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. V in W

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. V in W

let W be Subspace of V; :: thesis: 0. V in W

0. W in W ;

hence 0. V in W by Def2; :: thesis: verum

for W being Subspace of V holds 0. V in W

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. V in W

let W be Subspace of V; :: thesis: 0. V in W

0. W in W ;

hence 0. V in W by Def2; :: thesis: verum