let R be non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr ; :: thesis: for a being Element of R

for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R

for v, u, w being Element of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

let a be Element of R; :: thesis: for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R

for v, u, w being Element of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

let V be non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R; :: thesis: for v, u, w being Element of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

let v, u, w be Element of V; :: thesis: a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

thus a * (Sum <*v,u,w*>) = a * ((v + u) + w) by RLVECT_1:46

.= (a * (v + u)) + (a * w) by VECTSP_1:def 14

.= ((a * v) + (a * u)) + (a * w) by VECTSP_1:def 14 ; :: thesis: verum

for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R

for v, u, w being Element of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

let a be Element of R; :: thesis: for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R

for v, u, w being Element of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

let V be non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R; :: thesis: for v, u, w being Element of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

let v, u, w be Element of V; :: thesis: a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

thus a * (Sum <*v,u,w*>) = a * ((v + u) + w) by RLVECT_1:46

.= (a * (v + u)) + (a * w) by VECTSP_1:def 14

.= ((a * v) + (a * u)) + (a * w) by VECTSP_1:def 14 ; :: thesis: verum