 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 Th18:
  for p be odd prime Nat, m be positive Nat, k be Nat st 0<=k<=p-1
  for i,j be Nat st i in Seg (k+1) holds
  tau(j) divides
  (LBZ(Der1(INT.Ring),k,Product Del(ff_0(m,p),j),(tau(j))|^p))/.i
    proof
      let p be odd prime Nat, m be positive Nat, k be Nat;
      assume
A1:   0<=k<=p-1;
      set D = Der1(INT.Ring);
      set PR = Polynom-Ring INT.Ring;
      for i,j be Nat st i in Seg (k+1) holds
      tau(j) divides (LBZ(D,k,Product Del(ff_0(m,p),j),(tau(j))|^p))/.i
      proof
        let i,j be Nat;
        set f = Product ff_0(m,p);
        set xj = tau(j);
        set yj = Product Del(ff_0(m,p),j);
        assume
A2:     i in Seg(k+1);
A3:     1.PR = D.xj by Th13 .= (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
A4:     i in dom (LBZ(D,k,yj,xj|^p)) by A2,FINSEQ_1:def 3;
        set p1 = p - 1;
A5:     k+1 <= p-1 +1 by A1,XREAL_1:6;
A7:     p1+0 <= p1 +1 by XREAL_1:7;
A6:     1 <= i <= k+1 by A2,FINSEQ_1:1; then
A8:     1 <= i <= p by A5,XXREAL_0:2; then
A9:     i - 1 <= p -1 by XREAL_1:6;
        set i1 = i -' 1;
        set s = k choose (i-'1), t = eta(p,i1);
A10:    0 <= i1 <= p-1 by A6,XREAL_1:233,A9;
        per cases;
          suppose
            i1 <> 0; then
            1 <= i1 <= p by NAT_1:14,A7,A10,XXREAL_0:2; then
A13:        (D|^(i-'1)).(xj|^p)
           = (eta(p,i1))*(xj|^(p-'i1)) by A3,E_TRANS1:19;
           set tDx = t*(xj|^(p-'i1));
           set Dy = (D|^(k+1-'i)).yj;
A14:        (LBZ(D,k,yj,xj|^p)).i
            = (t*(xj|^(p-'i1)))*(s*((D|^(k+1-'i)).yj))
            by A13,A4,RINGDER1:def 4
            .= (xj|^(p-'i1))*(t*(s*((D|^(k+1-'i)).yj))) by RINGDER1:2;
set u = (xj|^(p-'i1)), v = (t*(s*((D|^(k+1-'i)).yj)));
reconsider w = (LBZ(D,k,yj,xj|^p)).i as Element of the carrier of PR
by A4, FINSEQ_2:11;
A15:        i-i <= p -i by A8,XREAL_1:6;
            p-'(i-'1) = p-(i-'1) by XREAL_1:233,A7,A10,XXREAL_0:2
            .= p-(i-1) by A6,XREAL_1:233 .= p - i +1; then
A17:        xj divides u by A15,Lm8;
            u divides w by A14,GCD_1:def 1; then
            xj divides w by A17, GCD_1:2;
            hence thesis by A4,PARTFUN1:def 6;
          end;
          suppose
            i1 = 0; then
            (D|^(i-'1)).(xj|^p)
            = (id PR).(xj|^p) by VECTSP11:18 .= xj|^p; then
A20:        (LBZ(D,k,yj,xj|^p)).i
            = (xj|^p)*(s*((D|^(k+1-'i)).yj)) by A4,RINGDER1:def 4;
            xj divides (xj|^p)*(s*((D|^(k+1-'i)).yj)) by GCD_1:7,Lm8;
            hence thesis by A20,A4,PARTFUN1:def 6;
          end;
        end;
        hence thesis;
      end;
