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

theorem Th12:
  for f being Monomorphism of K,F st K,F are_disjoint
  holds emb_iso f is multiplicative
  proof
    let f be Monomorphism of K,F;
    assume
AS: K,F are_disjoint;
set g = emb_iso f, R = embField f;
    now let a,b be Element of R;
    reconsider x = a, y = b as Element of carr f by defemb;
    per cases;
    suppose
A1:   x in [#]K & y in [#]K; then
      reconsider a1 = a, b1 = b as Element of K;
B:    a * b = a1 * b1 by Lm11;
      a in K & b in K by A1; then
C:    g.a = f.a & g.b = f.b by defiso;
      a * b in K by B;
      hence g.(a*b) = f.(a*b) by defiso .= g.a * g.b by C,B,GROUP_6:def 6;
      end;
    suppose
A2:    x = 0.K or y = 0.K;
B:     a*b = (multemb f).(x,y) by defemb .= multemb(f,x,y) by defmult
          .= 0.K by A2,defmultf;
X:     0.K in K;
       per cases by A2;
        suppose x = 0.K; then
C:       g.a = f.(0.K) by X,defiso .= 0.F by RING_2:6;
         thus g.(a*b) = f.(0.K) by B,X,defiso .= g.a * g.b by C,RING_2:6;
        end;
       suppose y = 0.K; then
C:      g.b = f.(0.K) by X,defiso .= 0.F by RING_2:6;
        thus g.(a*b) = f.(0.K) by B,X,defiso .= g.a * g.b by C,RING_2:6;
       end;
      end;
    suppose A: x in [#]K & x<>0.K &not y in [#]K; then
      reconsider a1 = a as Element of K;
      reconsider b1 = b as Element of F by A,Lm1;
      reconsider fa = f.a1 as Element of F;
B:    a * b = fa * b1 by AS,A,Lm10;
      a in K & not b in K by A; then
C:    g.a = f.a & g.b = b by defiso;
      not a * b in K by AS,A,Lm10;
      hence g.(a*b) = g.a * g.b by B,C,defiso;
      end;
    suppose A: y in [#]K & y<>0.K &not x in [#]K; then
      reconsider b1 = b as Element of K;
      reconsider a1 = a as Element of F by A,Lm1;
      reconsider fb = f.b1 as Element of F;
B:    a * b = a1 * fb by AS,A,Lm10;
      b in K & not a in K by A; then
C:    g.a = a & g.b = f.b by defiso;
      not a * b in K by AS,A,Lm10;
      hence g.(a*b) = g.a * g.b by B,C,defiso;
    end;
    suppose A: not x in [#]K & not y in [#]K & (the multF of F).(x,y) in rng f;
    then
    reconsider a1 = a, b1 = b as Element of F by Lm1;
C:    a * b = (f").(a1 * b1) by A,Lm13;
      not a in K & not b in K by A; then
D:    g.a = a & g.b = b by defiso;
      a1 * b1 in dom(f") by A,FUNCT_1:33; then
      (f").(a1 * b1) in rng(f") by FUNCT_1:3; then
      (f").(a1 * b1) in K;
      hence g.(a*b)=f.((f").(a1*b1)) by C,defiso .= g.a*g.b by D,A,FUNCT_1:35;
    end;
    suppose A: not x in [#]K & not y in [#]K &
      not((the multF of F).(x,y) in rng f); then
      reconsider a1 = a, b1 = b as Element of F by Lm1;
C:    a * b = a1 * b1 by AS,A,Lm12;
      not a in K & not b in K by A; then
D:    g.a = a & g.b = b by defiso;
      not a1 * b1 in K by AS,XBOOLE_0:def 4;
      hence g.(a*b) = g.a * g.b by D,C,defiso;
    end;
  end;
hence thesis;
end;
