reserve X for set;
reserve UN for Universe;

theorem Th100:
  for n being Nat holds (sequence_univers FinSETS).n = UNIVERSE n
  proof
    let n be Nat;
    defpred P[Nat] means (sequence_univers FinSETS).$1 = UNIVERSE $1;
A1: P[0] by Def9,CLASSES2:65;
A2: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
A3:  P[k];
      now
        thus (sequence_univers FinSETS).(k+1)
          = GrothendieckUniverse (sequence_univers FinSETS).k by Def9;
        succ Segm k = Segm(k + 1) by NAT_1:38;
        hence UNIVERSE (k+1)
          = Tarski-Class ((sequence_univers FinSETS).k) by A3,CLASSES2:66;
      end;
      hence thesis by Th98;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
