 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;

theorem Th1:
  for u being Vector of V holds
  ex l being Linear_Combination of V st l.u = 1 &
  for v being Vector of V st v <> u holds l.v = 0
  proof
    let u be Element of V;
    reconsider i0 = 0 as Element of INT by INT_1:def 2;
    deffunc F(Element of V) = i0;
    reconsider i1 = 1 as Element of INT by INT_1:def 2;
    ex f being Function of the carrier of V, INT st
    f.u = i1 & for v being Element of V st v <> u holds f.v = F(v)
    from FUNCT_2:sch 6;
    then consider f being Function of the carrier of V, INT such that
    A1: f.u = 1 and
    A2: for v being Element of V st v <> u holds f.v = 0;
    for v being Element of V holds not v in {u} implies v <> u by TARSKI:def 1;
    then
    A3: for v being Element of V holds not v in {u} implies f.v = 0.INT.Ring
      by A2;
    reconsider f as Element of Funcs(the carrier of V,
      the carrier of INT.Ring) by FUNCT_2:8;
    reconsider f as Linear_Combination of V by A3,VECTSP_6:def 1;
    take f;
    thus thesis by A1,A2;
  end;
