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 Th11:
  for V being RightMod of R, W being Submodule of V holds (Omega).
  V + W = the RightModStr of V & W + (Omega). V = the RightModStr of V
proof
  let V be RightMod of R, W be Submodule of V;
  consider W9 being strict Submodule of V such that
A1: the carrier of W9 = the carrier of (Omega).V;
A2: the carrier of W c= the carrier of W9 by A1,RMOD_2:def 2;
A3: W9 is Submodule of (Omega).V by Lm5;
  W + (Omega).V = W + W9 by A1,Lm4
    .= W9 by A2,Lm3
    .= the RightModStr of V by A1,A3,RMOD_2:31;
  hence thesis by Lm1;
end;
