reserve X for set;
reserve UN for Universe;

theorem Th104:
  for n being Nat holds
  (sequence_univers).(n+1) in GrothendieckUniverse sequence_univers
  proof
    let n be Nat;
    set SU = GrothendieckUniverse sequence_univers;
    now
      (n +1) in NAT;
      hence (n + 1) in dom sequence_univers by Def9;
      set f = sequence_univers;
A1:   dom f = NAT & f.0 = {} &
        for n be Nat holds f.(n+1) = GrothendieckUniverse (f.n) by Def9;
      thus f.1 = f.(0 + 1)
              .= GrothendieckUniverse ({}) by A1
              .= FinSETS by Th45,CLASSES2:56,CLASSES3:21;
    thus f.(n+1)=UNIVERSE n by Th102;
    end;
    then [(n + 1),(sequence_univers).(n + 1)] in sequence_univers &
      sequence_univers in SU & SU is axiom_GU1 by CLASSES3:def 4,FUNCT_1:1;
    then reconsider x = [(n+1),(sequence_univers).(n+1)] as pair Element of SU;
    x`2 is Element of SU;
    hence thesis;
  end;
