 reserve R for Ring;
 reserve x, y, y1 for set;
 reserve a, b for Element of R;
 reserve V for LeftMod of R;
 reserve v, w for Vector of V;
 reserve u,v,w for Vector of V;
 reserve F,G,H,I for FinSequence of V;
 reserve j,k,n for Nat;
 reserve f,f9,g for sequence of V;

theorem
  for V being LeftMod of INT.Ring, a being Element of INT.Ring,
      v being Vector of V holds
    (- a) * v = - a * v
  proof
    let V be LeftMod of INT.Ring,
        a be Element of INT.Ring,
        v be Vector of V;
    thus (- a) * v = a * (- v) by Th5
    .= - a * v by Th6;
  end;
