 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 Th30:
  for p be odd prime Nat, m be positive Nat,
  k,j be Nat st j in Seg m & p <= k
  holds ex u,v be Element of the carrier of Polynom-Ring INT.Ring st
  ((Der1(INT.Ring))|^k).f_0(m,p) = tau(j)*u + (p!)*v
    proof
      let p be odd prime Nat, m be positive Nat, k,j be Nat;
      set D = Der1(INT.Ring);
      set PR = Polynom-Ring INT.Ring;
      set tj = tau(j);
      set gj = Product Del(ff_0(m,p),j);
      assume
A1:   j in Seg m & p <= k;
      set p1 = p+1;
A2:   1 <= p <= k by A1,INT_2:def 4;
      1.PR = D.tj by Th27 .= (D|^1).tj by VECTSP11:19
      .= (D|^1).(tj|^1) by BINOM:8; then
A3:   (D|^((p+1) -' 1)).(tj|^p)
      = (eta(p,p))*(tj|^(p-'p)) by A2,E_TRANS1:19
      .= (eta(p,p))*(tj|^0) by XREAL_1:232
      .= (eta(p,p))* 1_PR by BINOM:8
      .= (p!) * 1.PR by Lm5;
A4:   len (LBZ(D,k,gj,tj|^p)) = k+1 by RINGDER1:def 4; then
A5:   dom (LBZ(D,k,gj,tj|^p)) = Seg (k+1) by FINSEQ_1:def 3;
A6:   0+ 1 <= p + 1 by XREAL_1:6;
A7:   k+1-'(p+1) = k+1 -(p+1) by A1,XREAL_1:6,XREAL_1:233
      .= k - p .= k -' p by A1,XREAL_1:233;
      1 <= p+1 <= k+1 by A1,XREAL_1:6,A6; then
A9:   p+1 in dom (LBZ(D,k,gj,tj|^p)) by A5;
reconsider lbz = (LBZ(D,k,gj,tj|^p))
      as non empty FinSequence of the carrier of Polynom-Ring INT.Ring by A4;
reconsider p9 = p! as Element of NAT by ORDINAL1:def 12;
A10:  (LBZ(D,k,gj,tj|^p)).(p+1)
      = ((k choose p)*((D|^(k-'p)).gj))*(p9 * 1.PR)
       by A7,A3,A9,RINGDER1:def 4
      .= ((k choose p)*((D|^(k-'p)).gj)* 1.PR)*p9 by BINOM:20
      .= (p9)*((k choose p)*((D|^(k-'p)).gj)) by BINOM:18;
      p <= len lbz by A1,A4,XREAL_1:39; then
A12:  len (lbz|p) = p by FINSEQ_1:59; then
A13:  dom (lbz|p) = Seg p by FINSEQ_1:def 3;
      set lbz1 = lbz|p;
A14:  p <= k+1 by A1,XREAL_1:39;
      for i be Nat st i in Seg p holds tau(j) divides lbz1/.i
      proof
        let i be Nat;
        assume
A16:    i in Seg p; then
        1 <= i <= p by FINSEQ_1:1; then
        1 <= i <= k+1 by A14,XXREAL_0:2; then
A18:    i in dom lbz by A5;
        i in dom lbz1 by A12,FINSEQ_1:def 3,A16; then
        lbz1/.i = lbz1.i by PARTFUN1:def 6
        .= lbz.i by A16,FUNCT_1:49
        .= lbz/.i by A18,PARTFUN1:def 6;
        hence thesis by A1,A16,Th28;
      end; then
      consider u be Element of PR such that
A20:  Sum lbz1 = tau(j)*u by GCD_1:def 1,A13,E_TRANS1:4;
A21:  len (lbz/^(p+1)) = (len lbz) -' (p+1) by RFINSEQ:29
      .= (k+1) -' (p+1) by RINGDER1:def 4;
A22:  dom (lbz/^(p+1)) = Seg ((k+1) -' (p+1)) by A21,FINSEQ_1:def 3;
      set kp1 = (k+1) -' (p+1);
A23:  for i1 be Nat st i1 in dom (lbz/^(p+1)) holds (lbz/^(p+1))/.i1 = 0.PR
      proof
        let i1 be Nat;
        assume
A24:    i1 in dom (lbz/^(p+1)); then
A25:    1 <= i1 <= ((k+1) -'(p+1)) by A22,FINSEQ_1:1;
        set pi1 = (p+1)+i1;
        (k+1) -'(p+1) = k+1 -(p+1) by A1,XREAL_1:6,XREAL_1:233; then
        pi1 <= (k+1) -(p+1) + (p+1) by A25,XREAL_1:6; then
        1 <= pi1 <= k+1 by A25,XREAL_1:39; then
A29:    pi1 in dom lbz by A5;
A30:    p+1 < p+1 +i1 by A25,XREAL_1:39;
        (lbz/^(p+1))/.i1 = lbz/.pi1 by A24,FINSEQ_5:27;
        hence thesis by A30,A29,Th29;
      end;
      (lbz/^(p+1)) = kp1|->0.PR
      proof
        for k be Nat st k in dom (lbz/^(p+1)) holds
        (lbz/^(p+1)).k = (kp1|-> 0.PR).k
        proof
          let k be Nat;
          assume
A31:      k in dom (lbz/^(p+1)); then
A32:      k in Seg kp1 by A21,FINSEQ_1:def 3;
          (lbz/^(p+1)).k = (lbz/^(p+1))/.k by A31,PARTFUN1:def 6
          .= 0.PR by A31,A23;
          hence thesis by A32,FINSEQ_2:57;
        end;
        hence thesis by A21,FINSEQ_1:def 3;
      end; then
      Sum (lbz/^(p+1)) = 0.PR by MATRIX_3:11; then
A34:  Sum (<*lbz/.(p+1)*>^(lbz/^(p+1))) = lbz/.(p+1) + 0.PR by FVSUM_1:72
      .= lbz/.(p+1);
      1 <= p1 & p1 <= len lbz by A1,XREAL_1:6,A6,A4; then
      lbz = (lbz|(p1-'1))^<*lbz.p1*>^(lbz/^p1) by FINSEQ_5:84
      .= (lbz|p)^<*lbz/.p1*>^(lbz/^p1) by A9,PARTFUN1:def 6
      .= (lbz|p)^(<*lbz/.p1*>^(lbz/^p1)) by FINSEQ_1:32; then
A36:  Sum lbz = tau(j)*u + lbz/.p1 by A34,A20,RLVECT_1:41
      .= tau(j)*u + (p!)*((k choose p)*((D|^(k-'p)).gj))
       by A10,A9,PARTFUN1:def 6;
      Sum lbz = (D|^k).(gj * (tj|^p)) by RINGDER1:25
      .= (D|^k).f_0(m,p) by A1,Lm9;
      hence thesis by A36;
    end;
