
theorem lemBas00:
for F being Field,
    E being FieldExtension of F
for a being Element of E
for b being Element of F st b = a^2 holds Ext_eval(X^2-b,a) = 0.E
proof
let F be Field, E be FieldExtension of F, a be Element of E;
let b be Element of F;
assume D: b = a^2;
A: X^2-b = <%-b,0.F,1.F%> by FIELD_9:def 7;
F is Subring of E by FIELD_4:def 1; then
C: 0.F = 0.E & 1.F = 1.E & -b = -(a^2) by D,C0SP1:def 3,FIELD_6:17;
X^2-b = <%-(a^2),0.E,1.E%> by A,C,FIELD_9:23 .= X^2-a^2 by FIELD_9:def 7;
hence Ext_eval(X^2-b,a) = eval(X^2-a^2,a) by FIELD_4:26
                      .= 0.E by FIELD_9:70;
end;
