 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 Th26:
  for p be odd prime Nat, m be positive Nat,
      k,j be Nat st k < p & j in Seg m
  holds eval(~(((Der1(INT.Ring))|^k).f_0(m,p)),In(j,INT.Ring)) = 0.INT.Ring
    proof
      let p be odd prime Nat, m be positive Nat;
      let k,j be Nat;
      assume
A1:   k < p & j in Seg m; then
      k < p-1 + 1; then
A3:   0 <= k <= p -1 by NAT_1:13;
      set D = Der1(INT.Ring);
      set tj = tau(j);
      set yj = Product Del(ff_0(m,p),j);
A4:   1 < p by INT_2:def 4;
      1+1 < p +1 by XREAL_1:6,INT_2:def 4; then
      2 <= p by NAT_1:13; then
      p -'2 = p - 2 & p -' 1 = p -1 by A4,XREAL_1:233; then
      reconsider p1 = p-'1 as non zero Element of NAT;
      set f = Product Del(ff_0(m,p),j);
A6:   len (LBZ(D,k,yj,tj|^p)) = k+1 by RINGDER1:def 4; then
A7:   dom (LBZ(D,k,yj,tj|^p)) = Seg (k+1) by FINSEQ_1:def 3;
      reconsider lbz = (LBZ(D,k,yj,tj|^p))
      as non empty FinSequence of the carrier of Polynom-Ring INT.Ring by A6;
A8:   for i be Nat st i in Seg (k+1) holds
      tj divides (LBZ(D,k,yj,tj|^p))/.i by A3,Th18;
      Sum lbz = (D|^k).(yj * (tj|^p)) by RINGDER1:25
      .= (D|^k).f_0(m,p) by A1,Lm9; then
      ex u be Element of Polynom-Ring INT.Ring st
      (D|^k).f_0(m,p) = tau(j)*u by GCD_1:def 1,A8,A7,E_TRANS1:4;
      hence thesis by Th25;
   end;
