
theorem Th28:
  for X being non empty TopSpace, Y being monotone-convergence
  T_0-TopSpace st oContMaps(X, Y) is complete continuous holds Omega Y is
  complete continuous
proof
  let X be non empty TopSpace, Y be monotone-convergence T_0-TopSpace such
  that
A1: oContMaps(X, Y) is complete continuous;
  set b = the Element of X;
A2: the TopStruct of Omega Y = the TopStruct of Y by WAYBEL25:def 2;
  consider F being directed-sups-preserving projection Function of oContMaps(X
  ,Y), oContMaps(X,Y) such that
A3: for f being continuous Function of X,Y holds F.f = X --> (f.b) and
  ex h being continuous Function of X,X st h = X --> b & F = oContMaps(h,
  Y) by Th27;
  oContMaps(X, Y) is full SubRelStr of (Omega Y)|^the carrier of X by
WAYBEL24:def 3;
  then reconsider imF = Image F as full non empty SubRelStr of (Omega Y)|^the
  carrier of X by YELLOW16:26;
A4: the carrier of imF = rng F by YELLOW_0:def 15;
A5: dom F = the carrier of oContMaps(X,Y) by FUNCT_2:52;
  now
    let a be set;
    hereby
      assume a is Element of imF;
      then consider h being object such that
A6:   h in dom F and
A7:   a = F.h by A4,FUNCT_1:def 3;
      reconsider h as continuous Function of X,Y by A6,Th2;
      reconsider x = h.b as Element of Omega Y by A2;
      a = X --> (h.b) by A3,A7
        .= (the carrier of X) --> x;
      hence ex x being Element of Omega Y st a = (the carrier of X) --> x;
    end;
    given x being Element of Omega Y such that
A8: a = (the carrier of X) --> x;
    a = X --> x by A8;
    then
A9: a is Element of oContMaps(X,Y) by Th1;
    then reconsider h = a as continuous Function of X,Y by Th2;
A10: X --> (h.b) = (the carrier of X) --> (h.b);
    h.b = x by A8,FUNCOP_1:7;
    then F.a = X --> x by A3,A10;
    hence a is Element of imF by A4,A5,A8,A9,FUNCT_1:def 3;
  end;
  then Omega Y, imF are_isomorphic by YELLOW16:50;
  then
A11: imF, Omega Y are_isomorphic by WAYBEL_1:6;
  Image F is complete continuous LATTICE by A1,WAYBEL15:15,YELLOW_2:35;
  hence thesis by A11,WAYBEL15:9,WAYBEL20:18;
end;
