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 Th8:
   n >=k implies n choose k <= (n|^k) / (k!)
proof
  defpred P[Nat] means $1 <= n implies n choose $1 <= (n|^$1) / ($1!);
  n|^0 = 1 & n choose 0 = 1 & 0! =1 by NEWTON:4,12,19;
  then
A1: P[0];
A2: P[i] implies P[i+1]
  proof
    set i1=i+1;
    assume
A3:   P[i] & i+1 <=n;
    then
A4:   i <n by NAT_1:13;
A5:   0 < n-i by XREAL_1:50,A4;
    n-i <= n-0 by XREAL_1:10;
    then (n-i)/i1 <= n /i1 by XREAL_1:72;
    then
A6:   ((n-i)/i1)*(n choose i) <= (n/i1)*((n|^i)/(i!))
      by A5,NAT_1:13,A3,XREAL_1:66;
    (n/i1)*((n|^i)/(i!)) = (n*(n|^i))/(i1*(i!)) by XCMPLX_1:76
    .= (n|^i1)/(i1*(i!)) by NEWTON:6
    .= (n|^i1)/(i1!) by NEWTON:15;
    hence thesis by A6,IRRAT_1:5;
  end;
  P[i] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
