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;

theorem
  x in (0).V iff x = 0.V
proof
  thus x in (0).V implies x = 0.V
  proof
    assume x in (0).V;
    then x in the carrier of (0).V;
    then x in {0.V} by Def3;
    hence thesis by TARSKI:def 1;
  end;
  thus thesis by Th17;
end;
