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

for w being Element of W st w = v holds

- 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

for w being Element of W st w = v holds

- v = - w

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

for w being Element of W st w = v holds

- v = - w

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

- v = - w

let w be Element of W; :: thesis: ( w = v implies - v = - w )

A1: ( - v = (- (1_ GF)) * v & - w = (- (1_ GF)) * w ) by VECTSP_1:14;

assume w = v ; :: thesis: - v = - w

hence - v = - w by A1, Th14; :: thesis: verum

for v being Element of V

for W being Subspace of V

for w being Element of W st w = v holds

- 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

for w being Element of W st w = v holds

- v = - w

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

for w being Element of W st w = v holds

- v = - w

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

- v = - w

let w be Element of W; :: thesis: ( w = v implies - v = - w )

A1: ( - v = (- (1_ GF)) * v & - w = (- (1_ GF)) * w ) by VECTSP_1:14;

assume w = v ; :: thesis: - v = - w

hence - v = - w by A1, Th14; :: thesis: verum