 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 Th21:
  for p be odd prime Nat, m be positive Nat, k be Nat st 0 <= k <= p-'2
  holds (((Der1(INT.Ring))|^k).f_0(m,p)).0 = 0.INT.Ring
    proof
      let p be odd prime Nat, m be positive Nat, k be Nat;
      set D = Der1(INT.Ring);
      set t0 = tau(0);
      assume
A1:   0 <= k <= p-'2;
      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
A3:   p -'2 = p - 2 & p -' 1 = p -1 by A2,XREAL_1:233; then
      reconsider p1 = p-'1 as non zero Element of NAT;
      p - 1 - 1 < p -1 by XREAL_1:44; then
A5:   k < p-'1 by A3,A1,XXREAL_0:2;
      set f = Product (x.(m,p));
A6:   (~((t0|^(p-'1))*f).k) = (~(t0|^(p-'1))*'(~f)).k by POLYNOM3:def 10
      .= ((~(t0|^(p-'1))*'(~f))|(Segm p1)).k by A5,NAT_1:44,FUNCT_1:49
      .= ((Segm p1) --> 0.INT.Ring).k by Lm6
      .= 0.INT.Ring by A5,NAT_1:44,FUNCOP_1:7;
      ((D|^k).F0).0 = (eta(0+k,k))*(F0.(0 +k)) by E_TRANS1:22
      .= (eta(0+k,k))*0.INT.Ring by A6,GROUP_4:6
      .= 0.INT.Ring;
      hence thesis;
   end;
