reserve n for Nat;

theorem Th7:
  for G be X_equal-in-line X_increasing-in-column 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)`1 = G*(i2,j2)`1 implies i1 = i2
proof
  let G be X_equal-in-line X_increasing-in-column 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)`1 = G*(i2,j2)`1 and
A4: i1 <> i2;
A5: 1 <= i1 & i1 <= len G by A1,MATRIX_0:32;
A6: 1 <= i2 & i2 <= len G by A2,MATRIX_0:32;
A7: 1 <= j2 & j2 <= width G by A2,MATRIX_0:32;
A8: i1 < i2 or i1 > i2 by A4,XXREAL_0:1;
  1 <= j1 & j1 <= width G by A1,MATRIX_0:32;
  then G*(i1,j1)`1 = G*(i1,1)`1 by A5,GOBOARD5:2
    .= G*(i1,j2)`1 by A5,A7,GOBOARD5:2;
  hence contradiction by A3,A5,A6,A7,A8,GOBOARD5:3;
end;
