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;
reserve RR for domRing;
reserve VV for RightMod of RR;
reserve LL for Linear_Combination of VV;
reserve aa for Scalar of RR;
reserve uu, vv for Vector of VV;

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,Def9
    .= 0.R by Th18;
  hence L2.v = - L1.v by RLVECT_1:6
    .= (- L1).v by Th50;
end;
