let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for a, b being Element of GF

for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

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

for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

let a, b be Element of GF; :: thesis: for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

let L be Linear_Combination of V; :: thesis: a * (b * L) = (a * b) * L

let v be Element of V; :: according to VECTSP_6:def 7 :: thesis: (a * (b * L)) . v = ((a * b) * L) . v

thus (a * (b * L)) . v = a * ((b * L) . v) by Def9

.= a * (b * (L . v)) by Def9

.= (a * b) * (L . v) by GROUP_1:def 3

.= ((a * b) * L) . v by Def9 ; :: thesis: verum

for a, b being Element of GF

for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

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

for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

let a, b be Element of GF; :: thesis: for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

let L be Linear_Combination of V; :: thesis: a * (b * L) = (a * b) * L

let v be Element of V; :: according to VECTSP_6:def 7 :: thesis: (a * (b * L)) . v = ((a * b) * L) . v

thus (a * (b * L)) . v = a * ((b * L) . v) by Def9

.= a * (b * (L . v)) by Def9

.= (a * b) * (L . v) by GROUP_1:def 3

.= ((a * b) * L) . v by Def9 ; :: thesis: verum