
theorem HAR:
  for n,k be non zero Nat holds (RHartr n).k = n*((n-1) choose (k-1))
  proof
    let n,k be non zero Nat;
    per cases;
    suppose k in dom RHartr n; then
      k <= len (RHartr n) by FINSEQ_3:25; then
      1 <= k <= n by CARD_1:def 7,NAT_1:14; then
      k in Seg len Newton_Coeff (n - 1); then
      A1: k in dom Newton_Coeff (n - 1) by FINSEQ_1:def 3; then
      A2: k in dom (n(#)Newton_Coeff (n - 1)) by VALUED_1:def 5;
      n*((n - 1) choose (k - 1)) = n*(Newton_Coeff (n - 1)).k
        by A1,NEWTON:def 5
      .= (n(#) Newton_Coeff (n - 1)).k by A2,VALUED_1:def 5;
      hence thesis;
    end;
    suppose
      B1: not k in dom RHartr n; then
      1 > k or k > len (RHartr n) by FINSEQ_3:25; then
      k > n by CARD_1:def 7,INT_1:74; then
      k - 1 > n - 1 by XREAL_1:9; then
      n*((n - 1) choose (k - 1)) = n*0 by NEWTON:def 3
      .= (RHartr n).k by B1,FUNCT_1:def 2;
      hence thesis;
    end;
  end;
