reserve X for set;
reserve UN for Universe;

theorem Th16:
  for n being Nat holds n in UN
  proof
    defpred P[Nat] means $1 in UN;
A1: P[0] by Th13;
    now
      let n be Nat;
      assume
A2:   P[n];
A3:   n (+) 1 = succ n by ORDINAL7:77;
      reconsider n9 = n as Element of UN by A2;
      succ n9 in UN;
      hence n + 1 in UN by A3,ORDINAL7:76;
    end;
    then
A4: for n be Nat st P[n] holds P[n + 1];
    for n be Nat holds P[n] from NAT_1:sch 2(A1,A4);
    hence thesis;
  end;
