reserve X for set;
reserve UN for Universe;

theorem Th103:
  FinSETS in GrothendieckUniverse sequence_univers &
  SETS in GrothendieckUniverse sequence_univers
  proof
    set SU = GrothendieckUniverse sequence_univers;
    now
      1 in NAT & 2 in NAT;
      hence 1 in dom sequence_univers & 2 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
A2:   f.1 = f.(0 + 1)
         .= GrothendieckUniverse ({}) by A1
         .=FinSETS by Th45,CLASSES2:56,CLASSES3:21;
      thus f.2 = f.(1 + 1)
              .= SETS by Th77,A2,Def9;
    end;
    then [1,FinSETS] in sequence_univers &
      [2,SETS] in sequence_univers & sequence_univers in SU &
      SU is axiom_GU1 by CLASSES3:def 4,FUNCT_1:1;
    then reconsider x = [1,FinSETS], y = [2,SETS] as pair Element of SU;
    x`2 is Element of SU & y`2 is Element of SU;
    hence thesis;
  end;
