reserve X for set;
reserve UN for Universe;

theorem Th94:
  {} in rng sequence_univers & FinSETS in rng sequence_univers &
  SETS in rng sequence_univers
  proof
    set f = sequence_univers {};
A1: dom f = NAT & f.0 = {} &
    for n be Nat holds f.(n+1) = GrothendieckUniverse (f.n) by Def9;
A2: f.1 = f.(0 + 1)
       .= GrothendieckUniverse ({}) by A1
       .=FinSETS by Th45,CLASSES2:56,CLASSES3:21;
    f.2 = f.(1 + 1)
       .= SETS by Th77,A2,Def9;
    hence thesis by A1,A2,FUNCT_1:3;
  end;
