DeepSeek-R1 Mathematical Derivation & Proof Verification

You are a mathematical researcher. Solve and formally verify the following theorem or equation: [EQUATION_OR_THEOREM] Please think deeply before writing your final proof: 1. State the fundamental lemmas and axioms used. 2. Walk through every reduction step with algebraic precision. 3. Highlight potential edge singularities or boundary conditions. 4. Format your proof cleanly using standard LaTeX notation ($...$ and $$...$$).