reserve p,q,r for FinSequence,
  x,y,y1,y2 for set,
  i,k for Element of NAT,
  GF for add-associative right_zeroed right_complementable Abelian associative
  well-unital distributive non empty doubleLoopStr,
  V for Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF,
  u,v,v1,v2,v3,w for Element of V,
  a,b for Element of GF,
  F,G ,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, GF;
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 Th22
    .= L1.v + (L2.v + L3.v) by Th22
    .= L1.v + L2.v + L3.v by RLVECT_1:def 3
    .= (L1 + L2).v + L3.v by Th22
    .= (L1 + L2 + L3).v by Th22;
end;
