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

theorem Th10:
for f being Monomorphism of K,F st K,F are_disjoint
    holds embField f is almost_left_invertible
  proof
    let f be Monomorphism of K,F;
    assume AS: K,F are_disjoint;
    now let a be Element of embField f;
      assume a <> 0.(embField f); then
X:    a <> 0.K by defemb;
    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 left_invertible by X,ALGSTR_0:def 39; then
        consider b1 being Element of K such that
B:      b1 * a1 = 1.K;
      reconsider y = b1 as Element of carr f by XBOOLE_0:def 3;
      reconsider b = y as Element of embField f by defemb;
        b*a = a*b by GROUP_1:def 12 .= a1*b1 by Lm11
        .= b1*a1 by GROUP_1:def 12 .= 1.(embField f) by B,defemb;
        hence a is left_invertible;
      end;
      suppose A: not x in [#]K; then
X:      x in [#]F & not x in rng f by Lm1;
      reconsider a1 = a as Element of F by A,Lm1;
Z:      dom f = [#]K by FUNCT_2:def 1;
        f.(0.K) = 0.F by RING_2:6; then
        0.F in rng f by Z,FUNCT_1:def 3; then
        a1 is left_invertible by X,ALGSTR_0:def 39; then
        consider b1 being Element of F such that
B:      b1 * a1 = 1.F;
U:      b1 <> 0.F & a1 <> 0.F by B;
        (the multF of F).(a1,b1) = a1 * b1
        .= 1_F by B,GROUP_1:def 12 .= f.(1_K) by GROUP_1:def 13; then
D:      (the multF of F).(a1,b1) in rng f by Z,FUNCT_1:3; then
D1:     not(the multF of F).(a1,b1) in [#]F \ 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;
C4:         bx <> 0.K by U,C1,RING_2:6;
            [#]embField f = carr f by defemb;
            hence a is left_invertible by Lm4,A,D1,C1,C4;
          end;
          suppose not b1 in rng f; then
            b1 in ([#]F) \ (rng f) by XBOOLE_0:def 5; then
            b1 is Element of carr f by XBOOLE_0:def 3; then
      reconsider b = b1 as Element of embField f by defemb;
E:          not b in [#]K by AS,XBOOLE_0:def 4;
            b*a = a*b by GROUP_1:def 12 .= f".(a1*b1) by A,D,E,Lm13
           .= f".(1_F) by B,GROUP_1:def 12 .= 1.K by Th3
           .= 1.(embField f) by defemb;
            hence a is left_invertible;
          end;
        end;
      end;
      hence thesis;
    end;
