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 v being Element of V

for W being Subspace of V holds v in v + 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 v being Element of V

for W being Subspace of V holds v in v + W

let v be Element of V; :: thesis: for W being Subspace of V holds v in v + W

let W be Subspace of V; :: thesis: v in v + W

( v + (0. V) = v & 0. V in W ) by Th17, RLVECT_1:4;

hence v in v + W ; :: thesis: verum

for v being Element of V

for W being Subspace of V holds v in v + 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 v being Element of V

for W being Subspace of V holds v in v + W

let v be Element of V; :: thesis: for W being Subspace of V holds v in v + W

let W be Subspace of V; :: thesis: v in v + W

( v + (0. V) = v & 0. V in W ) by Th17, RLVECT_1:4;

hence v in v + W ; :: thesis: verum