
theorem lem20:
for F being Field,
    E being FieldExtension of F
for p being Polynomial of F
for a being Element of F,
    x,b being Element of E st b = a
holds Ext_eval(a*p,x) = b * Ext_eval(p,x)
proof
let F be Field, E be FieldExtension of F, p be Polynomial of F;
let a be Element of F, x,b being Element of E;
assume AS: b = a;
reconsider p1 = p as Element of the carrier of Polynom-Ring F
    by POLYNOM3:def 10;
the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
    by FIELD_4:10; then
reconsider q1 = p1 as Element of the carrier of Polynom-Ring E;
H1: a * p1 is Element of the carrier of Polynom-Ring F
    by POLYNOM3:def 10;
H2: b * q1 is Element of the carrier of Polynom-Ring E
    by POLYNOM3:def 10;
Ext_eval(a*p1,x) = eval(b*q1,x) by H1,H2,FIELD_4:26,AS,FIELD_13:13
                .= b * eval(q1,x) by POLYNOM5:30
                .= b * Ext_eval(p1,x) by FIELD_4:26;
hence thesis;
end;
