reserve X for set;
reserve UN for Universe;

theorem Th105:
  for n being Nat holds (sequence_univers).n c= (sequence_univers).(n+1)
  proof
    defpred P[Nat] means (sequence_univers).$1 c= (sequence_univers).($1+1);
A1: P[0]
    proof
      {} c= (sequence_univers).1;
      hence thesis by Def9;
    end;
A2: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume P[k];
A3:   (sequence_univers).(k+1) = UNIVERSE k by Th102;
      Segm k c= Segm (k + 1) by NAT_1:11,39;
      then UNIVERSE k c= UNIVERSE (k+1) by CLASSES2:72;
      hence thesis by A3,Th102;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
