 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;
 reserve z0 for non zero Element of F_Real;

theorem Th38:
  for p,m,g holds
  Sum(delta_1(m,p,g)) in {In(p!,INT.Ring)}-Ideal
  proof
    let p,m,g;
    (for i being Nat st i in dom (^delta(m,p,g)) holds
    (^delta(m,p,g)).i in {In(p!,INT.Ring)}-Ideal)
    proof
      let i be Nat;
      assume
A1:   i in dom (^delta(m,p,g));
      len ^delta(m,p,g) = m by Def5; then
A2:   i in Seg m by A1,FINSEQ_1:def 3; then
      ('F'(f_0(m,p))).In(i,F_Real) in {In(p!,INT.Ring)}-Ideal by Th34;
        then
reconsider Ff = ('F'(f_0(m,p))).In(i,F_Real) as Element of INT.Ring;
      (g.i)*Ff in {In(p!,INT.Ring)}-Ideal by A2,Th34,IDEAL_1:def 2;
      hence thesis by A1,Def5;
    end; then
    Sum (^delta(m,p,g)) in {In(p!,INT.Ring)}-Ideal by E_TRANS1:3;
    hence thesis by LIOUVIL2:5,FIELD_4:2;
  end;
