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 Th61:
  b <> 0 implies a mod^ exp(b, c) in exp(b, c) proof assume that
A1: b <> 0;
    set n = a div^ exp(b, c);
    exp(b,c) <> 0 by A1; then
    consider d such that
A2: a = n*^exp(b, c)+^d & d in exp(b, c) by ORDINAL3:def 6;
    d = a -^ n*^exp(b, c) by A2,ORDINAL3:52
    .= a mod^ exp(b,c) by ORDINAL3:def 7;
    hence thesis by A2;
  end;
