 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 Th23:
  for p be odd prime Nat, m be positive Nat holds
  eval(~(((Der1(INT.Ring))|^(p-'1)).f_0(m,p)),0.INT.Ring)
    = ((p-'1)!)*(In((((-1)|^m)*(m!))|^p,INT.Ring))
    proof
      let p be odd prime Nat, m be positive Nat;
      set D = Der1(INT.Ring);
      set ProdX = Product (x.(m,p));
      set t0 = tau(0);
      set F0 = f_0(m,p);
A1:   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
      p -'2 = p - 2 & p -' 1 = p -1 by A1,XREAL_1:233; then
      reconsider p1 = p-'1 as non zero Element of NAT;
      reconsider f = Product (x.(m,p))
        as Element of the carrier of Polynom-Ring INT.Ring;
A3:   (eta(p1,p1)) = (p1!)/(0!) by XREAL_1:232 .= p1! by NEWTON:12;
A4:   (~((t0|^p1)*f).p1) = (~(t0|^(p-'1))*'f).(0+p1) by POLYNOM3:def 10
      .= (~f).0 by Lm6
      .= In((((-1)|^m)*(m!))|^p,INT.Ring) by Th20;
      ((D|^p1).F0).0 = (eta(0+p1,p1))*(F0.(0 +p1)) by E_TRANS1:22
      .= ((p-'1)!)*(In((((-1)|^m)*(m!))|^p,INT.Ring))
        by A3,A4,GROUP_4:6;
      hence thesis by POLYNOM5:31;
    end;
