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 Th33:
  for S being non empty RelStr, T being lower-bounded
  antisymmetric non empty RelStr holds Bottom (T |^ the carrier of S) = S -->
  Bottom T
proof
  let S be non empty RelStr, T be lower-bounded antisymmetric non empty RelStr;
  set L = (T |^ the carrier of S);
  reconsider f = S --> Bottom T as Element of L by Th19;
  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 Th19;
    assume b is_>=_than {};
    for x being Element of S holds f9.x <= b9.x
    proof
      let x be Element of S;
      f9. x = ((the carrier of S) --> Bottom T). x .= Bottom T by FUNCOP_1:7;
      hence thesis by YELLOW_0:44;
    end;
    then f9 <= b9 by YELLOW_2:9;
    hence thesis by WAYBEL10:11;
  end;
  f is_>=_than {};
  then f = "\/"({}, L) by A1,YELLOW_0:30;
  hence thesis by YELLOW_0:def 11;
end;
