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 Th1:
  K is non trivial & A is linearly-independent implies not 0.V in A
proof
  assume that
A1: K is non trivial and
A2: A is linearly-independent;
  0.K <> 1_K by A1,LMOD_6:def 1; then
  K is non degenerated;
  hence thesis by A2,VECTSP_7:2;
end;
