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 commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds

G . (d,(F "**" p)) = F "**" (G [;] (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 commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds

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

let F, G 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 & G is_distributive_wrt F holds

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

let p be FinSequence of 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,(F "**" p)) = F "**" (G [;] (d,p)) )

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

set e = the_unity_wrt F;

G . (d,(the_unity_wrt F)) = the_unity_wrt F by A1, A2, A3, FINSEQOP:66;

hence G . (d,(F "**" p)) = F "**" (G [;] (d,p)) by A1, A3, Th38; :: thesis: verum

for F, G 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 & G is_distributive_wrt F holds

G . (d,(F "**" p)) = F "**" (G [;] (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 commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds

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

let F, G 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 & G is_distributive_wrt F holds

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

let p be FinSequence of 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,(F "**" p)) = F "**" (G [;] (d,p)) )

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

set e = the_unity_wrt F;

G . (d,(the_unity_wrt F)) = the_unity_wrt F by A1, A2, A3, FINSEQOP:66;

hence G . (d,(F "**" p)) = F "**" (G [;] (d,p)) by A1, A3, Th38; :: thesis: verum