# Thread: Name of a rule

1. ## Name of a rule

What is the name of the rule which states:

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

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

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

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

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

The rule of substitution?

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

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

3. Originally Posted by TheKovachki
$x->\alpha(x)$ such that $\alpha^n(x)=x$ for some $n$.
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 $\alpha(x)$ to get a series of equations.