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