reserve S for Subset of TOP-REAL 2,
  C,C1,C2 for non empty compact Subset of TOP-REAL 2,
  p,q for Point of TOP-REAL 2;

theorem Th16:
  C is horizontal iff S-bound C = N-bound C
proof
  thus C is horizontal implies S-bound C = N-bound C
  proof
A1: N-min C in C by Th11;
A2: S-min C in C by Th12;
    assume
A3: C is horizontal;
    thus S-bound C = (S-min C)`2 by EUCLID:52
      .= (N-min C)`2 by A3,A2,A1
      .= N-bound C by EUCLID:52;
  end;
  assume
A4: S-bound C = N-bound C;
  let p,q;
  assume that
A5: p in C and
A6: q in C;
A7: p`2 <= N-bound C by A5,PSCOMP_1:24;
  S-bound C <= p`2 by A5,PSCOMP_1:24;
  then
A8: p`2 = S-bound C by A4,A7,XXREAL_0:1;
A9: q`2 <= N-bound C by A6,PSCOMP_1:24;
  S-bound C <= q`2 by A6,PSCOMP_1:24;
  hence thesis by A4,A9,A8,XXREAL_0:1;
end;
