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
  L + ZeroLC(V) = L & ZeroLC(V) + L = L
proof
  thus L + ZeroLC(V) = L
  proof
    let v;
    thus (L + ZeroLC(V)).v = L.v + ZeroLC(V).v by Th22
      .= L.v + 0.GF by Th3
      .= L.v by RLVECT_1:4;
  end;
  hence thesis;
end;
