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
  a in b implies n*^exp(omega, a) in exp(omega, b)
  proof
    assume a in b; then
    succ a c= b by ORDINAL1:21; then
A1: exp(omega, succ a) c= exp(omega, b) by ORDINAL4:27;
A2: exp(omega, succ a) = omega*^exp(omega, a) by ORDINAL2:44;
    n in omega by ORDINAL1:def 12; then
    n*^exp(omega, a) in omega*^exp(omega, a) by ORDINAL3:19,ORDINAL4:22;
    hence thesis by A1,A2;
  end;
