 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 Th36:
  for p,m,g holds Sum delta_1(m,p,g) in INT.Ring
  proof
    let p,m,g;
    for i being Nat st i in dom delta_1(m,p,g) holds
    (delta_1(m,p,g)).i in INT
    proof
      let i be Nat;
      assume i in dom delta_1(m,p,g); then
A2:   (delta_1(m,p,g)).i = (g.i)*(('F'(f_0(m,p))).(In(i,F_Real))) by Def5;
reconsider r = i as Element of F_Real by XREAL_0:def 1;
reconsider i0 = i as Element of INT.Ring by INT_1:def 2;
      ('F'(f_0(m,p))).(In(i,F_Real))
        = Sum eval('G'(f_0(m,p)),i0) by E_TRANS1:30;
      hence thesis by A2,INT_1:def 2;
    end;
    hence thesis by ZMATRLIN:42;
  end;
