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| - | ====== Theorem Name ====== | ||
| - | ===== Statement ===== | ||
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| - | **Theorem: | ||
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| - | ===== Prerequisites ===== | ||
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| - | To understand this proof, you should be familiar with: | ||
| - | * [[path: | ||
| - | * [[path: | ||
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| - | ===== Intuition ===== | ||
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| - | [Brief explanation of the intuitive meaning of the theorem and why it matters] | ||
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| - | ===== Proof ===== | ||
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| - | **Lemma 1:** [If needed, state any supporting lemmas] | ||
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| - | [Proof of supporting lemma] | ||
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| - | **Main Proof:** | ||
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| - | [Step-by-step proof using LaTeX for mathematical notation] | ||
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| - | Let $X$ be a set such that $X = \{x \in \mathbb{R} : x > 0\}$. | ||
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| - | We wish to prove that $\forall x \in X, \exists y \in \mathbb{R}$ such that $y^2 = x$. | ||
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| - | Step 1: [First step of proof] | ||
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| - | $$ | ||
| - | \text{Mathematical expression here} | ||
| - | $$ | ||
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| - | Step 2: [Second step of proof] | ||
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| - | Step n: [Final step of proof] | ||
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| - | Therefore, [conclusion]. ■ | ||
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| - | ===== Alternative Proofs ===== | ||
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| - | **Alternative Proof 1:** [If applicable] | ||
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| - | [Details of alternative approach] | ||
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| - | ===== Applications ===== | ||
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| - | [Discussion of important applications or consequences] | ||
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| - | ===== Historical Notes ===== | ||
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| - | [Brief historical context about the theorem and its development] | ||
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| - | ===== References ===== | ||
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| - | * [1] Author, A. (Year). *Title of Book*. Publisher. | ||
| - | * [2] Author, B. (Year). "Title of Paper." | ||
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| - | ===== Related Theorems ===== | ||
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| - | * [[path: | ||
| - | * [[path: | ||
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| - | ~~DISCUSSION~~ | ||