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
for R being Ring,
    E being R-homomorphic Ring,
    K being Subring of R,
    EK being K-homomorphic Ring
for f being Homomorphism of R,E st E = EK holds f|K is Homomorphism of K,EK
proof
let R be Ring, E be R-homomorphic Ring, K be Subring of R;
let EK be K-homomorphic Ring;
let f be Homomorphism of R,E;
the carrier of K c= the carrier of R by C0SP1:def 3; then
reconsider g = f|(the carrier of K) as Function of K,E by FUNCT_2:32;
g = f|K;
hence thesis by Th48,Th49,Th50;
end;
