reserve X1, X2, Y for non empty RelStr,
  f for Function of [:X1, X2:], Y,
  x for Element of X1,
  y for Element of X2;
reserve S for non empty RelStr,
  T for complete LATTICE;

theorem Th32:
  for S, T being complete Scott TopLattice, F being non empty
Subset of ContMaps (S, T) holds "\/"(F, (T |^ the carrier of S)) in the carrier
  of ContMaps (S, T)
proof
  let S, T be complete Scott TopLattice, F be non empty Subset of ContMaps (S,
  T);
  reconsider Ex = "\/"(F, (T |^ the carrier of S)) as Function of S, T by Th19;
  for X being Subset of S st X is non empty directed holds Ex
  preserves_sup_of X
  by YELLOW_0:17,Th31;
  then Ex is directed-sups-preserving;
  hence thesis by Def3;
end;
