 reserve R for Ring;
 reserve x, y, y1 for set;
 reserve a, b for Element of R;
 reserve V for LeftMod of R;
 reserve v, w for Vector of V;
 reserve u,v,w for Vector of V;
 reserve F,G,H,I for FinSequence of V;
 reserve j,k,n for Nat;
 reserve f,f9,g for sequence of V;
 reserve R for Ring;
 reserve V, X, Y for LeftMod of R;
 reserve u, u1, u2, v, v1, v2 for Vector of V;
 reserve a for Element of R;
 reserve V1, V2, V3 for Subset of V;
 reserve x for set;
 reserve W, W1, W2 for Submodule of V;
 reserve w, w1, w2 for Vector of W;
 reserve D for non empty set;
 reserve d1 for Element of D;
 reserve A for BinOp of D;
 reserve M for Function of [:the carrier of R,D:],D;
reserve B,C for Coset of W;
 reserve V for LeftMod of R;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve u, u1, u2, v, v1, v2 for Vector of V;
 reserve a, a1, a2 for Element of R;
 reserve X, Y, y, y1, y2 for set;
 reserve C for Coset of W;
 reserve C1 for Coset of W1;
 reserve C2 for Coset of W2;

theorem
  for V being Z_Module, W1, W2 being Submodule of V holds V
  is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1, C2 being Coset of W2
  ex v being Vector of V st C1 /\ C2 = {v} by VECTSP_5:46;
