### Simple Examples
This section introduces the usage of the LaTeX2Markdown tool, showing an example of the various environments available.
#### Theorem 1 (Euclid, 300 BC)
> There are infinitely many primes.
#### Proof
Suppose that $p_1 < p_2 < \dots < p_n$ are all of the primes. Let $P = 1 + \prod_{i=1}^n p_i$ and let $p$ be a prime dividing $P$.
Then $p$ can not be any of $p_i$, for otherwise $p$ would divide the difference $P - \left(\prod_{i=1}^n p_i \right) - 1$, which is impossible. So this prime $p$ is still another prime, and $p_1, p_2, \dots p_n$ cannot be all of the primes.
#### Exercise 1
> Give an alternative proof that there are an infinite number of prime numbers.
To solve this exercise, we first introduce the following lemma.
#### Lemma 1
> The Fermat numbers $F_n = 2^{2^{n}} + 1$ are pairwise relatively prime.
#### Proof
It is easy to show by induction that
\[ F_m - 2 = F_0 F_1 \dots F_{m-1}. \]
This means that if $d$ divides both $F_n$ and $F_m$ (with $n < m$), then $d$ also divides $F_m - 2$. Hence, $d$ divides 2. But every Fermat number is odd, so $d$ is necessarily one. This proves the lemma.
We can now provide a solution to the exercise.
#### Theorem 2 (Goldbach, 1750)
> There are infinitely many prime numbers.
#### Proof
Choose a prime divisor $p_n$ of each Fermat number $F_n$. By the lemma we know these primes are all distinct, showing there are infinitely many primes.
### Demonstration of the environments
We can format *italic text*, **bold text**, and `code` blocks.
1. A numbered list item
1. Another numbered list item
* A bulleted list item
* Another bulleted list item
#### Theorem 3
> This is a theorem. It contains an `align` block.
>
> All math environments supported by MathJaX should work with LaTeX - a full list is available on the MathJaX homepage.
>
> Maxwell's equations, differential form.
> \begin{align}
> \nabla \cdot \mathbf{E} &= \frac {\rho} {\varepsilon_0} \\\\
> \nabla \cdot \mathbf{B} &= 0 \\\\
> \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}} {\partial t} \\\\
> \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \\\\
> \end{align}
#### Theorem 4 (Theorem name)
> This is a named theorem.
#### Lemma 2
> This is a lemma.
#### Proposition 1
> This is a proposition
#### Proof
This is a proof.
This is a code listing.
One line of code
Another line of code