reserve n for Nat;

theorem Th6:
  for G be Y_equal-in-column Y_increasing-in-line Matrix of
TOP-REAL 2 for i1,i2,j1,j2 be Nat st [i1,j1] in Indices G & [i2,j2]
  in Indices G holds G*(i1,j1)`2 = G*(i2,j2)`2 implies j1 = j2
proof
  let G be Y_equal-in-column Y_increasing-in-line Matrix of TOP-REAL 2;
  let i1,i2,j1,j2 be Nat;
  assume that
A1: [i1,j1] in Indices G and
A2: [i2,j2] in Indices G and
A3: G*(i1,j1)`2 = G*(i2,j2)`2 and
A4: j1 <> j2;
A5: 1 <= j1 & j1 <= width G by A1,MATRIX_0:32;
A6: j1 < j2 or j1 > j2 by A4,XXREAL_0:1;
A7: 1 <= i2 & i2 <= len G by A2,MATRIX_0:32;
A8: 1 <= j2 & j2 <= width G by A2,MATRIX_0:32;
A9: 1 <= i1 & i1 <= len G by A1,MATRIX_0:32;
  then G*(i1,j2)`2 = G*(1,j2)`2 by A8,GOBOARD5:1
    .= G*(i2,j2)`2 by A7,A8,GOBOARD5:1;
  hence contradiction by A3,A9,A5,A8,A6,GOBOARD5:4;
end;
