reserve x for object;
reserve D for set;
reserve p for PartialPredicate of D;
reserve D for non empty set;
reserve p,q,r for PartialPredicate of D;

theorem
  PP_and(PP_BottomPred(D),PP_False(D)) = PP_False(D)
  proof
    set B = PP_BottomPred(D);
    set F = PP_False(D);
    set o = PP_and(B,F);
A1: dom(o) = {d where d is Element of D:
    d in dom B & B.d = FALSE or d in dom F & F.d = FALSE
    or d in dom B & B.d = TRUE & d in dom F & F.d = TRUE} by Th16;
    thus dom o = dom F
    proof
      thus dom o c= dom F;
      thus dom F c= dom o
      proof
        let x;
        assume
A2:     x in dom F;
        then F.x = FALSE by FUNCOP_1:7;
        hence thesis by A1,A2;
      end;
    end;
    let x;
    assume x in dom o;
    then per cases;
    suppose x in dom B & B.x = FALSE or
      x in dom B & B.x = TRUE & x in dom F & F.x = TRUE;
      hence thesis;
    end;
    suppose x in dom F & F.x = FALSE;
      hence thesis by Th19;
    end;
  end;
