reserve X for set;
reserve UN for Universe;

theorem Th64:
  for UN being non trivial Universe, X being non empty set st X in UN
  holds X* in UN
  proof
    let UN be non trivial Universe;
    let X be non empty set;
    assume
A1: X in UN;
    consider f be Function such that
A2: dom f = NAT and
A3: for n be Nat holds f.n = n-tuples_on X and
A4: union rng f = X* by Th63;
    now
      thus UN is FamUnion-closed;
      thus dom f = omega by A2;
      now
        let x be object;
        assume x in rng f;
        then consider y be object such that
A5:     y in dom f and
A6:     x = f.y by FUNCT_1:def 3;
        reconsider y as Nat by A5,A2;
        y-tuples_on X in UN by A1,Th58;
        hence x in UN by A6,A3;
      end;
      hence rng f c= UN;
    end;
    hence thesis by Def6,A4;
  end;
