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 Th58:
  for a, b, c being Ordinal
  holds 0 in c & c in b implies b -exponent(c*^exp(b,a)) = a
proof
  let a, b, c be Ordinal; assume
A1: 0 in c & c in b; then
A2: succ 0 c= c by ORDINAL1:21; then
A3: 1*^exp(b,a) = exp(b,a) & 1*^exp(b,a) c= c*^exp(b,a) by ORDINAL2:39,41;
A4: 1 in b & 0 in c*^exp(b,a) by A2,A1,ORDINAL1:12,ORDINAL3:8;
    now let d be Ordinal; assume
A5:   exp(b,d) c= c*^exp(b,a) & d c/= a; then
      succ a c= d by ORDINAL1:16, ORDINAL1:21; then
A6:   exp(b, succ a) c= exp(b, d) by A1,ORDINAL4:27;
      exp(b, succ a) = b*^exp(b, a) by ORDINAL2:44; then
      b*^exp(b, a) c= c*^exp(b,a) by A5,A6; then
      b c= c by ORDINAL3:35; then
      c in c by A1;
      hence contradiction;
    end;
    hence b-exponent(c*^exp(b,a)) = a by A3,A4,Def10;
  end;
