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 Th12:
  A<>{} & A is linearly-closed implies Sum(l) in A
proof
  assume
A1: A<>{} & A is linearly-closed;
  now
    per cases;
    suppose
      0.K<>1_K;
      hence thesis by A1,VECTSP_6:14;
    end;
    suppose
      0.K=1_K;
      then K is trivial;
      then Sum(l) = 0.V by Th5;
      hence thesis by A1,VECTSP_4:1;
    end;
  end;
  hence thesis;
end;
