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

theorem NIS:
  for a,b be Integer, k be Nat st k in dom (Newton_Coeff n) holds
    (Newton_Coeff n).k divides ((a,b) In_Power n).k
  proof
    let a,b be Integer, k be Nat such that
    A0: k in dom (Newton_Coeff n);
    A0a: k >= 1 by A0,FINSEQ_3:25; then
    reconsider m = k - 1 as Nat;
    A1: n+1 = len ((a,b) In_Power n) by NEWTON:def 4;
    len (Newton_Coeff n) = n+1 by NEWTON:def 5; then
    A1a: n+1 >= m+1 by A0,FINSEQ_3:25; then
    consider l such that
    A2: n = m + l by XREAL_1:6,NAT_1:10;
    A3: (Newton_Coeff n).k = (n choose m) by A0,NEWTON:def 5;
    A4: k in dom ((a,b) In_Power n) by A0a,A1,A1a,FINSEQ_3:25;
    l = n - m by A2; then
    ((a,b) In_Power n).k = (n choose m)*a|^l*b|^m by A4,NEWTON:def 4
    .= (Newton_Coeff n).k*(a|^l*b|^m) by A3;
    hence thesis;
  end;
