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
  for S, T being non empty Scott complete TopLattice holds Bottom
  ContMaps (S, T) = S --> Bottom T
proof
  let S, T be non empty Scott complete TopLattice;
  set L = ContMaps (S, T);
  reconsider f = S --> Bottom T as Element of L by Th21;
  reconsider f9 = f as Function of S, T;
A1: for b being Element of L st b is_>=_than {} holds f <= b
  proof
    let b be Element of L;
    reconsider b9 = b as Function of S, T by Th21;
    assume b is_>=_than {};
    for i being Element of S holds [f.i, b.i] in the InternalRel of T
    proof
      let i be Element of S;
      f. i = ((the carrier of S) --> Bottom T). i .= Bottom T by FUNCOP_1:7;
      then f9.i <= b9.i by YELLOW_0:44;
      hence thesis;
    end;
    hence thesis by Th20;
  end;
  f is_>=_than {};
  then f = "\/"({}, L) by A1,YELLOW_0:30;
  hence thesis by YELLOW_0:def 11;
end;
