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
  W1 is Submodule of W3 & W2 is Submodule of W3 implies W1 + W2 is
  Submodule of W3
proof
  assume
A1: W1 is Submodule of W3 & W2 is Submodule of W3;
  now
    let v;
    assume v in W1 + W2;
    then consider v1,v2 such that
A2: v1 in W1 & v2 in W2 and
A3: v = v1 + v2 by Th1;
    v1 in W3 & v2 in W3 by A1,A2,RMOD_2:8;
    hence v in W3 by A3,RMOD_2:20;
  end;
  hence thesis by RMOD_2:28;
end;
