We provide a proof of existence of symmetric equilibrium for symmetric bi-matrix games, a result implied by a more general result that was proved by John Nash. Our proof, unlike the original proof due to Nash, does not appeal to any fixed-point theorem. We prove that any solution to a certain specific quadratic programming problem, is a symmetric equilibrium for the associated symmetric bi-matrix game. We use no more than the continuity of real-valued multi-variable quadratic functions and the mean value theorem for real-valued quadratic functions of a single variable. This new proof does not require any fixed-point theorem and can be easily understood by anyone who is familiar with a beginner's course on real analysis. The implication of the results repoted here is that not just matrix games, as is traditionally the case, but also bi-matrix games become wholly a part of optimization theory and hence is within the scope of operations research.