Group/Examples/inv x = 1 - x/Lemma 2

Lemma for Group Example: inv x = 1 - x
Define $f: \left({0 \,.\,.\, 1}\right) \to \R$ by:


 * $f \left({x}\right) := \displaystyle \log \left({ \frac {1 - x}{x} }\right)$

and $g: \R \to \left({0 \,.\,.\, 1}\right)$:


 * $g \left({z}\right) := \dfrac 1 {1 + \exp z}$

Then:
 * $g \circ f \left({x}\right) = x$