 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 R being Ring,
      V being LeftMod of R, a being Element of R,
      v, u being Vector of V holds
    a * Sum<* v,u *> = a * v + a * u
  proof
    let R be Ring;
    let V be LeftMod of R, a be Element of R, v, u be Vector of V;
    thus a * Sum<* v,u *> = a * (v + u) by RLVECT_1:45
    .= a * v + a * u by VECTSP_1:def 14;
  end;
