10,GOBOARD5:1;
A13: G*(i1,j2)`1 = G*(i1,1)`1 by A4,A8,A11,A10,GOBOARD5:2
    .= G*(i1,j1)`1 by A4,A8,A6,A5,GOBOARD5:2;
  assume
A14: not thesis;
  per cases by A14,XXREAL_0:1;
  suppose
    i1 < i2;
    hence contradiction by A3,A4,A9,A11,A10,A13,GOBOARD5:3;
  end;
  suppose
    i1 > i2;
    hence contradiction by A3,A8,A7,A11,A10,A13,GOBOARD5:3;
  end;
  suppose
    j1 < j2;
    hence contradiction by A4,A8,A6,A10,A12,GOBOARD5:4;
  end;
  suppose
    j1 > j2;
    hence contradiction by A4,A8,A5,A11,A12,GOBOARD5:4;
  end;
end;
