Added a handful of examples.
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@@ -26,10 +26,15 @@
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\sigma_A(x) \subset \complex \setminus e^{i\theta}[0, \infty)
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\]
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then $x \in G_0(A)$.
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\begin{enumerate}
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\item There exists $y \in A$ such that $x = \exp(y)$.
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\item $x \in G_0(A)$.
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\end{enumerate}
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\end{proposition}
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\begin{proof}
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Since there exists a branch of the logarithm $\ell: \complex \setminus e^{i\theta}[0, \infty) \to \complex$, there exists $y \in A$ such that $x = \exp(y)$. In which case, $x \in G_0(A)$ by \autoref{proposition:functional-calculus-exp}.
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(1): Since there exists a branch of the logarithm $\ell: \complex \setminus e^{i\theta}[0, \infty) \to \complex$, there exists $y \in A$ such that $x = \exp(y)$.
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(2): \autoref{proposition:functional-calculus-exp}.
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\end{proof}
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