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

theorem Th7:
  for f being Monomorphism of K,T st K,T are_disjoint
  holds embField f is right_complementable
  proof
    let f be Monomorphism of K,T;
    assume
AS: K,T are_disjoint;
    now let a be Element of embField f;
    reconsider x = a as Element of carr f by defemb;
    per cases;
    suppose x in [#]K; then
      reconsider a1 = a as Element of K;
      a1 is right_complementable; then
      consider b1 being Element of K such that
B1:   a1 + b1 = 0.K;
      reconsider y = b1 as Element of carr f by XBOOLE_0:def 3;
      reconsider b = y as Element of embField f by defemb;
      a + b = a1 + b1 by Lm7 .= 0.(embField f) by B1,defemb;
      hence a is right_complementable;
      end;
    suppose
A2:   not x in [#]K; then
      reconsider a1 = a as Element of T by Lm1;
      a1 is right_complementable; then
      consider b1 being Element of T such that
B2:   a1 + b1 = 0.T;
      dom f = [#]K &
      (the addF of T).(a1,b1) = f.(0.K) by B2,RING_2:6,FUNCT_2:def 1; then
D0:   (the addF of T).(a1,b1) in rng f by FUNCT_1:3; then
D1:   not (the addF of T).(a1,b1) in [#]T \ rng f by XBOOLE_0:def 5;
      per cases;
      suppose b1 in rng f; then
        consider b1r being object such that
C1:     b1r in dom f & f.b1r = b1 by FUNCT_1:def 3;
        reconsider b1r as Element of K by C1;
        [#]embField f = carr f by defemb; then
        reconsider bx = b1r as Element of embField f by XBOOLE_0:def 3;
        reconsider y = bx as Element of carr f by defemb;
C2:     [#]embField f = carr f by defemb; then
        (the addF of T).(a1,f.bx) in [#]K by Lm2,A2,D1,C1;
        hence a is right_complementable by Lm2,A2,D1,C1,C2;
        end;
      suppose not b1 in rng f;
        then b1 in [#]T \ rng f by XBOOLE_0:def 5; then
        reconsider y = b1 as Element of carr f by XBOOLE_0:def 3;
        reconsider b = y as Element of embField f by defemb;
E1:     not b in [#]K by AS,XBOOLE_0:def 4;
Y1:     dom f = [#]K & f.(0.K) = 0.T by RING_2:6,FUNCT_2:def 1;
        a + b = f".(0.T) by A2,B2,D0,E1,Lm9
             .= 0.K by Y1,FUNCT_1:32 .= 0.(embField f) by defemb;
        hence a is right_complementable;
        end;
      end;
    end;
    hence thesis;
  end;
