reserve R for Ring,
  V for RightMod of R,
  W,W1,W2,W3 for Submodule of V,
  u,u1, u2,v,v1,v2 for Vector of V,
  x,y,y1,y2 for object;

theorem Th8:
  for W2 being strict Submodule of V holds W1 is Submodule of W2
  iff W1 + W2 = W2
proof
  let W2 be strict Submodule of V;
  thus W1 is Submodule of W2 implies W1 + W2 = W2
  proof
    assume W1 is Submodule of W2;
    then the carrier of W1 c= the carrier of W2 by RMOD_2:def 2;
    hence thesis by Lm3;
  end;
  thus thesis by Th7;
end;
