reserve x for set,
  K for Ring,
  r for Scalar of K,
  V for LeftMod of K,
  a,b,a1,a2 for Vector of V,
  A,A1,A2 for Subset of V,
  l for Linear_Combination of A,
  W for Subspace of V,
  Li for FinSequence of Submodules(V);

theorem Th2:
  not a in A implies l.a = 0.K
proof
  assume
A1: not a in A;
  Carrier l c= A by VECTSP_6:def 4;
  then not a in Carrier l by A1;
  hence thesis by VECTSP_6:2;
end;
