 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 Th32:
  for p be odd prime Nat, m be positive Nat,
      k,j be Nat st j in Seg m & p <= k
  holds eval(~(((Der1(INT.Ring))|^k).f_0(m,p)),In(j,INT.Ring))
    in {In(p!,INT.Ring)}-Ideal
    proof
      let p be odd prime Nat, m be positive Nat;
      let k,j be Nat;
      assume that
A1:   j in Seg m and
AA:   p <= k;
      consider u,v be Element of the carrier of Polynom-Ring INT.Ring such that
A2:   ((Der1(INT.Ring))|^k).f_0(m,p) = tau(j)*u + (p!)*v by AA,Th30,A1;
      reconsider z = p! as Element of INT.Ring by INT_1:def 2;
      set tu = tau(j)*u;
      set pv = (p!)*v;
      set r = In((p!),INT.Ring);
      set j1 = In(j,INT.Ring);
       eval(~(((Der1(INT.Ring))|^k).f_0(m,p)),j1)
       = eval(~(tu) + ~(pv),j1) by A2,POLYNOM3:def 10
      .= eval(~(tu),j1)+eval(~(pv),j1) by POLYNOM4:19
      .= 0.INT.Ring + eval(~(pv),j1) by Th25
      .= eval(r*(~v),j1) by Th6;
       hence thesis by Th31;
     end;
