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 W being Subspace of V holds (0). W = (0). 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 W being Subspace of V holds (0). W = (0). V
let W be Subspace of V; :: thesis: (0). W = (0). V
( the carrier of ((0). W) = {(0. W)} & the carrier of ((0). V) = {(0. V)} ) by Def3;
then A1: the carrier of ((0). W) = the carrier of ((0). V) by Def2;
(0). W is Subspace of V by Th26;
hence (0). W = (0). V by A1, Th29; :: thesis: verum