
theorem
  for I being domRing-like commutative Ring for F,F9 being
  add-associative right_zeroed right_complementable Abelian commutative
  associative well-unital distributive almost_left_invertible non degenerated
non empty doubleLoopStr for f being Function of I, F for f9 being Function of
I, F9 st I has_Field_of_Quotients_Pair F,f & I has_Field_of_Quotients_Pair F9,
  f9 holds F is_ringisomorph_to F9
proof
  let I be domRing-like commutative Ring;
  let F,F9 be add-associative right_zeroed right_complementable Abelian
  commutative associative well-unital distributive almost_left_invertible non
  degenerated non empty doubleLoopStr;
  let f be Function of I, F;
  let f9 be Function of I, F9;
  assume that
A1: I has_Field_of_Quotients_Pair F,f and
A2: I has_Field_of_Quotients_Pair F9,f9;
A3: (id F9) * f9 = f9 by FUNCT_2:17;
 f is RingMonomorphism by A1;
  then consider h2 being Function of F9, F such that
A4: h2 is RingHomomorphism & h2*f9 = f and
  for h9 being Function of F9, F st h9 is RingHomomorphism & h9*f9 = f
  holds h9 = h2 by A2;
  consider h3 being Function of F, F such that
  h3 is RingHomomorphism and
  h3*f = f and
A5: for h9 being Function of F, F st h9 is RingHomomorphism & h9*f = f
  holds h9 = h3 by A1;
A6: id F * f = f by FUNCT_2:17;
 f9 is RingMonomorphism by A2;
  then consider h1 being Function of F, F9 such that
A7: h1 is RingHomomorphism and
A8: h1*f = f9 and
  for h9 being Function of F, F9 st h9 is RingHomomorphism & h9*f = f9
  holds h9 = h1 by A1;
  (h2 * h1) * f = f & h2 * h1 is RingHomomorphism by A7,A8,A4,Th54,RELAT_1:36;
  then
A9: (h2 * h1) = h3 by A5
    .= id the carrier of F by A6,A5;
  consider h3 being Function of F9, F9 such that
  h3 is RingHomomorphism and
  h3*f9 = f9 and
A10: for h9 being Function of F9, F9 st h9 is RingHomomorphism & h9*f9 =
  f9 holds h9 = h3 by A2;
  (h1 * h2) * f9 = f9 & h1 * h2 is RingHomomorphism by A7,A8,A4,Th54,
RELAT_1:36;
  then h1 * h2 = h3 by A10
    .= id the carrier of F9 by A3,A10;
  then rng h1 = the carrier of F9 by FUNCT_2:18;
  then h1 is onto;
  then
A11: h1 is RingEpimorphism by A7;
  h1 is one-to-one by A9,FUNCT_2:31;
  then h1 is RingMonomorphism by A7;
  hence thesis by A11;
end;
