
theorem lemNor1cv:
for F being Field,
    E being FieldExtension of F
for p being Polynomial of F for q being Polynomial of E
for a being Element of F, b being Element of E
st p = q & a = b holds a * p = b * q
proof
let F be Field, E be FieldExtension of F;
let p be Polynomial of F; let q be Polynomial of E;
let a be Element of F, b be Element of E;
assume AS: p = q & a = b; then
A: (a|F) = (b|E) by FIELD_6:23;
thus a * p = (a|F) *' p by FIELD_8:2
          .= (b|E) *' q by A,AS,FIELD_4:17
          .= b * q by FIELD_8:2;
end;
