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 st v in W 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 st v in W holds

- v in v + W

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

- v in v + W

let W be Subspace of V; :: thesis: ( v in W implies - v in v + W )

assume v in W ; :: thesis: - v in v + W

then (- (1_ GF)) * v in v + W by Th58;

hence - v in v + W by VECTSP_1:14; :: thesis: verum

for v being Element of V

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

- v in v + W

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

- v in v + W

let W be Subspace of V; :: thesis: ( v in W implies - v in v + W )

assume v in W ; :: thesis: - v in v + W

then (- (1_ GF)) * v in v + W by Th58;

hence - v in v + W by VECTSP_1:14; :: thesis: verum