
theorem deg2evale:
for F being Field,
    E being FieldExtension of F
for p being Polynomial of F st deg p < 2
for a being Element of E
ex y,z being F-membered Element of E st Ext_eval(p,a) = y + a * z
proof
let F be Field, E be FieldExtension of F; let p be Polynomial of F;
assume A: deg p < 2;
let a be Element of E;
reconsider q = p as Polynomial of E by FIELD_4:8;
B: p is Element of the carrier of Polynom-Ring F &
   q is Element of the carrier of Polynom-Ring E by POLYNOM3:def 10; then
C: deg q = deg p by FIELD_4:20;
consider yF,zF being Element of F such that D: p = <%yF,zF%> by A,deg2;
consider y,z being Element of E such that E: q = <%y,z%> by A,C,deg2;
y = q.0 by E,POLYNOM5:38 .= yF by D,POLYNOM5:38; then
G1: y is F-membered by FIELD_7:def 5;
z = q.1 by E,POLYNOM5:38 .= zF by D,POLYNOM5:38; then
G2: z is F-membered by FIELD_7:def 5;
Ext_eval(p,a) = eval(q,a) by B,FIELD_4:26
             .= y + z * a by E,POLYNOM5:44
             .= y + a * z by GROUP_1:def 12;
hence thesis by G1,G2;
end;
