reserve R for Ring, S for R-monomorphic Ring,
        K for Field, F for K-monomorphic Field,
        T for K-monomorphic comRing;

theorem
for K,F being Field st K,F are_disjoint
holds F is K-monomorphic iff
      ex E being Field st E,F are_isomorphic & K is Subfield of E
proof
let K,F be Field;
assume AS: K,F are_disjoint;
now assume ex E being Field st E,F are_isomorphic & K is Subfield of E;
  then consider E being Field such that
  A: E,F are_isomorphic & K is Subfield of E;
  K is Subring of E by A,RING_3:43; then
  consider m being Function of K,E such that
  B: m is RingHomomorphism monomorphism by RING_3:def 3,RING_3:71;
  consider i being Function of E,F such that
  C: i is RingIsomorphism by A,QUOFIELD:def 23;
  set f = i * m;
  f is linear by B,C,RINGCAT1:1;
  hence F is K-monomorphic by B,C,RING_3:def 3;
  end;
hence thesis by AS,Th16;
end;
