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