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;

theorem Th31:
  for l being Linear_Combination of {}(the carrier of V) holds Sum (l) = 0.V
proof
  let l be Linear_Combination of {}(the carrier of V);
  l = ZeroLC(V) by Th21;
  hence thesis by Lm3;
end;
