let I be non degenerated commutative domRing-like Ring; :: thesis: the_Field_of_Quotients I is non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative distributive doubleLoopStr
A1: the_Field_of_Quotients I is almost_left_invertible

proof

let x be Element of ; :: according to ALGSTR_0:def 39 :: thesis: ( x = 0. (the_Field_of_Quotients I) or x is left_invertible )
assume x <> 0. (the_Field_of_Quotients I) ; :: thesis: x is left_invertible
then consider y being Element of such that
A2: x * y = 1. (the_Field_of_Quotients I) by Th48;
take y ; :: according to ALGSTR_0:def 27 :: thesis: y * x = 1. (the_Field_of_Quotients I)
thus y * x = 1. (the_Field_of_Quotients I) by A2; :: thesis: verum

end;

A3: ( q0. I <> q1. I & 0. (the_Field_of_Quotients I) = q0. I ) by Th21;
A4: ( 1. (the_Field_of_Quotients I) = q1. I & ( for x, y, z being Element of holds
( x * (y + z) = (x * y) + (x * z) & (y + z) * x = (y * x) + (z * x) ) ) ) by Th28, Th29;
( ( for x, y, z being Element of holds (x * y) * z = x * (y * z) ) & ( for u, v being Element of holds u + v = v + u ) ) by Th23, Th25;
hence the_Field_of_Quotients I is non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative distributive doubleLoopStr by A1, A3, A4, GROUP_1:def 4, RLVECT_1:def 5, STRUCT_0:def 8, VECTSP_1:def 18; :: thesis: verum