Mathematical Proof Template with Numbered Equations

Document rigorous mathematics directly inside your documentation hub. This template illustrates how theorem statements, lemmas, and numbered equations render cleanly on the DataLog theme.

Theorem

Theorem 1. Let $f : [a, b] \to \mathbb{R}$ be twice continuously differentiable with $f(a) = f(b) = 0$. Then there exists $c \in (a, b)$ such that $f''(c) + \frac{\pi^2}{(b-a)^2} f(c) = 0$.

Proof

We adapt the standard Wirtinger inequality. Consider the sine basis function $g(x) = \sin\left(\frac{\pi(x-a)}{b-a}\right)$ and define

Integration by parts shows that

Combining Equations \eqref{eq:inner-product} and \eqref{eq:ibp} yields the desired critical point when $f$ is not identically zero. $\square$

Notes for authors

• MathJax automatically numbers equation environments so you can reference them with \eqref{}.
• Inline math, such as $\int_a^b f(x)\,\mathrm{d}x$, remains crisp across light and dark modes.
• Use definition, lemma, and corollary blocks as needed—Markdown blockquotes keep the typography consistent.

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