
theorem Th46:
  for I being non degenerated domRing-like commutative Ring holds
the_Field_of_Quotients(I) is add-associative right_zeroed right_complementable
Abelian associative unital distributive almost_left_invertible non degenerated
  non empty doubleLoopStr
proof
  let I be non degenerated domRing-like commutative Ring;
A1: the_Field_of_Quotients(I) is almost_left_invertible
  proof
    let x be Element of the_Field_of_Quotients I;
    assume x <> 0.the_Field_of_Quotients I;
    then consider y being Element of the_Field_of_Quotients I such that
A2: x * y = 1.the_Field_of_Quotients I by Th45;
    take y;
    thus y * x = 1.the_Field_of_Quotients I by A2;
  end;
A3: q0.I <> q1.I & 0.the_Field_of_Quotients(I) = q0.I by Th19;
A4: 1.the_Field_of_Quotients(I) = q1.I & for x,y,z being Element
 of the_Field_of_Quotients(I) holds x * (y+z) = x*y + x*z & (y+z) * x =
  y*x + z* x by Th26,Th27;
  ( for x,y,z being Element of the_Field_of_Quotients(I)
  holds (x * y) * z = x * (y * z))& for u,v being Element of
  the_Field_of_Quotients(I) holds u + v = v + u by Th21,Th23;
  hence thesis by A1,A3,A4,GROUP_1:def 3,RLVECT_1:def 2,STRUCT_0:def 8
,VECTSP_1:def 7;
end;
