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 Th17:
  (for W1 being strict Submodule of V holds W1 is Submodule of W2
  implies W1 /\ W2 = W1) & for W1 st W1 /\ W2 = W1 holds W1 is Submodule of W2
proof
  thus for W1 being strict Submodule of V holds W1 is Submodule of W2 implies
  W1 /\ W2 = W1
  proof
    let W1 be strict Submodule of V;
    assume W1 is Submodule of W2;
    then
A1: the carrier of W1 c= the carrier of W2 by RMOD_2:def 2;
    the carrier of W1 /\ W2 = (the carrier of W1) /\ (the carrier of W2)
    by Def2;
    hence thesis by A1,RMOD_2:29,XBOOLE_1:28;
  end;
  thus thesis by Th16;
end;
