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 Th1:
  x in W1 + W2 iff ex v1,v2 st v1 in W1 & v2 in W2 & x = v1 + v2
proof
  thus x in W1 + W2 implies ex v1,v2 st v1 in W1 & v2 in W2 & x = v1 + v2
  proof
    assume x in W1 + W2;
    then x in the carrier of W1 + W2;
    then x in {v + u : v in W1 & u in W2} by Def1;
    then consider v2,v1 such that
A1: x = v1 + v2 & v1 in W1 & v2 in W2;
    take v1,v2;
    thus thesis by A1;
  end;
  given v1,v2 such that
A2: v1 in W1 & v2 in W2 & x = v1 + v2;
  x in {v + u : v in W1 & u in W2} by A2;
  then x in the carrier of W1 + W2 by Def1;
  hence thesis;
end;
