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
  for V being strict RightMod of R, W being strict Submodule of V holds
  (for v being Vector of V holds v in W) implies W = V
proof
  let V be strict RightMod of R, W be strict Submodule of V;
  assume for v being Vector of V holds v in W;
  then
A1: for v be Vector of V holds ( v in W iff v in V);
  V is Submodule of V by Th24;
  hence thesis by A1,Th30;
end;
