let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W being strict Subspace of M holds
( ((0). M) + W = W & W + ((0). M) = W )
let M 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 strict Subspace of M holds
( ((0). M) + W = W & W + ((0). M) = W )
let W be strict Subspace of M; :: thesis: ( ((0). M) + W = W & W + ((0). M) = W )
(0). M is Subspace of W by VECTSP_4:39;
then the carrier of ((0). M) c= the carrier of W by VECTSP_4:def 2;
hence ((0). M) + W = W by Lm3; :: thesis: W + ((0). M) = W
hence W + ((0). M) = W by Lm1; :: thesis: verum