 reserve n for Nat;
 reserve F for Field,
         p for irreducible Element of the carrier of Polynom-Ring F,
         f for Element of the carrier of Polynom-Ring F,
         a for Element of F;

theorem Th40:
  for F, p, a holds (emb p).a = Class(EqRel(Polynom-Ring F,{p}-Ideal),a|F)
   proof
     let F,p, a;
     reconsider pa = a|F as Element of Polynom-Ring F by POLYNOM3:def 10;
     dom(canHom F) = the carrier of F by FUNCT_2:def 1;
     hence (emb p).a = (canHom({p}-Ideal)).((canHom(F)).a) by FUNCT_1:13
           .= (canHom({p}-Ideal)).pa by RING_4:def 6
           .= Class(EqRel(Polynom-Ring F,{p}-Ideal),a|F) by RING_2:def 5;
   end;
