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
  0.V in A & A is linearly-closed implies A = [#]Lin(A)
proof
  assume
A1: 0.V in A & A is linearly-closed;
  thus A c= [#]Lin(A) by Th11;
  let x be object;
  assume x in [#]Lin(A);
  then x in Lin(A);
  then ex l st x = Sum(l) by MOD_3:4;
  hence thesis by A1,Th12;
end;
