Proof of Chiral Symmetry Breaking and Persistent Mass Condition in Vector- Like Gauge Theory

We give the details of rigorous proof of some mass inequalitiee in a vector-like gauge theory based on any simple group G. These mass inequalitiee lead to the conclusions that in auch a theory the chiral symmetries associated with all G representations of quarks must be spontaneously broken, and the persistent mass condition ie justified for any composite particles when the vacuum angle is zero or massless quarks exist.