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 V1 being Subset of V st V1 is linearly-closed holds

for v being Element of V st v in V1 holds

- v in V1

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 V1 being Subset of V st V1 is linearly-closed holds

for v being Element of V st v in V1 holds

- v in V1

let V1 be Subset of V; :: thesis: ( V1 is linearly-closed implies for v being Element of V st v in V1 holds

- v in V1 )

assume A1: V1 is linearly-closed ; :: thesis: for v being Element of V st v in V1 holds

- v in V1

let v be Element of V; :: thesis: ( v in V1 implies - v in V1 )

assume v in V1 ; :: thesis: - v in V1

then (- (1_ GF)) * v in V1 by A1;

hence - v in V1 by VECTSP_1:14; :: thesis: verum

for V1 being Subset of V st V1 is linearly-closed holds

for v being Element of V st v in V1 holds

- v in V1

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 V1 being Subset of V st V1 is linearly-closed holds

for v being Element of V st v in V1 holds

- v in V1

let V1 be Subset of V; :: thesis: ( V1 is linearly-closed implies for v being Element of V st v in V1 holds

- v in V1 )

assume A1: V1 is linearly-closed ; :: thesis: for v being Element of V st v in V1 holds

- v in V1

let v be Element of V; :: thesis: ( v in V1 implies - v in V1 )

assume v in V1 ; :: thesis: - v in V1

then (- (1_ GF)) * v in V1 by A1;

hence - v in V1 by VECTSP_1:14; :: thesis: verum