theorem
  for R being Ring, E being R-homomorphic Ring, K being Subring of R holds
  E is K-homomorphic
proof
let R be Ring,
    E be R-homomorphic Ring,
    K be Subring of R;
consider f being Function of R,E such that
A1: f is RingHomomorphism by RING_2:def 4;
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;
take g;
thus thesis by A1,Th48,Th49,Th50;
end;
