let D be non empty set ; :: thesis: for d being Element of D
for F, G being BinOp of D st F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
(G [;] (d,(id D))) . () = the_unity_wrt F

let d be Element of D; :: thesis: for F, G being BinOp of D st F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
(G [;] (d,(id D))) . () = the_unity_wrt F

let F, G be BinOp of D; :: thesis: ( F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F implies (G [;] (d,(id D))) . () = the_unity_wrt F )
assume that
A1: F is associative and
A2: F is having_a_unity and
A3: F is having_an_inverseOp and
A4: G is_distributive_wrt F ; :: thesis: (G [;] (d,(id D))) . () = the_unity_wrt F
set e = the_unity_wrt F;
set i = the_inverseOp_wrt F;
G . (d,()) = G . (d,(F . ((),()))) by
.= F . ((G . (d,())),(G . (d,()))) by ;
then the_unity_wrt F = F . ((F . ((G . (d,())),(G . (d,())))),( . (G . (d,())))) by A1, A2, A3, Th59;
then the_unity_wrt F = F . ((G . (d,())),(F . ((G . (d,())),( . (G . (d,())))))) by A1;
then the_unity_wrt F = F . ((G . (d,())),()) by A1, A2, A3, Th59;
then the_unity_wrt F = G . (d,()) by ;
then G . (d,((id D) . ())) = the_unity_wrt F ;
hence (G [;] (d,(id D))) . () = the_unity_wrt F by FUNCOP_1:53; :: thesis: verum