
theorem Th5:
  for P being set, f being Function st rng f c= union P
  ex p being Function st dom p = dom f & rng p c= P & f in product p
proof
  let P be set, f be Function;
  assume
A1: rng f c= union P;
  defpred P[object,object] means ex D2 being set st D2 = $2 & f.$1 in D2;
A2: for x being object st x in dom f ex a being object st a in P & P[x,a]
  proof
    let x be object;
    assume x in dom f;
    then f.x in rng f by FUNCT_1:def 3;
    then ex a being set st f.x in a & a in P by A1,TARSKI:def 4;
    hence thesis;
  end;
  consider p being Function such that
A3: dom p = dom f & rng p c= P and
A4: for x being object st x in dom f holds P[x,p.x] from FUNCT_1:sch 6(A2);
  take p;
   for x being object st x in dom f holds f.x in p.x
    proof let x be object;
     assume x in dom f;
      then P[x,p.x] by A4;
     hence f.x in p.x;
    end;
  hence thesis by A3,CARD_3:def 5;
end;
