reserve E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  i, j, m, n for Nat,
  p for Point of TOP-REAL 2;

theorem
  for P be Simple_closed_curve holds W-max(P) <> E-max(P)
proof
  let P be Simple_closed_curve;
  now
A1: |[E-bound P, upper_bound (proj2|E-most P)]|=E-max(P) by PSCOMP_1:def 23;
A2: |[W-bound P, upper_bound (proj2|W-most P)]|=W-max(P) by PSCOMP_1:def 20;
    assume W-max(P) = E-max(P);
    then W-bound P= E-bound P by A2,A1,SPPOL_2:1;
    hence contradiction by TOPREAL5:17;
  end;
  hence thesis;
end;
