reserve x,y,y1,y2 for object;
reserve R for Ring;
reserve a for Scalar of R;
reserve V,X,Y for RightMod of R;
reserve u,u1,u2,v,v1,v2 for Vector of V;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Submodule of V;
reserve w,w1,w2 for Vector of W;
reserve B,C for Coset of W;

theorem
  V1 is Coset of (0).V implies ex v st V1 = {v}
proof
  assume V1 is Coset of (0).V;
  then consider v such that
A1: V1 = v + (0).V by Def6;
  take v;
  thus thesis by A1,Th46;
end;
