let V be RealLinearSpace; :: thesis: for W being Subspace of V

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

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

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

reconsider C = v + W as Coset of W by RLSUB_1:def 6;

take C ; :: thesis: v in C

thus v in C by RLSUB_1:43; :: thesis: verum

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

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

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

reconsider C = v + W as Coset of W by RLSUB_1:def 6;

take C ; :: thesis: v in C

thus v in C by RLSUB_1:43; :: thesis: verum