# Double Induction Principle/Minimally Closed Class/Lemma

## Theorem

Let $M$ be a class which is closed under a progressing mapping $g$.

Let $b$ be an element of $M$ such that $M$ is minimally closed under $g$ with respect to $b$.

Let $\RR$ be a relation on $M$ which satisfies:

\((\text D_1)\) | $:$ | \(\ds \forall x \in M:\) | \(\ds \map \RR {x, b} \) | ||||||

\((\text D_2)\) | $:$ | \(\ds \forall x, y \in M:\) | \(\ds \map \RR {x, y} \land \map \RR {y, x} \implies \map \RR {x, \map g y} \) |

Let $x$ be a right normal element of $M$ with respect to $\RR$.

Then $x$ is also a left normal element of $M$ with respect to $\RR$.

## Proof

The proof proceeds by general induction.

Let $x \in M$ be right normal with respect to $\RR$

Let $\map P y$ be the proposition:

- $\map \RR {x, y}$ holds.

### Basis for the Induction

By condition $\text D_1$ of the definition of $\RR$:

- $\map \RR {x, b}$

for all $x \in M$.

Thus $\map P \O$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

Now it needs to be shown that if $\map P y$ is true, where $x \in M$, then it logically follows that $\map P {\map g y}$ is true.

So this is the induction hypothesis:

- $\map \RR {x, y}$ holds

from which it is to be shown that:

- $\map \RR {x, \map g y}$ holds

### Induction Step

This is the induction step:

Let $\map \RR {x, y}$ hold.

As $x$ is right normal with respect to $\RR$:

- $\map \RR {y, x}$ holds.

Thus by condition $\text D_2$ of the definition of $\RR$:

- $\map \RR {x, \map g y}$ holds.

So $\map P x \implies \map P {\map g x}$ and the result follows by the Principle of General Induction.

Therefore:

- $\forall x \in M$: if $x$ is a right normal element of $M$ with respect to $\RR$, then $x$ is a left normal element of $M$ with respect to $\RR$

$\blacksquare$