theorem Th56:
  for F being Field, E being F-homomorphic Field, K being Subfield of F holds
  E is K-homomorphic
proof
let F be Field,
    E be F-homomorphic Field,
    K be Subfield of F;
consider f being Function of F,E such that
A1: f is RingHomomorphism by RING_2:def 4;
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;
take g;
thus thesis by A1,Th53,Th54,Th55;
end;
