reserve a,b,i,j,k,l,m,n for Nat;

theorem ES:
  (n!)*(k!) <= (n+k)!
  proof
     set s = (n+k) choose n,l = n+k;
    k = l-n & n+k >= n+0 by XREAL_1:6; then
    A2: s*((n!)*(k!)) = (n+k)!/((n!)*(k!))*((n!)*(k!)) by NEWTON:def 3
    .= (n+k)! by XCMPLX_1:87;
    s*((n!)*(k!)) >= 1*((n!)*(k!)) by XREAL_1:64,NAT_1:14;
    hence thesis by A2;
  end;
