reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;
reserve R for Ring, F for Field;

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;
