reserve C for Simple_closed_curve,
  i, j, n for Nat,
  p for Point of TOP-REAL 2;

theorem Th5:
  for X being Subset of TOP-REAL 2 holds p in X & X is bounded
implies W-bound X <= p`1 & p`1 <= E-bound X & S-bound X <= p`2 & p`2 <= N-bound
  X
proof
  let X be Subset of TOP-REAL 2;
  assume that
A1: p in X and
A2: X is bounded;
  W-bound X = lower_bound (proj1|X) by PSCOMP_1:def 7;
  hence W-bound X <= p`1 by A1,A2,Lm7;
  E-bound X = upper_bound (proj1|X) by PSCOMP_1:def 9;
  hence E-bound X >= p`1 by A1,A2,Lm7;
  S-bound X = lower_bound (proj2|X) by PSCOMP_1:def 10;
  hence S-bound X <= p`2 by A1,A2,Lm8;
  N-bound X = upper_bound (proj2|X) by PSCOMP_1:def 8;
  hence thesis by A1,A2,Lm8;
end;
