reserve x for set,
  K for Ring,
  r for Scalar of K,
  V, M, M1, M2, N for LeftMod of K,
  a for Vector of V,
  m, m1, m2 for Vector of M,
  n, n1, n2 for Vector of N,
  A for Subset of V,
  l for Linear_Combination of A,
  W, W1, W2, W3 for Subspace of V;

theorem
  V is trivial iff the ModuleStr of V = (0).V
proof
  set X = the carrier of (0).V;
  reconsider W = the ModuleStr of V as strict Subspace of V by Th3;
  reconsider Z=(0).V as Subspace of W by VECTSP_4:39;
A1: now
    assume W <> Z;
    then consider a such that
A2: not a in Z by VECTSP_4:32;
    not a in X by A2;
    then not a in {0.V} by VECTSP_4:def 3;
    then a <> 0.V by TARSKI:def 1;
    hence V is non trivial;
  end;
  now
    assume V is non trivial;
    then consider a such that
A3: a <> 0.V;
    not a in {0.V} by A3,TARSKI:def 1;
    then not a in X by VECTSP_4:def 3;
    hence W <> (0).V;
  end;
  hence thesis by A1;
end;
