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 Th4:
  V is non trivial implies for A st A is base holds A <> {}
proof
  assume
A1: V is non trivial;
  let A such that
A2: A is base and
A3: A = {};
A4: A = {}(the carrier of V) by A3;
  the ModuleStr of V = Lin A by A2,VECTSP_7:def 3
    .= (0).V by A4,MOD_3:6;
  hence contradiction by A1,LMOD_6:7;
end;
