let D be non empty set ; :: thesis: for d being Element of D

for F being BinOp of D st F is associative & F is having_a_unity & F is having_an_inverseOp holds

(F * ((id D),(the_inverseOp_wrt F))) . (d,d) = the_unity_wrt F

let d be Element of D; :: thesis: for F being BinOp of D st F is associative & F is having_a_unity & F is having_an_inverseOp holds

(F * ((id D),(the_inverseOp_wrt F))) . (d,d) = the_unity_wrt F

let F be BinOp of D; :: thesis: ( F is associative & F is having_a_unity & F is having_an_inverseOp implies (F * ((id D),(the_inverseOp_wrt F))) . (d,d) = the_unity_wrt F )

assume A1: ( F is associative & F is having_a_unity & F is having_an_inverseOp ) ; :: thesis: (F * ((id D),(the_inverseOp_wrt F))) . (d,d) = the_unity_wrt F

set u = the_inverseOp_wrt F;

thus (F * ((id D),(the_inverseOp_wrt F))) . (d,d) = F . (d,((the_inverseOp_wrt F) . d)) by Th81

.= the_unity_wrt F by A1, Th59 ; :: thesis: verum

for F being BinOp of D st F is associative & F is having_a_unity & F is having_an_inverseOp holds

(F * ((id D),(the_inverseOp_wrt F))) . (d,d) = the_unity_wrt F

let d be Element of D; :: thesis: for F being BinOp of D st F is associative & F is having_a_unity & F is having_an_inverseOp holds

(F * ((id D),(the_inverseOp_wrt F))) . (d,d) = the_unity_wrt F

let F be BinOp of D; :: thesis: ( F is associative & F is having_a_unity & F is having_an_inverseOp implies (F * ((id D),(the_inverseOp_wrt F))) . (d,d) = the_unity_wrt F )

assume A1: ( F is associative & F is having_a_unity & F is having_an_inverseOp ) ; :: thesis: (F * ((id D),(the_inverseOp_wrt F))) . (d,d) = the_unity_wrt F

set u = the_inverseOp_wrt F;

thus (F * ((id D),(the_inverseOp_wrt F))) . (d,d) = F . (d,((the_inverseOp_wrt F) . d)) by Th81

.= the_unity_wrt F by A1, Th59 ; :: thesis: verum