
theorem :: HAT:
  for n,k be non zero Nat holds (Hartr n).k = 1/(n*((n-1) choose (k-1)))
  proof
    let n,k be non zero Nat;
    per cases;
    suppose
      k in dom RHartr n; then
      k in dom (Hartr n) by VALUED_1:def 7; then
      (Hartr n).k = ((RHartr n).k)" by VALUED_1:def 7
      .= (n*((n-1) choose (k - 1)))" by HAR
      .= 1/(n*((n - 1) choose (k -1))) by XCMPLX_1:215;
      hence thesis;
    end;
    suppose
      B1: not k in dom RHartr n; then
      not k in dom (Hartr n) by VALUED_1:def 7; then
      (Hartr n).k = 1/0 by FUNCT_1:def 2
      .= 1/((RHartr n).k) by B1,FUNCT_1:def 2
      .= 1/(n*((n-1) choose (k - 1))) by HAR;
      hence thesis;
    end;
  end;
