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 Th57:
  1 in b implies a in exp(b, succ(b-exponent(a))) proof assume
A1: 1 in b;
    per cases by ORDINAL3:8;
    suppose
A2:   0 in a; assume not thesis; then
      exp(b, succ(b-exponent(a))) c= a by ORDINAL1:16; then
      succ(b-exponent(a)) c= b-exponent(a) &
      b-exponent(a) in succ(b-exponent(a)) by A1,A2,Def10,ORDINAL1:6; then
      b-exponent(a) in b-exponent(a);
      hence contradiction;
    end;
    suppose
A3:   a = 0; then
      b-exponent(a) = 0 by Def10; then
      exp(b, succ(b-exponent(a))) = b by ORDINAL2:46;
      hence thesis by A1,A3,Lm1,ORDINAL1:10;
    end;
  end;
