
theorem TH39:
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 a being Element of F holds
emb(F,I,g).a = Class(EqRel(Polynom-Ring(card(nonConstantPolys F),F),I),
                     a|(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 a be Element of F;
set n = card(nonConstantPolys F), R = Polynom-Ring(n,F);
reconsider pa = a|(n,F) as Element of Polynom-Ring(n,F) by POLYNOM1:def 11;
dom(canHom(n,F)) = the carrier of F by FUNCT_2:def 1; then
emb(F,I,g).a
    = (canHom I).(canHom(card(nonConstantPolys F),F).a) by FUNCT_1:13
   .= (canHom I).(pa) by defcanhom
   .= Class(EqRel(R,I),pa) by RING_2:def 5;
hence thesis;
end;
