We consider the problem of multiple knife-edge diffraction estimation which is a fundamental task in many wireless communication applications. So far, one of the most accurate methods for this problem is the Vogler one whose recursive implementation is efficient to reduce the high computational complexity of the direct one. However, in the original report, Vogler only presented the final result of the recursive algorithm without a rigorous mathematical proof, thus making the method difficult to understand and implement in practice. To tackle this shortcoming, we first analyze the mathematical structure of the problem and then present a formal proof of the result. To gain intuition of the proof and the key steps, we provide a simplified study case of four knife-edges. The insight from our proposed analysis and proof can be used to obtain a comprehensive interpretation, initiate a practical implementation and develop new efficient algorithms with similar structure.