reserve X for set;
reserve UN for Universe;

theorem
  for X being Element of union rng sequence_univers
  for n being Nat st rank_univers X <= n holds
  X in (sequence_univers).n
  proof
    let X be Element of union rng sequence_univers;
    let n be Nat;
    assume
A1: rank_univers X <= n;
    defpred P[Nat] means X in (sequence_univers).$1;
A2: P[rank_univers X] by Def13;
A3: for j be Nat st rank_univers X <= j & P[j] holds P[j+1]
    proof
      let j be Nat;
      assume that rank_univers X <= j and
A4:   P[j];
      (sequence_univers).j c= (sequence_univers).(j+1) by Th105;
      hence thesis by A4;
    end;
    for i be Nat st rank_univers X <= i holds P[i] from NAT_1:sch 8(A2,A3);
    hence thesis by A1;
  end;
