 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 Th34:
  for p be odd prime Nat, m be positive Nat, j be Nat st j in Seg m holds
    ('F'(f_0(m,p))).In(j,F_Real) in {In(p!,INT.Ring)}-Ideal
    proof
      let p be odd prime Nat, m be positive Nat;
      set Gf = 'G'(f_0(m,p));
      set D = Der1(INT.Ring);
      let j be Nat;
      assume
A1:   j in Seg m;
A2:   len f_0(m,p) = len ~f_0(m,p) .= m*p + p by Th12; then
A3:   len Gf = m*p + p by E_TRANS1:def 8; then
A4:   1 <= p <= len Gf by INT_2:def 4,XREAL_1:38;
      set p1 = p-'1;
A5:   len (Gf|p) = p by XREAL_1:38,A3,FINSEQ_1:59; then
A6:   dom (Gf|p) = Seg p by FINSEQ_1:def 3;
A7:   dom (Gf|p) = dom(p|-> 0.INT.Ring) by A5,FINSEQ_1:def 3;
A8:   eval(Gf|p,In(j,INT.Ring)) = p|-> 0.INT.Ring
      proof
        for k be Nat st k in dom (eval(Gf|p,In(j,INT.Ring))) holds
        (eval(Gf|p,In(j,INT.Ring))).k = (p|-> 0.INT.Ring).k
        proof
          let k be Nat;
          assume
A9:       k in dom eval((Gf|p),In(j,INT.Ring)); then
A10:      k in dom (Gf|p) by E_TRANS1:def 7;
A11:      k in Seg p by A6,A9,E_TRANS1:def 7; then
A12:      1 <= k <= p by FINSEQ_1:1; then
          1 <= k <= len Gf by XREAL_1:38,A3; then
A13:      k in dom Gf by FINSEQ_3:25;
          k-1 < k by XREAL_1:44; then
          k-1 < p by A12,XXREAL_0:2; then
A15:      k-'1 < p by A12,XREAL_1:233;
A16:      (Gf|p)/.k = (Gf|p).k by A10,PARTFUN1:def 6
          .= Gf.k by FUNCT_1:49,A11
          .= (D|^(k-'1)).f_0(m,p) by A13,E_TRANS1:def 8;
          (eval((Gf|p),In(j,INT.Ring))).k
          = eval(~((D|^(k-'1)).f_0(m,p)),In(j,INT.Ring))
          by A16,A10,E_TRANS1:def 7
          .= 0.INT.Ring by A1,A15,Th26;
          hence thesis;
        end;
        hence thesis by E_TRANS1:def 7,A7;
      end;
A17:  (Eval(~^(Sum(Gf|p)))).(In(j,F_Real))
      = Sum (p|-> 0.INT.Ring) by A8,E_TRANS1:30
      .= 0.INT.Ring by MATRIX_3:11;
      len (Gf/^p) = (len Gf) - p by A4,RFINSEQ:def 1
      .= m*p+p - p by A2,E_TRANS1:def 8 .= m*p; then
A18:  dom (Gf/^p) = Seg (m*p) by FINSEQ_1:def 3;
      for k being Nat st k in dom (eval(Gf/^p,In(j,INT.Ring))) holds
      (eval(Gf/^p,In(j,INT.Ring))).k in {In(p!,INT.Ring)}-Ideal
      proof
        let k be Nat;
        assume
A19:    k in dom (eval(Gf/^p,In(j,INT.Ring))); then
A20:    k in dom (Gf/^p) by E_TRANS1:def 7;
A21:    k in Seg (m*p) by A18,A19,E_TRANS1:def 7; then
A22:    1 <= k <= m*p by FINSEQ_1:1; then
A23:    k+p <= m*p + p by XREAL_1:6;
        1 <= k+p by A22,XREAL_1:38; then
        k+p in Seg len Gf by A3,A23; then
A24:    k+p in dom Gf by FINSEQ_1:def 3;
        set kp = k+p;
A25:    kp -'1 = k + p -1 by A22,XREAL_1:38,233;
        1 <= k by FINSEQ_1:1,A21; then
A27:    0 + p <= k -1 +p by XREAL_1:6;
A28:    (Gf/^p)/.k = (Gf/^p).k by A20,PARTFUN1:def 6
        .= Gf.(p+k) by A4,A20,RFINSEQ:def 1
        .= (D|^(kp-'1)).f_0(m,p) by A24,E_TRANS1:def 8;
        (eval(~((D|^(kp-'1)).f_0(m,p)),In(j,INT.Ring)))
        in {In(p!,INT.Ring)}-Ideal by A1,A25,A27,Th32;
        hence thesis by A28,E_TRANS1:def 7,A20;
      end; then
A30:  Sum (eval(Gf/^p,In(j,INT.Ring))) in {In(p!,INT.Ring)}-Ideal
        by E_TRANS1:3;
      ^(Sum Gf) = ^(Sum ((Gf|p)^(Gf/^p)))
      .= ^(Sum (Gf|p) + Sum(Gf/^p)) by RLVECT_1:41
      .= ^(Sum (Gf|p)) + ^(Sum(Gf/^p)) by E_TRANS1:27; then
      ~^(Sum Gf) = ~^(Sum (Gf|p)) + ~^Sum(Gf/^p) by POLYNOM3:def 10; then
      Eval(~^(Sum Gf))
      = Eval(~^(Sum (Gf|p))) + Eval(~^Sum(Gf/^p)) by POLYDIFF:55; then
      (Eval(~^(Sum Gf))).In(j,F_Real)
      = 0.INT.Ring + (Eval(~^Sum(Gf/^p))).In(j,F_Real) by A17,VALUED_1:1
      .= (Eval(~^Sum(Gf/^p))).In(j,F_Real);
      hence thesis by A30,E_TRANS1:30;
    end;
