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) = 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) = 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) = 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) = the_unity_wrt F

set e = the_unity_wrt F;

set i = the_inverseOp_wrt F;

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

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

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

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

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

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

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

hence (G [;] (d,(id D))) . (the_unity_wrt F) = the_unity_wrt F by FUNCOP_1:53; :: 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 [;] (d,(id 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 [;] (d,(id 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 [;] (d,(id 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 [;] (d,(id D))) . (the_unity_wrt F) = the_unity_wrt F

set e = the_unity_wrt F;

set i = the_inverseOp_wrt F;

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

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

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

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

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

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

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

hence (G [;] (d,(id D))) . (the_unity_wrt F) = the_unity_wrt F by FUNCOP_1:53; :: thesis: verum