reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;

theorem
  for R being comRing for V being RightMod of R for L being
  Linear_Combination of V holds L + ZeroLC(V) = L & ZeroLC(V) + L = L
proof
  let R be comRing;
  let V be RightMod of R;
  let L be Linear_Combination of V;
  thus L + ZeroLC(V) = L
  proof
    let v be Vector of V;
    thus (L + ZeroLC(V)).v = L.v + ZeroLC(V).v by Def9
      .= L.v + 0.R by Th18
      .= L.v by RLVECT_1:4;
  end;
  hence thesis by Th39;
end;
