reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem
  for V being free Z_Module, I being Basis of V, v being Vector of V
  st v in I holds
  Lin(I \ {v}) is free & Lin{v} is free
  proof
    let V be free Z_Module, I be Basis of V,
    v be Vector of V such that
    A1: v in I;
    A2: I is linearly-independent by VECTSP_7:def 3;
    then I \ {v} is linearly-independent by XBOOLE_1:36,ZMODUL02:56;
    hence Lin(I \ {v}) is free;
    {v} is linearly-independent by A1,A2,ZFMISC_1:31,ZMODUL02:56;
    hence Lin{v} is free;
  end;
