
theorem
  for X being non empty TopSpace st for N being eventually-directed net
  of Omega X holds ex_sup_of N & sup N in Lim N holds X is monotone-convergence
proof
  let X be non empty TopSpace such that
A1: for N being eventually-directed net of Omega X holds ex_sup_of N &
  sup N in Lim N;
  let D be non empty directed Subset of Omega X;
  Net-Str D = NetStr (#D, (the InternalRel of (Omega X))|_2 D, (id the
    carrier of (Omega X))|D#) by WAYBEL17:def 4;
  then
A2: rng the mapping of Net-Str D = D by YELLOW14:2;
  ex_sup_of Net-Str D by A1;
  hence ex_sup_of D,Omega X by A2;
  let V be open Subset of X such that
A3: sup D in V;
  reconsider Z = V as Subset of Omega X by Lm1;
A4: sup Net-Str D = Sup the mapping of Net-Str D by WAYBEL_2:def 1
    .= sup rng the mapping of Net-Str D;
  the TopStruct of X = the TopStruct of Omega X by Def2;
  then Int Z = Int V by TOPS_3:77;
  then sup Net-Str D in Int Z by A2,A3,A4,TOPS_1:23;
  then
A5: Z is a_neighborhood of sup Net-Str D by CONNSP_2:def 1;
  sup Net-Str D in Lim Net-Str D by A1;
  then Net-Str D is_eventually_in V by A5,YELLOW_6:def 15;
  then consider i being Element of Net-Str D such that
A6: for j being Element of Net-Str D st i <= j holds (Net-Str D).j in V;
  now
    take a = (Net-Str D).i;
    dom the mapping of Net-Str D = the carrier of Net-Str D by FUNCT_2:def 1;
    hence a in D by A2,FUNCT_1:def 3;
    thus a in V by A6;
  end;
  hence thesis by XBOOLE_0:3;
end;
