
theorem FPD:
  for n,k be Nat holds (n!)|^k divides (n*k)!
  proof
    let n be Nat;
    defpred  P[Nat] means (n!)|^$1 divides (n*$1)!;
    A1: P[0]
    proof
      (n!)|^0 = 1 by NEWTON:4;
      hence thesis by INT_2:12;
    end;
    A2: for m be Nat st P[m] holds P[m + 1]
    proof
      let m be Nat such that
      B1: (n!)|^m divides (n*m)!;
      (n*(m +1))! = ((n*m)!*(n!))* ((n*m + n)!/((n*m)!*(n!)))
        by XCMPLX_1:87; then
      B2: ((n*m)!*(n!)) divides (n*(m +1))!;
      (n!)|^m*(n!) divides (n*m)!*(n!) by B1,INT_6:8; then
      (n!)|^m*(n!) divides (n*(m +1))! by B2,INT_2:9;
      hence thesis by NEWTON:6;
    end;
    for c be Nat holds P[c] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
