reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;

theorem Th4:
  S1^S2 is Ordinal-yielding implies
  S1 is Ordinal-yielding & S2 is Ordinal-yielding
  proof
    given a such that
A1: rng (S1^S2) c= a;
    thus S1 is Ordinal-yielding
    proof
      take a;
      rng S1 c= rng(S1^S2) by Th3;
      hence thesis by A1;
    end;
    take a;
    rng S2 c= rng(S1^S2) by Th3;
    hence thesis by A1;
  end;
