
theorem TH40:
for F being Field
for g being bijective Function of (nonConstantPolys F),card(nonConstantPolys F)
for I being maxIdeal of nonConstantPolys(g,F)-Ideal
for p being Element of the carrier of Polynom-Ring F
for n being Element of NAT holds
((PolyHom emb(F,I,g)).p).n =
    Class(EqRel(Polynom-Ring(card(nonConstantPolys F),F),I),
          (p.n)|(card(nonConstantPolys F),F))
proof
let F be Field;
let g be bijective Function of (nonConstantPolys F),card(nonConstantPolys F);
let I be maxIdeal of nonConstantPolys(g,F)-Ideal;
let p be Element of the carrier of Polynom-Ring F;
let m be Nat;
set n = card(nonConstantPolys F), R = Polynom-Ring(n,F);
 ((PolyHom emb(F,I,g)).p).m
       = emb(F,I,g).(p.m) by FIELD_1:def 2
      .= Class(EqRel(R,I),(p.m)|(n,F)) by TH39;
hence thesis;
end;
