theorem Th57:
for F being Field,
    E being F-homomorphic Field,
    K being Subfield of F,
    EK being K-homomorphic Field
for f being Homomorphism of F,E st E = EK holds f|K is Homomorphism of K,EK
proof
let F be Field, E being F-homomorphic Field,K be Subfield of F;
let EK be K-homomorphic Field;
let f be Homomorphism of F,E;
the carrier of K c= the carrier of F by EC_PF_1:def 1; then
reconsider g = f|(the carrier of K) as Function of K,E by FUNCT_2:32;
g = f|K;
hence thesis by Th53,Th54,Th55;
end;
