Fixed Point of Mappings is Fixed Point of Composition

Theorem
Let $S$ be a set.

Let $f, g: S \to S$ be mappings.

Let $x \in S$ be a fixed point of both $f$ and $g$.

Then $x$ is also a fixed point of $f \circ g$, the composition of $f$ and $g$.

Proof
Since $x$ is a fixed point of $g$:


 * $\map g x = x$

Thus:


 * $\map f {\map g x} = \map f x$

Since $x$ is a fixed point of $f$:


 * $\map f x = x$

It follows that:


 * $\map {\paren {f \circ g} } x = \map f {\map g x} = x$

Thus $x$ is a fixed point of $f \circ g$.