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 Th63:
  1 in c & c-exponent b in c-exponent a implies b in a proof assume
A1: 1 in c & c-exponent b in c-exponent a; then
    succ(c-exponent(b)) c= c-exponent(a) by ORDINAL1:21; then
A2: exp(c, succ(c-exponent(b))) c= exp(c, c-exponent(a)) by A1,ORDINAL4:27;
A3: 0 in a by A1,Def10;
    b in exp(c, succ(c-exponent(b))) by A1,Th57; then
    b in exp(c, c-exponent(a)) & exp(c, c-exponent(a)) c= a by A1,A2,A3,Def10;
    hence thesis;
  end;
