reserve i,j,n,n1,n2,m,k,l,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat,
        F for XFinSequence,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

theorem Th13:
  k <= n implies n choose k <= n|^k
proof
  defpred P[Nat] means $1 <= n implies n choose $1 <= n|^$1;
  n choose 0 = 1 by NEWTON:19;
  then
A1: P[0] by NEWTON:4;
A2: P[m] implies P[m+1]
  proof
    assume
A3:   P[m];
    set m1=m+1;
    assume
A4:   m1<=n;
    then m <n by NAT_1:13;
    then
A5:   n-m >m-m by XREAL_1:14;
A6:   n-m <= n-0 by XREAL_1:10;
A7:   n choose m1 = ((n-m)/m1) * (n choose m) by IRRAT_1:5;
    (n-m)/m1 <= (n-m)/1 by A5, XREAL_1:118,NAT_1:11;
    then (n-m)/m1 <= n by A6,XXREAL_0:2;
    then
A8: n choose m1 <= n * (n choose m) by A7,XREAL_1:64;
    n * (n choose m) <= n * (n|^m) by A3,A4,NAT_1:13,XREAL_1:64;
    then n choose m1 <= n * (n|^m) by A8,XXREAL_0:2;
    hence thesis by NEWTON:6;
  end;
  P[m] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
