
theorem
  for X being non empty TopSpace for L being non trivial complete Scott
  TopLattice holds oContMaps(X, L) is complete continuous iff InclPoset the
  topology of X is continuous & L is continuous
proof
  let X be non empty TopSpace;
  let L be non trivial complete Scott TopLattice;
A1: L is Scott TopAugmentation of L by YELLOW_9:44;
  Omega L = the TopRelStr of L by WAYBEL25:15;
  then
A2: the RelStr of Omega L = the RelStr of L;
A3: L is monotone-convergence by WAYBEL25:29;
  hereby
    assume
A4: oContMaps(X, L) is complete continuous;
    hence InclPoset the topology of X is continuous by A3,Th26;
    Omega L is continuous by A1,A4,Th28;
    hence L is continuous by A2,YELLOW12:15;
  end;
  thus thesis by A1,Th36;
end;
