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 Th62:
  0 in a implies
  ex n,c st a = n*^exp(omega, omega-exponent(a))+^c & 0 in Segm n &
  c in exp(omega, omega-exponent(a))
  proof assume
A1: 0 in a;
    set c = omega-exponent a;
    set n = a div^ exp(omega, c);
    set b = a mod^ exp(omega, c);
    n in omega  by A1,Th59; then
    reconsider n as Nat;
    take n,b;
    thus a = n*^exp(omega, c)+^b by ORDINAL3:65;
    thus 0 in Segm n by A1,Th60;
    thus b in exp(omega, c) by Th61;
  end;
