reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem Th24:
  b >= 2 implies b! < b|^b
  proof
    defpred P[Nat] means $1! < $1|^$1;
    A1: P[2]
    proof
      2|^(2) = 2*2 by NEWTON:81;
      hence thesis by NEWTON:14;
    end;
    A2: for k be Nat st k >= 2 & P[k] holds P[k+1]
    proof
      let k be Nat such that
      B1: k >= 2 & k! < k|^k;
      B2: k >= 1 by B1,XXREAL_0:2;
      k+1 > k +0 by XREAL_1:6; then
      (k+1)|^k > k|^k by B2,PREPOWER:10; then
      B3: (k+1)|^k*(k+1) > k|^k*(k+1) by XREAL_1:68;
      k!*(k+1) < k|^k*(k+1) by B1,XREAL_1:68; then
      k!*(k+1) < (k+1)|^k*(k+1) by B3,XXREAL_0:2;then
      (k+1)! < (k+1)|^k*(k+1) by NEWTON:15;
      hence thesis by NEWTON:6;
    end;
    for x be Nat st x >= 2 holds P[x] from NAT_1:sch 8(A1,A2);
    hence thesis;
  end;
