theorem Th22:
  L1 + (L2 + L3) = L1 + L2 + L3
proof
  let v;
  thus (L1 + (L2 + L3)).v = L1.v + (L2 + L3).v by Def8
    .= L1.v + (L2.v + L3.v) by Def8
    .= L1.v + L2.v + L3.v
    .= (L1 + L2).v + L3.v by Def8
    .= (L1 + L2 + L3).v by Def8;
end;
