 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 Th28:
  for p be odd prime Nat, m be positive Nat,
      j,k be Nat st j in Seg m & p <= k holds
  for i be Nat st i in Seg p 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, j,k be Nat;
      assume
A1:   j in Seg m & p <= k;
      set D = Der1(INT.Ring);
      set PR = Polynom-Ring INT.Ring;
A2:   p-'1 = p-1 by XREAL_1:233,NAT_1:14;
      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 p1 = p-'1 as non zero Nat by A2;
      (for i be Nat st i in Seg p holds
      tau(j) divides
      (LBZ(Der1(INT.Ring),k,Product Del(ff_0(m,p),j),(tau(j))|^p))/.i )
      proof
        let i be 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 n0 = j as Nat;
        assume
A4:     i in Seg p; then
A5:     1 <= i <= p by FINSEQ_1:1;
set f = Product ff_0(m,p);
set xj = tau(j);
set yj = Product Del(ff_0(m,p),j);
A6:     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
A7:     dom (LBZ(D,k,yj,xj|^p)) = Seg (k+1) by FINSEQ_1:def 3;
        p < k+1 by A1,NAT_1:13; then
A8:     Seg p c= Seg(k+1) by FINSEQ_1:5;
reconsider i1 = i-1 as Nat by A5;
A9:     i-'1 = i- 1 by A5,XREAL_1:233;
A10:    xj divides (D|^(i -' 1)).(xj|^p)
        proof
          per cases;
            suppose
              i-'1 = 0; then
              (D|^(i -' 1)).(xj|^p) = (xj|^p) by Lm11;
              hence thesis by Lm8;
            end;
            suppose
A13:          i-'1 <> 0;
A14:          i-'1 < i by A9,XREAL_1:44; then
A15:          1 <= i-'1 < p by A13,NAT_1:14,A5,XXREAL_0:2; then
A16:          p - (i -' 1) > (i -' 1) - (i -' 1) by XREAL_1:6;
A17:          p-(i-'1) = p - (i - 1) by A5,XREAL_1:233 .= p - i +1; then
A18:          p-'(i -' 1)
              = p-i+1 by A14,A5,XXREAL_0:2,XREAL_1:233;
reconsider p9 = p-i+1 as non zero Nat by A17,A16,INT_1:3,ORDINAL1:def 12;
reconsider s = (eta(p,(i -' 1))) as Element of NAT;
A19:          xj divides s*(xj|^p9) by Lm8,E_TRANS1:7;
              (D|^(i -' 1)).(xj|^p)
              = s*(xj|^p9) by A18,A15,A6,E_TRANS1:19;
              hence thesis by A19;
            end;
          end;
reconsider u = (k choose (i-'1))*((D|^(k+1-'i)).yj)
as Element of the carrier of PR;
A20:      xj divides ((D|^(i -' 1)).(xj|^p))*u by A10,GCD_1:7;
          (LBZ(D,k,yj,xj|^p)).i
          = ((D|^(i -' 1)).(xj|^p))*u by A7,A4,A8,RINGDER1:def 4;
          hence thesis by A20,A7,A4,A8,PARTFUN1:def 6;
        end;
        hence thesis;
      end;
