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 Th16:
  s! is Element of NAT
proof
A0: s is Nat by TARSKI:1;
  defpred P[Nat] means $1! is Element of NAT;
A1: now
    let s be Nat;
    assume P[s];
    then reconsider k=s! as Element of NAT;
    (s+1)! = (s+1) * k by Th15;
    hence P[s+1];
  end;
A2: P[0] by RVSUM_1:94;
  for s be Nat holds P[s] from NAT_1:sch 2(A2,A1);
  hence thesis by A0;
end;
