let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of 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 VectSpStr of 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 10 :: thesis: ((0. GF) * L) . v = (ZeroLC V) . v
thus ((0. GF) * L) . v = (0. GF) * (L . v) by Def12
.= 0. GF by VECTSP_1:39
.= (ZeroLC V) . v by Th22 ; :: thesis: verum