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 Th27:
  for n,k being Nat holds exp(n,k) = n|^k
  proof let n be Nat;
    defpred P[Nat] means exp(n,$1) = n|^$1;
    exp(n,0) = 1 by ORDINAL2:43; then
A1: P[0] by NEWTON:4;
A2: now
      let k be Nat such that
A3:   P[k];
      reconsider n9 = n, nk = n|^k as Element of NAT by ORDINAL1:def 12;
      Segm(k+1) = succ Segm k by NAT_1:38; then
      exp(n,k+1) = n *^ exp(n,k) by ORDINAL2:44 .= n9 * nk by A3,CARD_2:37;
      hence P[k+1] by NEWTON:6;
    end;
    thus for k being Nat holds P[k] from NAT_1:sch 2(A1,A2);
  end;
