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 Th35: :: Lemma 2.4, p. 115
  for S, T being complete Scott TopLattice holds ContMaps (S, T)
  is sups-inheriting SubRelStr of (T |^ the carrier of S)
proof
  let S, T be complete Scott TopLattice;
  set L = T |^ the carrier of S;
  reconsider CS = ContMaps (S, T) as full SubRelStr of L by Def3;
  now
    let X be Subset of CS;
    assume ex_sup_of X,L;
    per cases;
    suppose
      X is non empty;
      hence "\/"(X,L) in the carrier of CS by Th32;
    end;
    suppose
      X is empty;
      then "\/"(X,L) = Bottom L by YELLOW_0:def 11;
      then "\/"(X,L) = S --> Bottom T by Th33;
      hence "\/"(X,L) in the carrier of CS by Def3;
    end;
  end;
  hence thesis by YELLOW_0:def 19;
end;
