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 <> {} & V1 is linearly-closed holds

0. 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 <> {} & V1 is linearly-closed holds

0. V in V1

let V1 be Subset of V; :: thesis: ( V1 <> {} & V1 is linearly-closed implies 0. V in V1 )

assume that

A1: V1 <> {} and

A2: V1 is linearly-closed ; :: thesis: 0. V in V1

set x = the Element of V1;

reconsider x = the Element of V1 as Element of V by A1, TARSKI:def 3;

(0. GF) * x in V1 by A1, A2;

hence 0. V in V1 by VECTSP_1:14; :: thesis: verum

for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds

0. 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 <> {} & V1 is linearly-closed holds

0. V in V1

let V1 be Subset of V; :: thesis: ( V1 <> {} & V1 is linearly-closed implies 0. V in V1 )

assume that

A1: V1 <> {} and

A2: V1 is linearly-closed ; :: thesis: 0. V in V1

set x = the Element of V1;

reconsider x = the Element of V1 as Element of V by A1, TARSKI:def 3;

(0. GF) * x in V1 by A1, A2;

hence 0. V in V1 by VECTSP_1:14; :: thesis: verum