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

theorem D1:
  for k,l be Nat holds
    k in dom (Newton_Coeff l) implies (Newton_Coeff l).k is non zero
  proof
    let k,l be Nat;
    assume
    k in dom (Newton_Coeff l); then
    A1: len (Newton_Coeff l) = l+1 &
    len (Newton_Coeff l) >= k & k >= 1 by FINSEQ_3:25,NEWTON:def 5; then
    reconsider i = k-1 as Nat;
    consider m be Nat such that
    A2: (l+1) = k + m by A1,NAT_1:10;
    (Newton_Coeff (m+i)).(i+1) is non zero;
    hence thesis by A2;
  end;
