reserve X for set;
reserve UN for Universe;

theorem
  for X being set for n being Nat holds
  (sequence_univers X).(n+1) is epsilon-transitive &
  Tarski-Class ((sequence_univers X).(n+1))
    = GrothendieckUniverse ((sequence_univers X).(n+1))
  proof
    let X be set;
    let n be Nat;
    (sequence_univers X).(n+1) = GrothendieckUniverse ((sequence_univers X).n)
      by Def9;
    hence thesis by CLASSES3:22;
  end;
