let G be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; :: thesis: 0. G is_a_unity_wrt the addF of G

now :: thesis: for x being Element of G holds
( the addF of G . ((0. G),x) = x & the addF of G . (x,(0. G)) = x )

let x be Element of G; :: thesis: ( the addF of G . ((0. G),x) = x & the addF of G . (x,(0. G)) = x )
thus the addF of G . ((0. G),x) = (0. G) + x
.= x by RLVECT_1:4 ; :: thesis: the addF of G . (x,(0. G)) = x
thus the addF of G . (x,(0. G)) = x + (0. G)
.= x by RLVECT_1:4 ; :: thesis: verum

end;

hence 0. G is_a_unity_wrt the addF of G by BINOP_1:3; :: thesis: verum