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

theorem ILS:
  for a,b be non negative Real holds
    ((a,b) In_Power n).i >= ((a,b) Subnomial n).i
proof
  let a,b be non negative Real;
  per cases;
  suppose i in dom (Newton_Coeff n); then
    (Newton_Coeff n).i <> 0 by D1; then
    (Newton_Coeff n).i * ((a,b) Subnomial n).i >= 1*((a,b)Subnomial n).i
      by NAT_1:14,XREAL_1:64; then
    ((Newton_Coeff n)(#)((a,b) Subnomial n)).i >= ((a,b) Subnomial n).i
      by VALUED_1:5;
    hence thesis by INS;
  end;
  suppose not i in dom (Newton_Coeff n); then
    not i in dom ((a,b)In_Power n) /\ dom (Newton_Coeff n); then
    not i in dom ((a,b) Subnomial n) by VALUED_1:16;
    hence thesis by FUNCT_1:def 2;
  end;
end;
