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 Th31:
  for W1 being strict Submodule of V holds W1 is Submodule of W3
  implies W1 + (W2 /\ W3) = (W1 + W2) /\ W3
proof
  let W1 be strict Submodule of V;
  assume
A1: W1 is Submodule of W3;
  thus (W1 + W2) /\ W3 = W3 /\ (W1 + W2) by Th14
    .= (W1 /\ W3) + (W3 /\ W2) by A1,Lm11,RMOD_2:29
    .= W1 + (W3 /\ W2) by A1,Th17
    .= W1 + (W2 /\ W3) by Th14;
end;
