
theorem mmo:
for F being Field,
    E being FieldExtension of F,
    K being E-extending FieldExtension of F
for p being Element of the carrier of Polynom-Ring F,
    q being Element of the carrier of Polynom-Ring E
st p = q holds Roots(K,p) = Roots(K,q)
proof
let F be Field, E be FieldExtension of F,
    K be E-extending FieldExtension of F;
let p be Element of the carrier of Polynom-Ring F,
    q be Element of the carrier of Polynom-Ring E;
H: Roots(K,p) = {a where a is Element of K : a is_a_root_of p,K} &
   Roots(K,q) = {a where a is Element of K : a is_a_root_of q,K}
   by FIELD_4:def 4;
assume AS: p = q;
Z: now let o be object;
   assume o in Roots(K,p); then
   consider a being Element of K such that
   A: o = a & a is_a_root_of p,K by H;
   Ext_eval(q,a) = Ext_eval(p,a) by AS,FIELD_8:6 .= 0.K by A,FIELD_4:def 2;
   then a is_a_root_of q,K by FIELD_4:def 2;
   hence o in Roots(K,q) by A,H;
   end;
now let o be object;
   assume o in Roots(K,q); then
   consider a being Element of K such that
   B: o = a & a is_a_root_of q,K by H;
   Ext_eval(p,a) = Ext_eval(q,a) by AS,FIELD_8:6 .= 0.K by B,FIELD_4:def 2;
   then a is_a_root_of p,K by FIELD_4:def 2;
   hence o in Roots(K,p) by B,H;
   end;
hence thesis by Z,TARSKI:2;
end;
