If you are asking whether:
\int fr(...) cos\theta_1Le_1 > \int fr(...)cos\theta_1 fr(...)cos\theta_2 Le_2
(simplified terms from the render equation you posted above), I don't think you could prove this. Le_2 could just be much greater than Le_1.
For example you could have a set of mirrors which only hit a light source after two bounces. Then ...
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