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 ex C being Coset of W st v in C

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 ex C being Coset of W st v in C

let v be Element of V; :: thesis: for W being Subspace of V ex C being Coset of W st v in C

let W be Subspace of V; :: thesis: ex C being Coset of W st v in C

reconsider C = v + W as Coset of W by Def6;

take C ; :: thesis: v in C

thus v in C by Th44; :: thesis: verum

for v being Element of V

for W being Subspace of V ex C being Coset of W st v in C

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 ex C being Coset of W st v in C

let v be Element of V; :: thesis: for W being Subspace of V ex C being Coset of W st v in C

let W be Subspace of V; :: thesis: ex C being Coset of W st v in C

reconsider C = v + W as Coset of W by Def6;

take C ; :: thesis: v in C

thus v in C by Th44; :: thesis: verum