
theorem
  for S being monotone-convergence T_0-TopSpace, T being T_0-TopSpace st
  S, T are_homeomorphic holds T is monotone-convergence
proof
  let S be monotone-convergence T_0-TopSpace, T be T_0-TopSpace;
  given h being Function of S, T such that
A1: h is being_homeomorphism;
 the carrier of S = the carrier of Omega S by Lm1;
  then reconsider f = h as Function of Omega S, Omega T by Lm1;
  f is isomorphic by A1,Th18;
  then
A2: rng f = the carrier of Omega T by WAYBEL_0:66;
  let D be non empty directed Subset of Omega T;
A3: f" is isomorphic by A1,Th18,YELLOW14:10;
  then f" is sups-preserving by WAYBEL13:20;
  then
A4: f" preserves_sup_of D;
A5: rng h = [#]T by A1;
A6: h is one-to-one by A1;
A7: h is onto by A5,FUNCT_2:def 3;
  f".:D is directed by A3,YELLOW_2:15;
  then
A8: f"D is non empty directed Subset of Omega S by A2,A5,A6,TOPS_2:55;
  then ex_sup_of f"D,Omega S by Def4;
  then ex_sup_of f.:f"D,Omega T by A1,Th18,YELLOW14:16;
  hence
A9: ex_sup_of D,Omega T by A2,FUNCT_1:77;
  let V be open Subset of T;
  assume sup D in V;
  then h".sup D in h".:V by FUNCT_2:35;
  then h".sup D in h"V by A5,A6,TOPS_2:55;
  then h qua Function".sup D in h"V by A7,A6,TOPS_2:def 4;
  then f".sup D in h"V by A2,A6,A7,A5,TOPS_2:def 4;
  then sup (f".:D) in h"V by A9,A4;
  then
A10: sup (f"D) in h"V by A2,A5,A6,TOPS_2:55;
  h"V is open by A1,TOPGRP_1:26;
  then f"D meets h"V by A8,A10,Def4;
  then consider a being object such that
A11: a in f"D and
A12: a in h"V by XBOOLE_0:3;
  reconsider a as Element of S by A12;
  now
    take b = h.a;
    thus b in D & b in V by A11,A12,FUNCT_2:38;
  end;
  hence thesis by XBOOLE_0:3;
end;
