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 Th5:
  K is trivial implies (for r holds r = 0.K) & for a holds a = 0.V
proof
  assume K is trivial;
  then
A1: 0.K = 1_K;
A2: now
    let a;
    thus a = 1_K*a
      .= 0.V by A1,VECTSP_1:14;
  end;
  now
    let r;
    thus r = r*1_K
      .= 0.K by A1;
  end;
  hence thesis by A2;
end;
