reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;

theorem Th38:
  l! >= l
proof
  defpred P[Nat] means $1! >= $1;
A1: for l st P[l] holds P[l+1]
  proof
    let l;
    assume
A2: l!>=l;
A3: l!*(l+1)=(l+1)! by Th15;
    l=0 & (l+1)!>=(l+1) or l>=1 & (l+1)!>=(l+1)
    proof
      per cases by NAT_1:14;
      case
        l=0;
        hence thesis by FINSEQ_2:50;
      end;
      case
A4:     l>=1;
        (l+1)!>=(l+1)*l by A2,A3,XREAL_1:64;
        hence thesis by A4,Th34;
      end;
    end;
    hence thesis;
  end;
A5: P[0];
  for l being Nat holds P[l] from NAT_1:sch 2(A5,A1);
  hence thesis;
end;
