Theorem: [Formal statement of the theorem]
To understand this proof, you should be familiar with:
[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]
~~DISCUSSION~~