During the study of real quadratic rational maps as dynamical systems, I have found the general form of real quadratic rational maps with multiple fixed points. My goal after that is to study those maps with a non-repelling fixed point. After my previous presentation of the general form of real quadratic rational maps with a non-repelling fixed point with a coefficient family, I have found a fundamental domain for this coefficient family. In this presentation, I will prove the domain I took satisfies the axioms of a fundamental domain.