
theorem Th31:
  for X being monotone-convergence non empty TopSpace, N being
  eventually-directed prenet of Omega X holds ex_sup_of N
proof
  let X be monotone-convergence non empty TopSpace, N be eventually-directed
  prenet of Omega X;
  rng netmap (N,Omega X) is directed by WAYBEL_2:18;
  hence ex_sup_of rng the mapping of N,Omega X by Def4;
end;
