reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem Th40:
  L1 + (L2 + L3) = L1 + L2 + L3
proof
  let v;
  thus (L1 + (L2 + L3)).v = L1.v + (L2 + L3).v by Def10
    .= L1.v + (L2.v + L3.v) by Def10
    .= L1.v + L2.v + L3.v
    .= (L1 + L2).v + L3.v by Def10
    .= (L1 + L2 + L3).v by Def10;
end;
