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