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 Th7:
  W1 is Submodule of W1 + W2 & W2 is Submodule of W1 + W2
proof
  the carrier of W1 c= the carrier of W1 + W2 by Lm2;
  hence W1 is Submodule of W1 + W2 by RMOD_2:27;
  the carrier of W2 c= the carrier of W2 + W1 by Lm2;
  then the carrier of W2 c= the carrier of W1 + W2 by Lm1;
  hence thesis by RMOD_2:27;
end;
