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 Th47:
  PP_and(p,PP_False(D)) = PP_False(D)
  proof
    set q = PP_False(D);
    set f = PP_and(p,q);
A1: dom f = {d where d is Element of D:
             d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
          or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE} by Th16;
    thus
A3: dom f = dom q
    proof
      thus dom f c= dom q;
      let x;
      assume x in dom q;
      then reconsider d = x as Element of D;
      q.d = FALSE;
      hence thesis by A1;
    end;
    let x;
    assume
A5: x in dom f;
    then q.x = FALSE by FUNCOP_1:7;
    hence f.x = q.x by A3,A5,Th19;
  end;
