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 = ZeroLC(V) implies L2 = - L1
proof
  assume
A1: L1 + L2 = ZeroLC(V);
  let v;
  L1.v + L2.v = ZeroLC(V).v by A1,Th22
    .= 0.GF by Th3;
  hence L2.v = - L1.v by RLVECT_1:6
    .= (- L1).v by Th36;
end;
