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 Th43:
  0.V in v + W iff v in W
proof
  thus 0.V in v + W implies v in W
  proof
    assume 0.V in v + W;
    then consider u such that
A1: 0.V = v + u and
A2: u in W;
    v = - u by A1,VECTSP_1:16;
    hence thesis by A2,Th22;
  end;
  assume v in W;
  then
A3: - v in W by Th22;
  0.V = v + (- v) by VECTSP_1:19;
  hence thesis by A3;
end;
