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 Th22:
  b >= 2 implies (b+1)! > 2|^b
  proof
    defpred P[Nat] means ($1+1)! > 2|^$1;
    A1: P[2]
    proof
      B1: (2+1)! = 2*3 by NEWTON:14,15;
      2|^(2) = 2*2 by NEWTON:81;
      hence thesis by B1;
    end;
    A2: for k be Nat st k >= 2 & P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
      B1: k>=2 & (k+1)! > 2|^k; then
      k+2 > 2+0 by XREAL_1:6; then
      (k+1)!*(k+1+1) > 2|^k*2 by B1,XREAL_1:96; then
      (k+1)!*(k+1+1) > 2|^(k+1) by NEWTON:6;
      hence thesis by NEWTON:15;
    end;
    for x be Nat st x >= 2 holds P[x] from NAT_1:sch 8(A1,A2);
    hence thesis;
  end;
