 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 Th25:
  for j be Nat, u be Element of the carrier of Polynom-Ring INT.Ring
  holds eval(~((tau(j))*u),In(j,INT.Ring)) = 0.INT.Ring
    proof
      let j be Nat, u be Element of the carrier of Polynom-Ring INT.Ring;
      Roots tau(j) = {In(j,INT.Ring)} by Lm4; then
      In(j,INT.Ring) in Roots tau(j) by TARSKI:def 1; then
      In(j,INT.Ring) is_a_root_of tau(j) by POLYNOM5:def 10; then
A1:   eval(tau(j),In(j,INT.Ring)) = 0.INT.Ring by POLYNOM5:def 7;
      eval(~((tau(j))*u),In(j,INT.Ring))
      = eval((~(tau(j)))*'(~u),In(j,INT.Ring)) by POLYNOM3:def 10
      .= 0.INT.Ring * eval((~u),In(j,INT.Ring)) by A1,POLYNOM4:24
      .= 0.INT.Ring;
      hence thesis;
    end;
