# Thread: Name of a rule

1. ## Name of a rule

What is the name of the rule which states:

Let $\displaystyle \alpha(x)$ be a function which $\displaystyle x->\alpha(x)$ such that $\displaystyle \alpha^n(x)=x$ for some $\displaystyle n$. Given an equation:

$\displaystyle A(x)f(x) + B(x)f(\alpha(x)) = C(x)$

by repeatedly apply $\displaystyle \alpha$ to $\displaystyle x$ we get a series of equation:

$\displaystyle A(x)f(x) + B(x)f(\alpha(x)) = C(x)$
$\displaystyle A(\alpha(x))f(\alpha(x)) + B(\alpha(x))f(\alpha^2(x)) = C(\alpha(x))$
$\displaystyle A(\alpha^2(x))f(\alpha^2(x)) + B(\alpha^2(x))f(\alpha^3(x)) = C(\alpha^2(x))$

2. I'm not aware of the rule you are referring to.

The rule of substitution?

For a general equality with free variable $\displaystyle x$,
$\displaystyle F(x) = G(x)$ means it is true for all values $\displaystyle x$.
For convenience, since $\displaystyle x$ is a free variable, we can rename it to $\displaystyle t$.
$\displaystyle F(t) = G(t)$, now substitute $\displaystyle \alpha(x)$ for $\displaystyle t$.
$\displaystyle F(\alpha(x)) = G(\alpha(x))$

You can repeat this to show:
$\displaystyle F(\alpha^n(x)) = G(\alpha^n(x))$

3. Originally Posted by TheKovachki
$\displaystyle x->\alpha(x)$ such that $\displaystyle \alpha^n(x)=x$ for some $\displaystyle n$.
Why is this fact important? Where is it being used?

4. Never mind, I guess there is no real rule to it, just continually plunging in $\displaystyle \alpha(x)$ to get a series of equations.