
theorem lift5a:
for F being Field
for E being FieldExtension of F holds id F is Monomorphism of F,E
proof
let F be Field, E be FieldExtension of F;
I: F is Subfield of E by FIELD_4:7; then
H: the carrier of F c= the carrier of E by EC_PF_1:def 1;
rng(id F) c= the carrier of E by I,EC_PF_1:def 1; then
reconsider f = id F as Function of F,E by FUNCT_2:6;
f is additive multiplicative unity-preserving monomorphism
  proof
  now let a,b be Element of F;
    reconsider a1 = a, b1 = b as Element of E by H;
    K: [a,b] in [:the carrier of F,the carrier of F:] by ZFMISC_1:def 2;
    thus f.(a+b) = ((the addF of E)||(the carrier of F)).(a,b)
                   by I,EC_PF_1:def 1
                .= f.a + f.b by K,FUNCT_1:49;
    end;
  hence K1: f is additive;
  now let a,b be Element of F;
    reconsider a1 = a, b1 = b as Element of E by H;
    K: [a,b] in [:the carrier of F,the carrier of F:] by ZFMISC_1:def 2;
    thus f.(a*b) = ((the multF of E)||(the carrier of F)).(a,b)
                   by I,EC_PF_1:def 1
                .= f.a * f.b by K,FUNCT_1:49;
    end;
  hence K2: f is multiplicative;
  thus f is unity-preserving by I,EC_PF_1:def 1;
  hence f is monomorphism by K1,K2;
  end;
hence thesis;
end;
