
theorem e1a:
for F1,F2 being Field,
    E1 being FieldExtension of F1, E2 being FieldExtension of F2
for K1 being E1-extending FieldExtension of F1,
    K2 being E2-extending FieldExtension of F2
for h1 being Function of F1,F2, h2 being Function of E1,E2,
    h3 being Function of K1,K2 
st h2 is h1-extending & h3 is h2-extending holds h3 is h1-extending
proof
let F1,F2 be Field,
    E1 be FieldExtension of F1, E2 be FieldExtension of F2;
let K1 be E1-extending FieldExtension of F1,
    K2 be E2-extending FieldExtension of F2;
let h1 be Function of F1,F2, h2 be Function of E1,E2,
    h3 be Function of K1,K2;
assume AS: h2 is h1-extending & h3 is h2-extending;
now let a be Element of F1;
  thus h3.a = h3.@(a,E1) by FIELD_7:def 4
           .= h2.@(a,E1) by AS
           .= h2.a by FIELD_7:def 4
           .= h1.a by AS;
  end;
hence thesis;
end;
