 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem VECT9Th26:
  for G be Field, V be VectSp of G, A being Subset of V
  st A is linearly-independent holds
  A is Basis of Lin(A)
  proof
    let G be Field, V be VectSp of G;
    let A be Subset of V such that
    A1: A is linearly-independent;
    set W = Lin(A);
    for x being object st x in A holds x in the carrier of W
    by STRUCT_0:def 5,VECTSP_7:8;
    then reconsider B = A as linearly-independent Subset of W
    by A1,VECTSP_9:12,TARSKI:def 3;
    W = Lin(B) by VECTSP_9:17;
    hence thesis by VECTSP_7:def 3;
  end;
