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 Th28:
  for n,k being Nat st n > 1 holds n |^|^ k > k
  proof
    let n,k be Nat such that
A1: n > 1;
    defpred H[Nat] means n |^|^ $1 > $1;
A2: H[0] by Th13;
A3: now
      let k be Nat such that
A4:   H[k];
      succ Segm k = Segm(k+1) by NAT_1:38; then
      n |^|^ (k+1) = exp(n, n |^|^ k) by Th14 .= n |^ (n |^|^ k); then
A5:   n |^|^ (k+1) > n |^ k by A1,A4,PEPIN:66;
      n |^ k > k by A1,NAT_4:3; then
      n |^ k >= k+1 by NAT_1:13;
      hence H[k+1] by A5,XXREAL_0:2;
    end;
    for k being Nat holds H[k] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
