
theorem Th59:
  for F being non degenerated almost_left_invertible domRing-like
  commutative Ring holds F is_ringisomorph_to the_Field_of_Quotients(F)
proof
  let F be non degenerated almost_left_invertible domRing-like commutative
  Ring;
A1: 0.F <> 1.F;
A2: dom canHom(F) = the carrier of F by FUNCT_2:def 1;
A3: for x being object holds x in the carrier of the_Field_of_Quotients(F)
  implies x in rng canHom(F)
  proof
    let x be object;
    assume x in the carrier of the_Field_of_Quotients(F);
    then reconsider x as Element of Quot.F;
    consider u being Element of Q.F such that
A4: x = QClass.u by Def5;
    consider a,b being Element of F such that
A5: u = [a,b] and
A6: b <> 0.F by Def1;
    consider z being Element of F such that
A7: z * b = 1.F by A6,VECTSP_1:def 9;
    reconsider t = [a*z,1.F] as Element of Q.F by A1,Def1;
A8: for x being object holds x in QClass.t implies x in QClass.u
    proof
      let x be object;
      assume
A9:   x in QClass.t;
      then reconsider x as Element of Q.F;
      x`1 = x`1 * 1.F
        .= x`1 * t`2
        .= x`2 * t`1 by A9,Def4
        .= x`2 * (a * z);
      then x`1 * u`2 = (x`2 * (a * z)) * b by A5
        .= x`2 * ((a * z) * b) by GROUP_1:def 3
        .= x`2 * (a * 1.F) by A7,GROUP_1:def 3
        .= x`2 * a
        .= x`2 * u`1 by A5;
      hence thesis by Def4;
    end;
    for x being object holds x in QClass.u implies x in QClass.t
    proof
      let x be object;
      assume
A10:  x in QClass.u;
      then reconsider x as Element of Q.F;
      x`1 * t`2 = x`1 * (b * z) by A7
        .= (x`1 * b) * z by GROUP_1:def 3
        .= (x`1 * u`2) * z by A5
        .= (x`2 * u`1) * z by A10,Def4
        .= (x`2 * a) * z by A5
        .= (x`2 * (a * z)) by GROUP_1:def 3
        .= x`2 * t`1;
      hence thesis by Def4;
    end;
    then
A11: QClass.t = QClass.u by A8,TARSKI:2;
    (canHom(F)).(a*z) = QClass.(quotient(a*z,1.F)) by Def21
      .= x by A1,A4,A11,Def20;
    hence thesis by A2,FUNCT_1:def 3;
  end;
  for x being object holds x in rng canHom(F) implies x in the carrier of
  the_Field_of_Quotients(F);
  then rng canHom(F) = the carrier of the_Field_of_Quotients(F) by A3,TARSKI:2;
  then
A12: canHom(F) is onto;
A13: canHom(F) is embedding by Th57;
  then canHom(F) is RingHomomorphism;
  then canHom(F) is RingEpimorphism by A12;
  hence thesis by A13;
end;
