theorem
  (for X st X in F holds X is natural-membered) implies union F is
  natural-membered
proof
  assume
A1: for X st X in F holds X is natural-membered;
  let x;
  assume x in union F;
  then consider X such that
A2: x in X and
A3: X in F by TARSKI:def 4;
  X is natural-membered by A1,A3;
  hence thesis by A2;
end;
