
theorem
for F being Field,
    p being irreducible Element of the carrier of Polynom-Ring F
for E being polynomial_disjoint Field
st E = embField(emb p) holds F is polynomial_disjoint
proof
let F be Field, p be irreducible Element of the carrier of Polynom-Ring F;
let E be polynomial_disjoint Field;
assume AS1: E = embField(emb p);
assume AS2: F is non polynomial_disjoint;
H: F is Subfield of E by AS1,FIELD_2:17; then
reconsider K = E as FieldExtension of F by FIELD_4:7;
A: the carrier of F c= the carrier of K by H, EC_PF_1:def 1;
set o = the Element of (the carrier of F) /\ (the carrier of Polynom-Ring F);
[#]F /\ [#]Polynom-Ring F <> {} by AS2,FIELD_3:def 3; then
B: o in (the carrier of F) & o in (the carrier of Polynom-Ring F)
   by XBOOLE_0:def 4;
the carrier of Polynom-Ring F c= the carrier of Polynom-Ring K by FIELD_4:10;
then o in [#]K /\ [#]Polynom-Ring K by A,B,XBOOLE_0:def 4;
hence thesis by FIELD_3:def 3;
end;
