
theorem exteval2:
for F being Field
for a,b being Element of F
for E being FieldExtension of F
for x being Element of E holds Ext_eval(<%a,b%>,x) = @(a,E) + @(b,E) * x
proof
let F be Field, a,b be Element of F, E be FieldExtension of F,
    x be Element of E;
A: F is Subring of E by FIELD_4:def 1;
B: <%a,b%> is Element of the carrier of Polynom-Ring F &
   <%@(a,E),@(b,E)%> is Element of the carrier of Polynom-Ring E
   by POLYNOM3:def 10;
a = @(a,E) & b = @(b,E) by FIELD_7:def 4; then
<%@(a,E),@(b,E)%> = <%a,b%> by A,FIELD_6:20;
hence Ext_eval(<%a,b%>,x) = eval(<%@(a,E),@(b,E)%>,x) by B,FIELD_4:26
                         .= @(a,E) + @(b,E) * x by POLYNOM5:44;
end;
