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

theorem
  for a,b be positive Real, n be non zero Nat holds
    Sum ((a+b,0)Subnomial (n+1)) > Sum ((a,b)Subnomial (n+1))
  proof
    let a,b be positive Real, n be non zero Nat;
    (a+b - 0)*Sum (((a+b),0)Subnomial (n+1)) =
      (a+b)|^((n+1)+1) - 0|^((n+1)+1)  by SumS; then
    (a+b)*Sum (((a+b),0)Subnomial (n+1)) = (a+b)*(a+b)|^(n+1) by NEWTON:6; then
    Sum (((a+b),0)Subnomial (n+1)) = (a+b)|^(n+1) by XCMPLX_1:5; then
    Sum (((a+b),0)Subnomial (n+1)) = Sum ((a,b) In_Power (n+1)) by NEWTON:30;
    hence thesis by SSI;
  end;
