let V be Z_Module; :: thesis: for W being Submodule of V
for v being VECTOR of V ex C being Coset of W st v in C
let W be Submodule 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 Def13;
take C ; :: thesis: v in C
thus v in C by Th58; :: thesis: verum