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 being Element of GF

for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)

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 being Element of GF

for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)

let a be Element of GF; :: thesis: for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)

let L1, L2 be Linear_Combination of V; :: thesis: a * (L1 + L2) = (a * L1) + (a * L2)

let v be Element of V; :: according to VECTSP_6:def 7 :: thesis: (a * (L1 + L2)) . v = ((a * L1) + (a * L2)) . v

thus (a * (L1 + L2)) . v = a * ((L1 + L2) . v) by Def9

.= a * ((L1 . v) + (L2 . v)) by Th22

.= (a * (L1 . v)) + (a * (L2 . v)) by VECTSP_1:def 7

.= ((a * L1) . v) + (a * (L2 . v)) by Def9

.= ((a * L1) . v) + ((a * L2) . v) by Def9

.= ((a * L1) + (a * L2)) . v by Th22 ; :: thesis: verum

for a being Element of GF

for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)

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 being Element of GF

for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)

let a be Element of GF; :: thesis: for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)

let L1, L2 be Linear_Combination of V; :: thesis: a * (L1 + L2) = (a * L1) + (a * L2)

let v be Element of V; :: according to VECTSP_6:def 7 :: thesis: (a * (L1 + L2)) . v = ((a * L1) + (a * L2)) . v

thus (a * (L1 + L2)) . v = a * ((L1 + L2) . v) by Def9

.= a * ((L1 . v) + (L2 . v)) by Th22

.= (a * (L1 . v)) + (a * (L2 . v)) by VECTSP_1:def 7

.= ((a * L1) . v) + (a * (L2 . v)) by Def9

.= ((a * L1) . v) + ((a * L2) . v) by Def9

.= ((a * L1) + (a * L2)) . v by Th22 ; :: thesis: verum