Task: Translation
Input (English): English sentence: Klein initiated a correspondence with Poincaré, and soon a friendly rivalry ensued as both sought to formulate and prove a grand uniformization theorem that would serve as a capstone to this theory.
Output (Tagalog): Klein na sinimulan ng isang sulat na may Poincaré, at sa lalong madaling panahon ng isang friendly agawan ensued bilang parehong sought na bumalangkas at patunayan ng isang grand uniformization teorama na magsilbi bilang isang capstone sa teorya na ito.