let C, D be non empty set ; :: thesis: for B being Element of Fin C

for F being BinOp of D

for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp holds

(the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f))

let B be Element of Fin C; :: thesis: for F being BinOp of D

for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp holds

(the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f))

let F be BinOp of D; :: thesis: for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp holds

(the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f))

let f be Function of C,D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp implies (the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f)) )

assume that

A1: ( F is commutative & F is associative & F is having_a_unity ) and

A2: F is having_an_inverseOp ; :: thesis: (the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f))

set e = the_unity_wrt F;

set u = the_inverseOp_wrt F;

the_inverseOp_wrt F is_distributive_wrt F by A1, A2, FINSEQOP:63;

then A3: 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, FINSEQOP:61;

hence (the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f)) by A1, A3, Th16; :: thesis: verum

for F being BinOp of D

for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp holds

(the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f))

let B be Element of Fin C; :: thesis: for F being BinOp of D

for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp holds

(the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f))

let F be BinOp of D; :: thesis: for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp holds

(the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f))

let f be Function of C,D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp implies (the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f)) )

assume that

A1: ( F is commutative & F is associative & F is having_a_unity ) and

A2: F is having_an_inverseOp ; :: thesis: (the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f))

set e = the_unity_wrt F;

set u = the_inverseOp_wrt F;

the_inverseOp_wrt F is_distributive_wrt F by A1, A2, FINSEQOP:63;

then A3: 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, FINSEQOP:61;

hence (the_inverseOp_wrt F) . (F $$ (B,f)) = F $$ (B,((the_inverseOp_wrt F) * f)) by A1, A3, Th16; :: thesis: verum