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The following lemmas are styled using the CSS preamble.

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Let $p$ be a prime number, and assume $p$ divides the product of two integers $a$ and $b$.
Then $p|a$ or $p|b$ (or both).

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Let $D = \\{z : |z| \\lt 1\\}$ be the open unit disk in the complex plane $\\mathbb{C}$ centered at the origin and let $f : D \\to D$ be a holomorphic map such that $f(0) = 0$.

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Then, $|f(z)| \\leq |z|$ for all $z$ in $D$ and $|f'(0)| \\leq 1$.

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Moreover, if $|f(z)| = |z|$ for some non-zero $z$ or $|f'(0)| = 1$, then $f(z) = az$ for some $a$ in $\\mathbb{C}$ with $|a| = 1$.

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", "variables": {}, "tags": ["demo"], "metadata": {"description": "

Defines a CSS class in the preamble which styles the \"Lemma\" environment, used in the statement.

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