theorem
  (L1 + L2) * a = L1 * a + L2 * a
proof
  let v;
  thus ((L1 + L2) * a).v = (L1 + L2).v * a by Def10
    .= (L1.v + L2.v) * a by Def9
    .= L1.v * a + L2.v * a by VECTSP_1:def 7
    .= (L1 * a).v + L2.v * a by Def10
    .= (L1 * a).v + (L2 * a). v by Def10
    .= ((L1 * a) + (L2 * a)).v by Def9;
end;
