reserve X for set;
reserve UN for Universe;

theorem
  for f being Function of NAT,union rng sequence_univers holds
  f in super_univers
  proof
    let f be Function of NAT,union rng sequence_univers;
A1: super_univers is non trivial;
    now
      dom sequence_univers = NAT & super_univers is non trivial by Def9;
      hence dom sequence_univers in super_univers;
      now
        let x be object;
        assume x in rng sequence_univers;
        then consider y be object such that
A2:     y in dom sequence_univers and
A3:     x = (sequence_univers).y by FUNCT_1:def 3;
        reconsider y as Nat by A2;
        per cases;
        suppose
A4:       y = 0;
          (sequence_univers).0 = {} by Def9;
          hence x in super_univers by A4,A3,Th13;
        end;
        suppose y <> 0;
          then consider m be Nat such that
A5:       y = m + 1 by NAT_1:6;
          thus x in super_univers by A5,A3,Th104;
        end;
      end;
      hence rng sequence_univers c= super_univers;
    end;
    then union rng sequence_univers in super_univers by CLASSES3:def 3;
    then [:NAT,union rng sequence_univers:] in super_univers by A1,CLASSES2:61;
    hence thesis by Th13;
  end;
