let D be non empty set ; :: thesis: for F being BinOp of D
for p being FinSequence of D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp holds
(the_inverseOp_wrt F) . (F "**" p) = F "**" ((the_inverseOp_wrt F) * p)
let F be BinOp of D; :: thesis: for p being FinSequence of D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp holds
(the_inverseOp_wrt F) . (F "**" p) = F "**" ((the_inverseOp_wrt F) * p)
let p be FinSequence of D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp implies (the_inverseOp_wrt F) . (F "**" p) = F "**" ((the_inverseOp_wrt F) * p) )
assume that
A1: ( F is commutative & F is associative ) and
A2: F is having_a_unity and
A3: F is having_an_inverseOp ; :: thesis: (the_inverseOp_wrt F) . (F "**" p) = F "**" ((the_inverseOp_wrt F) * p)
set e = the_unity_wrt F;
set u = the_inverseOp_wrt F;
the_inverseOp_wrt F is_distributive_wrt F by A1, A2, A3, FINSEQOP:63;
then A4: for d1, d2 being Element of D holds (the_inverseOp_wrt F) . (F . (d1,d2)) = F . (((the_inverseOp_wrt F) . d1),((the_inverseOp_wrt F) . d2)) ;
(the_inverseOp_wrt F) . (the_unity_wrt F) = the_unity_wrt F by A1, A2, A3, FINSEQOP:61;
hence (the_inverseOp_wrt F) . (F "**" p) = F "**" ((the_inverseOp_wrt F) * p) by A2, A4, Th28; :: thesis: verum