
theorem Th40:
  for X being non empty TopSpace, Y being monotone-convergence
  T_0-TopSpace, N being net of ContMaps(X,Omega Y), f, g being Function of X,
  Omega Y st f = "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X) &
  ex_sup_of rng the mapping of N, (Omega Y) |^ the carrier of X & g in rng the
  mapping of N holds g <= f
proof
  let X be non empty TopSpace, Y be monotone-convergence T_0-TopSpace, N be
  net of ContMaps(X,Omega Y), f, g be Function of X, Omega Y;
  set m = the mapping of N, L = (Omega Y) |^ the carrier of X, s = "\/"(rng m,
  L);
  assume that
A1: f = "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X) and
A2: ex_sup_of rng m,L and
A3: g in rng the mapping of N;
  reconsider g1 = g as Element of L by WAYBEL24:19;
  rng m is_<=_than s by A2,YELLOW_0:def 9;
  then g1 <= s by A3;
  hence thesis by A1,WAYBEL10:11;
end;
