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 holds V is Subspace of 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: V is Subspace of V

A1: the lmult of V = the lmult of V | [: the carrier of GF, the carrier of V:] by RELSET_1:19;

( 0. V = 0. V & the addF of V = the addF of V || the carrier of V ) by RELSET_1:19;

hence V is Subspace of V by A1, Def2; :: thesis: verum

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: V is Subspace of V

A1: the lmult of V = the lmult of V | [: the carrier of GF, the carrier of V:] by RELSET_1:19;

( 0. V = 0. V & the addF of V = the addF of V || the carrier of V ) by RELSET_1:19;

hence V is Subspace of V by A1, Def2; :: thesis: verum