 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem Th17:
  for k,j be Nat st j <= k holds (j!)*(k choose j) = eta(k,j)
  proof
    let k,j be Nat;
    assume
A1: j <= k; then
    0 <= k - j by XREAL_1:48; then
    reconsider kj1 = k - j as Nat by INT_1:3,ORDINAL1:def 12;
    (j!)*(k choose j)
     = (j!)*((k!)/((j!) * (kj1!))) by A1,NEWTON:def 3
    .= ((j!)*(k!))/((j!) * (kj1!))
    .= (k!)/(kj1!) by XCMPLX_1:91
    .= eta(k,j) by A1,XREAL_1:233;
    hence thesis;
  end;
