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 L being Linear_Combination of V holds (0. GF) * L = ZeroLC V

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 L being Linear_Combination of V holds (0. GF) * L = ZeroLC V

let L be Linear_Combination of V; :: thesis: (0. GF) * L = ZeroLC V

let v be Element of V; :: according to VECTSP_6:def 7 :: thesis: ((0. GF) * L) . v = (ZeroLC V) . v

thus ((0. GF) * L) . v = (0. GF) * (L . v) by Def9

.= 0. GF

.= (ZeroLC V) . v by Th3 ; :: thesis: verum

for L being Linear_Combination of V holds (0. GF) * L = ZeroLC V

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 L being Linear_Combination of V holds (0. GF) * L = ZeroLC V

let L be Linear_Combination of V; :: thesis: (0. GF) * L = ZeroLC V

let v be Element of V; :: according to VECTSP_6:def 7 :: thesis: ((0. GF) * L) . v = (ZeroLC V) . v

thus ((0. GF) * L) . v = (0. GF) * (L . v) by Def9

.= 0. GF

.= (ZeroLC V) . v by Th3 ; :: thesis: verum