theorem Th45:
  (a + b) * L = a * L + b * L
proof
  let v;
  thus ((a + b) * L).v = (a + b) * L.v by Def11
    .= a * L.v + b * L.v
    .= (a * L).v + b * L.v by Def11
    .= (a * L).v + (b * L). v by Def11
    .= ((a * L) + (b * L)).v by Def10;
end;
