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
  v in W1 or v in W2 implies v in W1 + W2
proof
  assume
A1: v in W1 or v in W2;
  now
    per cases by A1;
    suppose
A2:   v in W1;
      v = v + 0.V & 0.V in W2 by RLVECT_1:def 4,RMOD_2:17;
      hence thesis by A2,Th1;
    end;
    suppose
A3:   v in W2;
      v = 0.V + v & 0.V in W1 by RLVECT_1:def 4,RMOD_2:17;
      hence thesis by A3,Th1;
    end;
  end;
  hence thesis;
end;
