reserve x,y for object,
  R for Ring,
  V for LeftMod of R,
  L for Linear_Combination of V,
  a for Scalar of R,
  v,u for Vector of V,
  F,G for FinSequence of the carrier of V,
  C for finite Subset of V;
reserve X,Y,Z for set,
  A,B for Subset of V,
  T for finite Subset of V,
  l for Linear_Combination of A,
  f,g for Function of the carrier of V,the carrier of R;
reserve R for Skew-Field;
reserve a,b for Scalar of R;
reserve V for LeftMod of R;
reserve v,v1,v2,u for Vector of V;
reserve f for Function of the carrier of V, the carrier of R;

theorem
  for V being LeftMod of R for A being Subset of V st Lin(A) = V
  holds ex I being Basis of V st I c= A
proof
  let V be LeftMod of R;
  let A be Subset of V;
  assume Lin(A) = V;
  then consider B being Subset of V such that
A1: B c= A and
A2: B is base by Th19;
  reconsider B as Basis of V by A2;
  take B;
  thus thesis by A1;
end;
