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 Th39:
  for R being comRing for V being RightMod of R for L1, L2 being
  Linear_Combination of V holds L1 + L2 = L2 + L1
proof
  let R be comRing;
  let V be RightMod of R;
  let L1, L2 be Linear_Combination of V;
  let v be Vector of V;
  thus (L1 + L2).v = L2.v + L1.v by Def9
    .= (L2 + L1).v by Def9;
end;
