
theorem Th15:
  for T being non empty TopSpace holds for f being Function of T,
  R^1 holds for P being Subset of T holds P <> {} & P is compact & f is
continuous implies ex x2 being Point of T st x2 in P & f.x2 = lower_bound(f.:P)
proof
  let T be non empty TopSpace;
  let f be Function of T,R^1;
  let P be Subset of T;
  assume P <> {} & P is compact & f is continuous;
  then consider x1,x2 being Point of T such that
  x1 in P and
A1: x2 in P and
  f.x1 = upper_bound [#](f.:P) and
A2: f.x2 = lower_bound [#](f.:P) by Lm1;
  take x2;
  thus thesis by A1,A2;
end;
