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;

theorem
  L.v = 0.R iff not v in Carrier(L)
proof
  thus L.v = 0.R implies not v in Carrier(L)
  proof
    assume
A1: L.v = 0.R;
    assume not thesis;
    then ex u st u = v & L.u <> 0.R;
    hence thesis by A1;
  end;
  assume not v in Carrier(L);
  hence thesis;
end;
