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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $W \\subseteq U$ be subspaces of $V$.\nThen $U / W$ is a subspace of $V / W$, and the map\n$$(V / W) / (U / W) \\to V / U, \\qquad (v + W) + U/W \\mapsto v + U,$$\nis well defined, and it is a linear isomorphism of [[quotient spaces|0BR3FY5Q8M]].\n\n## Remarks\n\nThe proof applies the [[first isomorphism theorem|0MTBGVZ4XJ]] to the surjective map $V / W \\to V / U$, $v + W \\mapsto v + U$, whose kernel is $U / W$.\nIn finite dimensions, both sides have dimension $\\dim V - \\dim U$.\n\n## Explanations\n\n### L1 human\n\nDividing first by a small subspace and then by a larger one is the same as dividing by the larger one at once.\nThis works like cancelling in fractions.\nThe same theorem holds for groups, rings and modules.\n",
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