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
  v in W implies - v in v + W
proof
  assume v in W;
  then v * (- 1_R) in v + W by Th55;
  hence thesis by VECTSP_2:32;
end;
