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
  L1 + (L2 + L3) = L1 + L2 + L3
proof
  let v;
  thus (L1 + (L2 + L3)).v = L1.v + (L2 + L3).v by Def9
    .= L1.v + (L2.v + L3.v) by Def9
    .= L1.v + L2.v + L3.v by RLVECT_1:def 3
    .= (L1 + L2).v + L3.v by Def9
    .= (L1 + L2 + L3).v by Def9;
end;
