reserve n for Nat,
        i,j,i1,i2,i3,i4,i5,i6 for Element of n,
        p,q,r for n-element XFinSequence of NAT;
reserve i,j,n,n1,n2,m,k,l,u,e,p,t for Nat,
        a,b for non trivial Nat,
        x,y for Integer,
        r,q for Real;

theorem Th10:
  k <=n implies n! <= k! * n |^ (n-'k)
proof
  assume
A1: k <=n;
  then reconsider nk=n-k as Nat by NAT_1:21;
  defpred P[Nat] means (k+$1)! <= k! * (k+$1) |^ $1;
  (k+0) |^ 0 = 1 by NEWTON:4;
  then
A2: P[0];
A3: P[i] implies P[i+1]
  proof
    set ki1=k+i+1;
    assume
A4:   P[i];
A5:   (k+i) |^0 = 1 & ki1 |^0 = 1 by NEWTON:4;
A6:   ki1 |^ i * ki1 = ki1 |^(i+1) by NEWTON:6;
    k+i < ki1 & (i>=1 or i=0) by NAT_1:13,14;
    then (k+i) |^ i <= ki1 |^i by A5,PREPOWER:10;
    then k! * (k+i) |^ i <= k! *(ki1 |^i) by XREAL_1:66;
    then
A7:   (k+i)! <= k! *(ki1 |^i) by A4,XXREAL_0:2;
    ki1! = (k+i)!* ki1 by NEWTON:15;
    then ki1! <= (k! * ki1 |^ i) * ki1 by A7,XREAL_1:66;
    hence thesis by A6;
  end;
  P[i] from NAT_1:sch 2(A2,A3);
  then P[nk];
  hence thesis by A1,XREAL_1:233;
end;
