reserve X for set;
reserve UN for Universe;

theorem Th101:
  for n being Nat holds
  GrothendieckUniverse((sequence_univers {}).n)
    = (sequence_univers GrothendieckUniverse {}).n
  proof
    defpred P[Nat] means GrothendieckUniverse ((sequence_univers {}).$1)
    = (sequence_univers GrothendieckUniverse {}).$1;

A1: P[0]
    proof
      GrothendieckUniverse ((sequence_univers {}).0)
        = GrothendieckUniverse {} &
      (sequence_univers GrothendieckUniverse {}).0 = GrothendieckUniverse {}
        by Def9;
      hence thesis;
    end;
A2: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
A3:   P[k];
      GrothendieckUniverse ((sequence_univers {}).(k+1))
        = GrothendieckUniverse (GrothendieckUniverse (sequence_univers {}).k) &
      (sequence_univers GrothendieckUniverse {}).(k+1)
        = GrothendieckUniverse (sequence_univers (GrothendieckUniverse {})).k
        by Def9;
      hence thesis by A3;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2 (A1,A2);
    hence thesis;
  end;
