theorem Th44:
  |.p-q.|^2 = |.p.|^2 - 2*|(q, p)| + |.q.|^2
proof
  |.p-q.|^2 = |(p-q, p-q)| by Th34
    .= |(p, p)| - 2*|(q, p)| + |(q, q)| by Th29
    .= |.p.|^2 - 2*|(q, p)| + |(q, q)| by Th34
    .= |.p.|^2 - 2*|(q, p)| + |.q.|^2 by Th34;
  hence thesis;
end;
