 reserve x, y, y1, y2 for set;
 reserve V for Z_Module;
 reserve u, v, w for Vector of V;
 reserve F, G, H, I for FinSequence of V;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve KL1, KL2 for Linear_Combination of V;
 reserve X for Subset of V;

theorem Th14:
  for V being free Z_Module for I being Basis of V, v being Vector of V
    holds v in Lin(I)
  proof
    let V be free Z_Module;
    let I be Basis of V, v be Vector of V;
A1: v in the ModuleStr of V;
    for I being Subset of V holds
    I is base iff
    I is linearly-independent & Lin(I) = the ModuleStr of V;
    hence thesis by A1;
  end;
