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Don't understand use of auxiliary function in proving theorems

How was the auxiliary function to be used in proving the theorem below determined? I don't understand why this function was chosen, and how the author arrived at the decision of choosing this function? This is the first time I came across the use of auxiliary functions in proving a theorem, so I am a bit confused whether the function is chosen by pure intuition or is there a method behind this.

http://www.mathhelpforum.com/math-he...3&d=1290060131

Thanks a lot for reading.

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