
theorem
  for V being non trivial torsion-free Z_Module, v being non zero Vector of V,
      A being Subset of V, a being Element of INT.Ring
  st a in NAT & A is linearly-independent & a > 0 & a * v in Lin(A)
  holds ex L being Linear_Combination of A, u being Vector of V
  st a * v = Sum(L) & u in A & L.u <> 0
  proof
    let V be non trivial torsion-free Z_Module, v be non zero Vector of V,
    A be Subset of V, a be Element of INT.Ring such that
    A1: a in NAT & A is linearly-independent & a > 0 & a * v in Lin(A);
a1: a <> 0.INT.Ring by A1;    
    consider L be Linear_Combination of A such that
    A2: a * v = Sum(L) by A1,ZMODUL02:64;
    Carrier(L) <> {}
    proof
      assume Carrier(L) = {};
      then Sum(L) = 0.V by ZMODUL02:23;
      hence contradiction by a1,A2,ZMODUL01:def 7;
    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= A by VECTSP_6:def 4;
    take L, u;
    thus thesis by A2,A3,A4,A5;
  end;
