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

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