
theorem
  for X being non empty TopSpace, Y being monotone-convergence
T_0-TopSpace holds ContMaps(X,Omega Y) is directed-sups-inheriting SubRelStr of
  (Omega Y) |^ the carrier of X
proof
  let X be non empty TopSpace, Y be monotone-convergence T_0-TopSpace;
  set L = (Omega Y) |^ the carrier of X;
  reconsider C = ContMaps(X,Omega Y) as non empty full SubRelStr of L by
WAYBEL24:def 3;
  C is directed-sups-inheriting
  proof
    let D be directed Subset of C such that
A1: D <> {} and
    ex_sup_of D,L;
    reconsider D as non empty directed Subset of C by A1;
    set N = Net-Str D;
A2: the TopStruct of X = the TopStruct of X;
    for x being Point of X holds commute(N,x,Omega Y) is eventually-directed;
    then
A3: "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X) is
    continuous Function of X, Y by Th42;
    the TopStruct of Y = the TopStruct of Omega Y by Def2;
    then "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X) is
    continuous Function of X, Omega Y by A2,A3,YELLOW12:36;
    then
A4: "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X) in the
    carrier of C by WAYBEL24:def 3;
    Net-Str D = NetStr (#D, (the InternalRel of C)|_2 D, (id the carrier
      of C)|D#) by WAYBEL17:def 4;
    hence thesis by A4,YELLOW14:2;
  end;
  hence thesis;
end;
