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

theorem Th15:
  for f being Monomorphism of K,F, E being Field st E = embField f holds
    K is Subfield of E
    proof
      let f be Monomorphism of K,F, E be Field;
A2:   [#]embField f = carr f by defemb;
      assume
A1:   E = embField f; then
      [#]E = carr f by defemb; then
A3:   [#]K c= [#]E by XBOOLE_0:def 3;
A4:   dom the addF of E = [:[#]E,[#]E:] by FUNCT_2:def 1;
      set g1 = (the addF of K), g2 = (the addF of E)|[:[#]K,[#]K:];
A5:   dom g2 = dom(the addF of E) /\ [:[#]K,[#]K:] by RELAT_1:61
      .= [:[#]K,[#]K:] by A3,ZFMISC_1:96,XBOOLE_1:28,A4
      .= dom g1 by FUNCT_2:def 1;
      now let x be set;
        assume x in dom the addF of K; then
        consider x1,x2 being object such that
A6:     x1 in [#]K & x2 in [#]K & x = [x1,x2] by ZFMISC_1:def 2;
  reconsider a = x1, b = x2 as Element of K by A6;
A7:     [a,b] in [:[#]K,[#]K:] by ZFMISC_1:def 2;
  reconsider a1=x1,b1=x2 as Element of embField f by A6,A2,XBOOLE_0:def 3;
        ((the addF of E)||[#]K).(a,b)=a1+b1 by A1,A7,FUNCT_1:49 .= a+b by Lm7;
        hence (the addF of K).x = g2.x by A6;
      end; then
A8:   the addF of K = (the addF of E) || [#]K by A5;
      set g1 = (the multF of K), g2 = (the multF of E)|[:[#]K,[#]K:];
A9:   dom(the multF of E) = [:[#]E,[#]E:] by FUNCT_2:def 1;
A10:  dom g2 = dom(the multF of E) /\ [:[#]K,[#]K:] by RELAT_1:61
      .= [:[#]K,[#]K:] by A3,ZFMISC_1:96,XBOOLE_1:28,A9
      .= dom g1 by FUNCT_2:def 1;
      now let x be set;
        assume x in dom (the multF of K); then
        consider x1,x2 being object such that
A11:    x1 in [#]K & x2 in [#]K & x = [x1,x2] by ZFMISC_1:def 2;
  reconsider a = x1, b = x2 as Element of K by A11;
A12:    [a,b] in [:[#]K,[#]K:] by ZFMISC_1:def 2;
  reconsider a1=x1,b1=x2 as Element of embField f by A11,A2,XBOOLE_0:def 3;
        ((the multF of E)||[#]K).(a,b)=a1*b1 by A1,A12,FUNCT_1:49.=a*b by Lm11;
        hence (the multF of K).x = g2.x by A11;
      end; then
A13:  the multF of K = (the multF of E) || [#]K by A10;
      1.E = 1.K & 0.E = 0.K by defemb,A1;
      hence thesis by A3,A8,A13,EC_PF_1:def 1;
    end;
