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 [:] ((id D),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 [:] ((id D),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 [:] ((id D),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 [:] ((id D),d)) . () = the_unity_wrt F
set e = the_unity_wrt F;
set i = the_inverseOp_wrt F;
G . ((),d) = G . ((F . ((),())),d) 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 . (((id D) . ()),d) = the_unity_wrt F ;
hence (G [:] ((id D),d)) . () = the_unity_wrt F by FUNCOP_1:48; :: thesis: verum