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

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

(the_inverseOp_wrt F) . ((the_inverseOp_wrt F) . d) = d

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

(the_inverseOp_wrt F) . ((the_inverseOp_wrt F) . d) = d

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

assume A1: ( F is having_a_unity & F is associative & F is having_an_inverseOp ) ; :: thesis: (the_inverseOp_wrt F) . ((the_inverseOp_wrt F) . d) = d

then F . (d,((the_inverseOp_wrt F) . d)) = the_unity_wrt F by Th59;

hence (the_inverseOp_wrt F) . ((the_inverseOp_wrt F) . d) = d by A1, Th60; :: thesis: verum

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

(the_inverseOp_wrt F) . ((the_inverseOp_wrt F) . d) = d

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

(the_inverseOp_wrt F) . ((the_inverseOp_wrt F) . d) = d

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

assume A1: ( F is having_a_unity & F is associative & F is having_an_inverseOp ) ; :: thesis: (the_inverseOp_wrt F) . ((the_inverseOp_wrt F) . d) = d

then F . (d,((the_inverseOp_wrt F) . d)) = the_unity_wrt F by Th59;

hence (the_inverseOp_wrt F) . ((the_inverseOp_wrt F) . d) = d by A1, Th60; :: thesis: verum