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