 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;
 reserve g for non zero Polynomial of INT.Ring;
 reserve f for Element of the carrier of Polynom-Ring INT.Ring;

theorem Th24:
  for p be odd prime Nat, m be positive Nat,
      k be non zero Nat st p <= k holds
  eval(~(((Der1(INT.Ring))|^k).f_0(m,p)), 0.INT.Ring)
    = (k!)*(~(Product (x.(m,p))).(k -' (p-'1)))
    proof
      let p be odd prime Nat, m be positive Nat;
      let k be non zero Nat;
      set D = Der1(INT.Ring);
      set t0 = tau(0);
      assume
A1:   p <= k;
      set F0 = f_0(m,p);
A2:   1 < p by INT_2:def 4;
      1+1 < p +1 by INT_2:def 4,XREAL_1:6; then
      2 <= p by NAT_1:13; then
A5:   p -'2 = p - 2 & p -' 1 = p -1 by A2,XREAL_1:233;
      reconsider f = Product (x.(m,p))
        as Element of the carrier of Polynom-Ring INT.Ring;
      reconsider p1 = p-'1 as non zero Element of NAT by A5;
A6:   1 = p -(p -1) .= p-(p-'1) by XREAL_1:233,A2; then
      1 <= k -(p-'1) by A1,XREAL_1:6; then
      reconsider kp = k - (p-'1) as Element of NAT by INT_1:3;
A7:   p - 1 < p by XREAL_1:44;
A8:   (eta(k,k)) = (k!)/(0!) by XREAL_1:232 .= k! by NEWTON:12;
A9:   (~((t0|^(p-'1))*f).k) = (~(t0|^p1)*'f).(kp + p1) by POLYNOM3:def 10
      .= f.kp by Lm6;
      ((D|^k).F0).0 = (eta(0+k,k))*(F0.(0 +k)) by E_TRANS1:22
      .= (eta(k,k))*(f.kp) by A9,GROUP_4:6
      .= (k!)*(f.(k -' (p-'1))) by A8,XREAL_1:233,A7,A6,A1,XXREAL_0:2;
      hence thesis by POLYNOM5:31;
    end;
