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 (the_Field_of_Quotients I); :: according to ALGSTR_0:def 38 :: 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 (the_Field_of_Quotients I) such that
A2: x * y = 1. (the_Field_of_Quotients I) by Th45;
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 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 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 3, RLVECT_1:def 2, STRUCT_0:def 8, VECTSP_1:def 7; :: thesis: verum