
theorem lift5:
for F1,F2 being Field, E being F1-algebraic FieldExtension of F1
st F1 == F2 holds E is F2-algebraic FieldExtension of F2
proof
let F1,F2 be Field, E be F1-algebraic FieldExtension of F1;
assume AS: F1 == F2; then
reconsider E3 = E as FieldExtension of F2 by lift9;
now let a be Element of E3;
    reconsider a1 = a as Element of E;
    consider p being non zero Polynomial of F1 such that
    B: Ext_eval(p,a1) = 0.E by FIELD_6:43;
    reconsider p1 = p as non zero Polynomial of F2 by AS,lift6;
    Ext_eval(p1,a1) = 0.E3 by B,AS,lift7;
    hence a is F2-algebraic by FIELD_6:43;
    end;
hence thesis by FIELD_7:def 11;
end;
