
theorem Kr1:
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 non constant Element of the carrier of Polynom-Ring F
holds KrRoot(I,g.p) is_a_root_of (PolyHom emb(F,I,g)).p
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 non constant Element of the carrier of Polynom-Ring F;
set n = card(nonConstantPolys F),
    R = Polynom-Ring(card(nonConstantPolys F),F);
rng g c= card(nonConstantPolys F) &
dom g = nonConstantPolys F &
p in nonConstantPolys F by FUNCT_2:def 1,RELAT_1:def 19; then
A0: g.p in n by FUNCT_1:3;
reconsider q = Poly(g.p,p) as
              Element of the carrier of Polynom-Ring(n,F) by POLYNOM1:def 11;
A1: q in nonConstantPolys(g,F);
A3: nonConstantPolys(g,F) c= nonConstantPolys(g,F)-Ideal by IDEAL_1:def 14;
    nonConstantPolys(g,F)-Ideal c= I by defideal; then
A2: q - 0.(Polynom-Ring(n,F)) in I by A1,A3;
eval((PolyHom emb(F,I,g)).p,KrRoot(I,g.p))
     = Class(EqRel(Polynom-Ring(n,F),I), q) by A0,Kr2
    .= Class(EqRel(Polynom-Ring(n,F),I), 0.Polynom-Ring(n,F)) by A2,RING_1:6
    .= 0.KroneckerField(F,g,I) by RING_1:def 6;
hence thesis by POLYNOM5:def 7;
end;
