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 W2 implies W1 /\ W3 is Submodule of W2 /\ W3
proof
  set A1 = the carrier of W1;
  set A2 = the carrier of W2;
  set A3 = the carrier of W3;
  set A4 = the carrier of W1 /\ W3;
  assume W1 is Submodule of W2;
  then A1 c= A2 by RMOD_2:def 2;
  then A1 /\ A3 c= A2 /\ A3 by XBOOLE_1:26;
  then A4 c= A2 /\ A3 by Def2;
  then A4 c= (the carrier of W2 /\ W3) by Def2;
  hence thesis by RMOD_2:27;
end;
