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

for d being Element of D

for F, G 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 & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))

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

for F, G 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 & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))

let d be Element of D; :: thesis: for F, G 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 & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))

let F, G 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 & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * 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 & G is_distributive_wrt F implies (G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f)) )

assume that

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

A2: F is having_an_inverseOp and

A3: G is_distributive_wrt F ; :: thesis: (G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))

set e = the_unity_wrt F;

set u = G [;] (d,(id D));

G [;] (d,(id D)) is_distributive_wrt F by A3, FINSEQOP:54;

then A4: for d1, d2 being Element of D holds (G [;] (d,(id D))) . (F . (d1,d2)) = F . (((G [;] (d,(id D))) . d1),((G [;] (d,(id D))) . d2)) ;

(G [;] (d,(id D))) . (the_unity_wrt F) = the_unity_wrt F by A1, A2, A3, FINSEQOP:69;

hence (G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f)) by A1, A4, Th16; :: thesis: verum

for d being Element of D

for F, G 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 & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))

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

for F, G 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 & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))

let d be Element of D; :: thesis: for F, G 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 & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))

let F, G 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 & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * 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 & G is_distributive_wrt F implies (G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f)) )

assume that

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

A2: F is having_an_inverseOp and

A3: G is_distributive_wrt F ; :: thesis: (G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))

set e = the_unity_wrt F;

set u = G [;] (d,(id D));

G [;] (d,(id D)) is_distributive_wrt F by A3, FINSEQOP:54;

then A4: for d1, d2 being Element of D holds (G [;] (d,(id D))) . (F . (d1,d2)) = F . (((G [;] (d,(id D))) . d1),((G [;] (d,(id D))) . d2)) ;

(G [;] (d,(id D))) . (the_unity_wrt F) = the_unity_wrt F by A1, A2, A3, FINSEQOP:69;

hence (G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f)) by A1, A4, Th16; :: thesis: verum