 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 Th18:
  for V being free Z_Module for I being Basis of V
  for A being non empty Subset of V st A misses I
  for B being Subset of V st B = I \/ A holds B is linearly-dependent
  proof
    let V be free Z_Module;
    let I be Basis of V;
    let A be non empty Subset of V such that
    A1: A misses I;
    consider v being object such that
    A2: v in A by XBOOLE_0:def 1;
    let B be Subset of V such that
    A3: B = I \/ A;
    A4: A c= B by A3,XBOOLE_1:7;
    reconsider v as Vector of V by A2;
    reconsider Bv = B \ {v} as Subset of V;
    A5: I \ {v} c= B \ {v} by A3,XBOOLE_1:7,33;
    not v in I by A1,A2,XBOOLE_0:3;
    then I c= Bv by A5,ZFMISC_1:57;
    then
    A6: Lin(I) is Submodule of Lin(Bv) by ZMODUL02:70;
    assume A7: B is linearly-independent;
    v in Lin(I) by Th14;
    hence contradiction by A7,A2,A4,A6,Th17,ZMODUL01:23;
  end;
