Make The Condition Satisfied

Algebra Level 4

Let $$P(x)=\ln\dfrac{1-x}{1+x}$$. For some $$a$$ and $$b$$ The condition $$P(a)+P(b)=P \left (\dfrac{a+b}{1+ab} \right )$$ is satisfied.

Given that $$a\in (f,g)$$ and $$b\in (m,n)$$. $$f<g$$ and $$m<n$$.

Find $$(g-f)\times(n-m)$$

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