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) = 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) = 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) = 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) = the_unity_wrt F

set e = the_unity_wrt F;

set i = the_inverseOp_wrt F;

G . ((the_unity_wrt F),d) = G . ((F . ((the_unity_wrt F),(the_unity_wrt F))),d) by A2, SETWISEO:15

.= F . ((G . ((the_unity_wrt F),d)),(G . ((the_unity_wrt F),d))) by A4, BINOP_1:11 ;

then the_unity_wrt F = F . ((F . ((G . ((the_unity_wrt F),d)),(G . ((the_unity_wrt F),d)))),((the_inverseOp_wrt F) . (G . ((the_unity_wrt F),d)))) by A1, A2, A3, Th59;

then the_unity_wrt F = F . ((G . ((the_unity_wrt F),d)),(F . ((G . ((the_unity_wrt F),d)),((the_inverseOp_wrt F) . (G . ((the_unity_wrt F),d)))))) by A1;

then the_unity_wrt F = F . ((G . ((the_unity_wrt F),d)),(the_unity_wrt F)) by A1, A2, A3, Th59;

then the_unity_wrt F = G . ((the_unity_wrt F),d) by A2, SETWISEO:15;

then G . (((id D) . (the_unity_wrt F)),d) = the_unity_wrt F ;

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

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) = 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) = 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) = 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) = the_unity_wrt F

set e = the_unity_wrt F;

set i = the_inverseOp_wrt F;

G . ((the_unity_wrt F),d) = G . ((F . ((the_unity_wrt F),(the_unity_wrt F))),d) by A2, SETWISEO:15

.= F . ((G . ((the_unity_wrt F),d)),(G . ((the_unity_wrt F),d))) by A4, BINOP_1:11 ;

then the_unity_wrt F = F . ((F . ((G . ((the_unity_wrt F),d)),(G . ((the_unity_wrt F),d)))),((the_inverseOp_wrt F) . (G . ((the_unity_wrt F),d)))) by A1, A2, A3, Th59;

then the_unity_wrt F = F . ((G . ((the_unity_wrt F),d)),(F . ((G . ((the_unity_wrt F),d)),((the_inverseOp_wrt F) . (G . ((the_unity_wrt F),d)))))) by A1;

then the_unity_wrt F = F . ((G . ((the_unity_wrt F),d)),(the_unity_wrt F)) by A1, A2, A3, Th59;

then the_unity_wrt F = G . ((the_unity_wrt F),d) by A2, SETWISEO:15;

then G . (((id D) . (the_unity_wrt F)),d) = the_unity_wrt F ;

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