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 Th15:
  C is vertical iff W-bound C = E-bound C
proof
  thus C is vertical implies W-bound C = E-bound C
  proof
A1: E-min C in C by Th14;
A2: W-min C in C by Th13;
    assume
A3: C is vertical;
    thus W-bound C = (W-min C)`1 by EUCLID:52
      .= (E-min C)`1 by A3,A2,A1
      .= E-bound C by EUCLID:52;
  end;
  assume
A4: W-bound C = E-bound C;
  let p,q;
  assume that
A5: p in C and
A6: q in C;
A7: p`1 <= E-bound C by A5,PSCOMP_1:24;
  W-bound C <= p`1 by A5,PSCOMP_1:24;
  then
A8: p`1 = W-bound C by A4,A7,XXREAL_0:1;
A9: q`1 <= E-bound C by A6,PSCOMP_1:24;
  W-bound C <= q`1 by A6,PSCOMP_1:24;
  hence thesis by A4,A9,A8,XXREAL_0:1;
end;
