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@@ -8,7 +8,7 @@
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\begin{enumerate}
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\item[(a)] $\bigcup_{i \in I}U_i = X$.
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\item[(b)] For each $i, j \in I$, either $U_i \cap U_j = \emptyset$, or $f_i|_{U_i \cap U_j} = f_j|_{U_i \cap U_j}$.
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\end\{enumerate\}
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\end{enumerate}
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then there exists a unique $f: X \to Y$ such that $f|_{U_i} = f_i$ for all $i \in I$.
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\end{lemma}
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@@ -17,7 +17,7 @@
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\begin{enumerate}
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\item By assumption (a), $\bracs{x|(x, y) \in \Gamma} = \bigcup_{i \in I}U_i = X$.
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\item For any $x \in X$, there exists $y \in Y$ with $(x, y) \in \Gamma$, and $i \in I$ such that $(x, y) \in \Gamma_i$. If $(x, y') \in \Gamma_j \subset \Gamma$, then $x \in U_i \cap U_j \ne \emptyset$. By assumption (b), $y = y'$.
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\end\{enumerate\}
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\end{enumerate}
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Thus $\Gamma$ is the graph of a function $f: X \to Y$ with $f|_{U_i} = f_i$ for all $i \in I$.
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\end{proof}
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@@ -29,7 +29,7 @@
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\item[(a)] $\bigcup_{V \in \fF}V = E$.
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\item[(b)] For each $V, W \in \fF$, $T_V|_{V \cap W} = T_W|_{V \cap W}$.
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\item[(c)] $\fF$ is upward-directed with respect to includion.
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\end\{enumerate\}
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\end{enumerate}
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then there exists a unique $T \in \hom(E; F)$ such that $T|_{V} = T_V$ for all $V \in \fF$.
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\end{lemma}
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