theorem Th43:
  |.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 Th28
    .= |.p.|^2 + 2*|(q, p)| + |(q, q)| by Th34
    .= |.p.|^2 + 2*|(q, p)| + |.q.|^2 by Th34;
  hence thesis;
end;
