reserve a, b, c, d, x, y, z for Complex;

theorem Th47:
  (x-y).|.(x-y) = x.|.x - x.|.y - y.|.x + y.|.y
proof
  (x - y).|.(x - y) =x.|.(x - y) - y.|.(x - y)
    .=x.|.x - x.|.y - y.|.(x - y) by Th45
    .=x.|.x - x.|.y - (y.|.x - y.|.y) by Th45
    .=x.|.x - x.|.y - y.|.x + y.|.y;
  hence thesis;
end;
