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

theorem NCI:
  (Newton_Coeff n).(i+1) = n choose i
  proof
    per cases;
    suppose
      B1: i+1 in dom Newton_Coeff n; then
      1 <= i+1 <= len (Newton_Coeff n) by FINSEQ_3:25; then
      1 <= i+1 <= n+1 by NEWTON:def 5; then
      reconsider k = (n+1)-(i+1) as Element of NAT by NAT_1: 21;
      n = i+k & i=(i+1)-1;
      hence thesis by B1,NEWTON:def 5;
    end;
    suppose
      B1: not (i+1) in dom Newton_Coeff n; then
      not 1 <= i+1 <= len (Newton_Coeff n) by FINSEQ_3:25; then
      not 0+1 <= i+1 <= n+1 by NEWTON:def 5; then
      not 0 <= i <= n by XREAL_1:6; then
      (n choose i) = 0 by NEWTON:def 3;
      hence thesis by B1,FUNCT_1:def 2;
    end;
  end;
