{
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  "body": "\n## Statement\n\nLet $G$ be a [[flow network|ZSNXTW5E2K]] on a finite [[directed graph|0R5HTV7REV]]. It has a source $s$, a sink $t$, and a capacity $c(e) \\ge 0$ on each edge $e$.\n\nLet $\\phi$ be a flow in $G$. That is, $0 \\le \\phi(e) \\le c(e)$ for each edge $e$, and at each vertex other than $s$ and $t$ the flow in equals the flow out.\n\nLet $V = L \\cup U$ be a cut. That is, $L$ and $U$ are disjoint, $s \\in L$ and $t \\in U$. The capacity of the cut is the sum of $c(e)$ over the edges that go from $L$ to $U$.\n\nThen the value of $\\phi$ is at most the capacity of the cut.\n",
  "commit": "bdea0329dc328964ec62566e23a0159c6ce93f2d",
  "front": {
    "kind": "statement",
    "msc": [
      "05C21",
      "90B10",
      "90C35"
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    "name": "Flow value is at most cut capacity",
    "role": "theorem",
    "sources": [
      {
        "locator": "source/s_networkflow_flows-cuts.ptx:36",
        "url": "https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_networkflow_flows-cuts.ptx#L36"
      }
    ],
    "tag": "7TQK5EC8B8",
    "uses": [
      "0R5HTV7REV",
      "ZSNXTW5E2K"
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  "link": "https://math2.tamnd.com/node/7TQK5EC8B8/",
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      "target": "0R5HTV7REV"
    },
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      "target": "ZSNXTW5E2K"
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  "markdown": "https://math2.tamnd.com/node/7TQK5EC8B8.md",
  "msc": [
    "05C21",
    "90B10",
    "90C35"
  ],
  "name": "Flow value is at most cut capacity",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7TQ/7TQK5EC8B8.md",
  "status": null,
  "tag": "7TQK5EC8B8",
  "uri": "m2://7TQK5EC8B8",
  "url": "https://math2.tamnd.com/wiki/flow-value-is-at-most-cut-capacity/",
  "writing": {
    "tag": "7TQK5EC8B8",
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      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
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        "what": "edge proves in",
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