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 Th14:
  u > n|^k & n >= k & k > i implies (n choose i)*(u |^i) < (u|^k)/n
proof
  assume
A1: u > n|^k & n >= k & k > i;
  then
A2: k>=i+1 by NAT_1:13;
A3: u >=1 by A1,NAT_1:14;
A4: n >0 by A1;
  then n >=1 by NAT_1:14;
  then n|^(i+1) <= n|^k by A2,PREPOWER:93;
  then n|^(i+1) < u by A1,XXREAL_0:2;
  then
A5: n|^(i+1) * u|^i < u * (u|^i) by XREAL_1:68;
  u*(u|^i) = u|^(i+1) by NEWTON:6;
  then u*(u|^i) <= u |^k by A3,A2,PREPOWER:93;
  then n|^(i+1) * u|^i < u|^k & n|^(i+1) = n*n|^i by A5,XXREAL_0:2,NEWTON:6;
  then n*(n|^i * u|^i) < u|^k; then
A6: n|^i * (u|^i)< (u|^k) / n by A4, XREAL_1:81;
  i <= n by A1,XXREAL_0:2;
  then n choose i <= n|^i by Th13;
  then (n choose i) * (u|^i) <= (n|^i)* (u|^i) by XREAL_1:64;
  hence thesis by A6,XXREAL_0:2;
end;
