In triangle LMP, LP^2=LM^2-MP^2 and in triangle LNP, LP^2=LN^2-NP^2,
so (1) LM^2-MP^2=LN^2-NP^2. Therefore, MP^2-NP^2=LM^2-LN^2.
Also, in triangle LMN, MN^2=LM^2+LN^2, and (2) LM^2=MN^2-LN^2.
Substituting for LM in (1): MN^2-LN^2-MP^2=LN^2-NP^2. And MN^2-2LN^2=MP^2-NP^2. This is the same as MP^2-PN^2=MN^2-2LN^2. QED.