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

for F, G being BinOp of D

for p being FinSequence of D st F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p)

let d be Element of D; :: thesis: for F, G being BinOp of D

for p being FinSequence of D st F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p)

let F, G be BinOp of D; :: thesis: for p being FinSequence of D st F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p)

let p be FinSequence of D; :: thesis: ( F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F implies (G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p) )

assume that

A1: F is associative and

A2: F is having_a_unity and

A3: F is having_an_inverseOp and

A4: G is_distributive_wrt F ; :: thesis: (G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p)

set e = the_unity_wrt F;

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

G [;] (d,(id D)) is_distributive_wrt F by A4, FINSEQOP:54;

then A5: 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, A4, FINSEQOP:69;

hence (G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p) by A2, A5, Th28; :: thesis: verum

for F, G being BinOp of D

for p being FinSequence of D st F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p)

let d be Element of D; :: thesis: for F, G being BinOp of D

for p being FinSequence of D st F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p)

let F, G be BinOp of D; :: thesis: for p being FinSequence of D st F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds

(G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p)

let p be FinSequence of D; :: thesis: ( F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F implies (G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p) )

assume that

A1: F is associative and

A2: F is having_a_unity and

A3: F is having_an_inverseOp and

A4: G is_distributive_wrt F ; :: thesis: (G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p)

set e = the_unity_wrt F;

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

G [;] (d,(id D)) is_distributive_wrt F by A4, FINSEQOP:54;

then A5: 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, A4, FINSEQOP:69;

hence (G [;] (d,(id D))) . (F "**" p) = F "**" ((G [;] (d,(id D))) * p) by A2, A5, Th28; :: thesis: verum