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

theorem INS:
  for a,b be Real, n be Nat holds
    (a,b) In_Power n = (Newton_Coeff n) (#) ((a,b) Subnomial n)
  proof
    let a,b be Real, n be Nat;
    A0a: dom ((a,b) In_Power n) = dom (Newton_Coeff n) by DOMN; then
    A0: dom ((a,b) In_Power n) =
      dom (Newton_Coeff n) /\ dom ((a,b) In_Power n); then
    A1: dom ((a,b) In_Power n) = dom ((a,b) Subnomial n) by VALUED_1:16;
    for c be object st c in dom ((a,b) In_Power n) holds
    ((a,b) In_Power n).c = (Newton_Coeff n).c * ((a,b) Subnomial n).c
    proof
      let c be object such that
      B1: c in dom ((a,b) In_Power n);
      reconsider c as positive Nat by B1,FINSEQ_3:25;
      set m = c-1;
      c <= len (Newton_Coeff (n+1-1)) by B1,A0a,FINSEQ_3:25; then
      consider k be Nat such that
      B2: n+1 = c + k by NAT_1:10;
      B3: n = m + k & k = n - m by B2;
      (Newton_Coeff (m+k)).(m+1) * ((a,b) Subnomial (m+k)).(m+1)
      = ((m+k) choose m)* ((a,b) Subnomial (m+k)).(m+1)
        by B2,B1,A0a,NEWTON:def 5
      .= ((m+k) choose m)*((a|^k)*(b|^m)) by B1,A1,B3,Def2
      .= ((m+k) choose m)*(a|^k)*(b|^m)
      .= ((a,b) In_Power (m+k)).(m+1) by NEWTON:def 4,B1,B3;
      hence thesis by B2;
    end;
    hence thesis by A0,A1,VALUED_1:def 4;
  end;
