
theorem LmND1:
  for V being non trivial free Z_Module, v being non zero Vector of V,
  I being Basis of V holds
  ex L being Linear_Combination of I, u being Vector of V
  st v = Sum(L) & u in I & L.u <> 0
  proof
    let V be non trivial free Z_Module, v be non zero Vector of V,
    I be Basis of V;
    A1: I is linearly-independent & (Omega).V = Lin(I) by VECTSP_7:def 3;
    v in Lin(I) by A1;
    then consider L be Linear_Combination of I such that
    A2: v = Sum(L) by ZMODUL02:64;
    Carrier(L) <> {}
    proof
      assume Carrier(L) = {};
      then Sum(L) = 0.V by ZMODUL02:23;
      hence contradiction by A2;
    end;
    then consider uu be object such that
    A3: uu in Carrier(L) by XBOOLE_0:def 1;
    consider u be Vector of V such that
    A4: u = uu & L.u <> 0 by A3;
    A5: Carrier(L) c= I by VECTSP_6:def 4;
    take L, u;
    thus thesis by A2,A3,A4,A5;
  end;
