Now we will introduce some lemmas for the proof of the global existence theorem for the nonlinear wave equations with quadratic nonlinearities. The exsistence theorem which was proven before gives $n >5$ for the quadratic nonlinearities. But in fact this is not optimal. The theorem which will be introduced in the next seminars shows that $n > 3$ is sufficient for this case. To prove this, we have to apply some lemmas using Lorentz group operators in the Minkowski space.