
theorem
  for n be Nat holds
    Parity ((n+2)!) = 2*Parity(Triangle (n+1))*Parity (n!)
  proof
    let n be Nat;
    ((n+1)+1)! = (n+1)!*((n+1)+1) by NEWTON:15
    .= (n+2)*(n!*(n+1)) by NEWTON:15
    .= 2*((n+1)*((n+1)+1)/2)*(n!)
    .= 2*Triangle(n+1)*(n!) by NUMPOLY1:19; then
    Parity ((n+2)!) = (Parity (2*Triangle(n+1)))*(Parity (n!)) by NEWTON05:25
    .= ((Parity 2)*(Parity (Triangle (n+1))))*(Parity (n!)) by NEWTON05:25;
    hence thesis;
  end;
