theorem Th46:
  a * (L1 + L2) = a * L1 + a * L2
proof
  let v;
  thus (a * (L1 + L2)).v = a * (L1 + L2).v by Def11
    .= a * (L1.v + L2.v) by Def10
    .= a * L1.v + a * L2.v
    .= (a * L1).v + a * L2.v by Def11
    .= (a * L1).v + (a * L2). v by Def11
    .= ((a * L1) + (a * L2)).v by Def10;
end;
