 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 Th29:
  for p be odd prime Nat, m be positive Nat, k,j be Nat
  for i be Nat st p+1 < i &
  i in dom(LBZ(Der1(INT.Ring),k,Product Del(ff_0(m,p),j),(tau(j))|^p)) holds
  (LBZ(Der1(INT.Ring),k,Product Del(ff_0(m,p),j),(tau(j))|^p))/.i
  = 0.Polynom-Ring INT.Ring
    proof
      let p be odd prime Nat, m be positive Nat, k,j be Nat;
      1 < p by INT_2:def 4; then
      1+1 <= p by INT_1:7; then
      2 - 2 <= p - 2 by XREAL_1:6; then
      p-2 in NAT by INT_1:3; then
reconsider p2 = p-2 as Nat;
reconsider n0 = j as Nat;
set D = Der1(INT.Ring);
set PR = Polynom-Ring INT.Ring;
set f = Product ff_0(m,p);
set xj = tau(j);
set yj = Product Del(ff_0(m,p),j);
reconsider u = (k choose p)*((D|^(k -' p)).yj) as Element of the carrier of PR;
A2:   1.PR = D.xj by Th27 .= (D|^1).xj by VECTSP11:19
      .= (D|^1).(xj|^1) by BINOM:8;
      len (LBZ(D,k,yj,xj|^p)) = k+1 by RINGDER1:def 4; then
A3:   dom (LBZ(D,k,yj,xj|^p)) = Seg (k+1) by FINSEQ_1:def 3;
      for i be Nat st p+1 < i & i in dom (LBZ(D,k,yj,xj|^p)) holds
      (LBZ(D,k,yj,xj|^p))/.i = 0.PR
      proof
        let i be Nat;
        assume
A4:     p+1 < i & i in dom (LBZ(D,k,yj,xj|^p));
        set u = (k choose (i-'1))*((D|^(k+1-'i)).yj);
A5:     1 <= i <= k+1 by A3,A4,FINSEQ_1:1;
        p+1 -1 < i - 1 by A4,XREAL_1:6; then
A7:     p < i-'1 by A5,XREAL_1:233;
        (LBZ(D,k,yj,xj|^p)).i = u*((D|^(i-'1)).(xj|^p)) by A4,RINGDER1:def 4
        .= u*(0.PR) by A7,A2,E_TRANS1:21 .= 0.PR;
        hence thesis by A4,PARTFUN1:def 6;
      end;
      hence thesis;
    end;
