Theorem Name

Theorem: [Formal statement of the theorem]

To understand this proof, you should be familiar with:

• Prerequisite 1
• Prerequisite 2

[Brief explanation of the intuitive meaning of the theorem and why it matters]

Lemma 1: [If needed, state any supporting lemmas]

[Proof of supporting lemma]

Main Proof:

[Step-by-step proof using LaTeX for mathematical notation]

Let $X$ be a set such that $X = \{x \in \mathbb{R} : x > 0\}$.

We wish to prove that $\forall x \in X, \exists y \in \mathbb{R}$ such that $y^2 = x$.

Step 1: [First step of proof]

$$ \text{Mathematical expression here} $$

Step 2: [Second step of proof]

Step n: [Final step of proof]

Therefore, [conclusion]. ■

Alternative Proof 1: [If applicable]

[Details of alternative approach]

[Discussion of important applications or consequences]

[Brief historical context about the theorem and its development]

• [1] Author, A. (Year). *Title of Book*. Publisher.
• [2] Author, B. (Year). “Title of Paper.” *Journal*, Volume(Issue), Pages.

• Related Theorem 1
• Related Theorem 2

~~DISCUSSION~~