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  "body": "\n## Proof\n\nWe use [[double counting|0F8YBHF2W3]] of vertex–edge incidences.\n\n1. The set $V$ is finite. The [[edge set|0WNGQBPPQX]] $E$ consists of $2$-element subsets of $V$, so $E$ is finite too.\n2. Let $R = \\set{(v, e) \\in V \\times E : v \\in e}$, the set of incidences.\n3. Fix $v \\in V$. The number of $e \\in E$ with $(v, e) \\in R$ is the number of edges that contain $v$. By the definition of the [[degree|0T0F6CCS9W]], this is $\\deg(v)$.\n4. Fix $e \\in E$. The number of $v \\in V$ with $(v, e) \\in R$ is $\\abs{e} = 2$, since each edge of a [[graph|0JD5C82QMB]] is a $2$-element subset of $V$.\n5. Counting $R$ by its first coordinate gives $\\abs{R} = \\sum_{v \\in V} \\deg(v)$, by step 3.\n6. Counting $R$ by its second coordinate gives $\\abs{R} = \\sum_{e \\in E} 2 = 2 \\abs{E}$, by step 4.\n7. Comparing steps 5 and 6 gives $\\sum_{v \\in V} \\deg(v) = 2 \\abs{E}$.\n\n## Idea\n\nEach edge has exactly two ends. Adding up the degrees counts every edge once at each end, so every edge is counted twice.\n",
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