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  "body": "\n## Statement\n\nLet $F$ be the [[free group|0KTW26C80J]] on a set $X$, and let $r \\in F$ be a cyclically reduced word.\nLet $G = \\langle X \\mid r \\rangle$ be the [[one-relator group|68AFPHR4GN]] it defines.\nLet $Y$ be a subset of $X$ that does not contain some generator that occurs in $r$.\nThen the [[subgroup|0X1JXT47GA]] of $G$ generated by the images of the elements of $Y$ is free, and these images form a free basis of it.\n\n## Remarks\n\nMagnus proved the theorem by induction on the length of $r$.\nAfter a change of generators, $G$ is written as an [[HNN extension|YEYERRY37K]] or an amalgamated product of one-relator groups with shorter relators, and the induction applies to them.\nThe same method shows that the [[word problem|PBGTTAVMMT]] of one-relator groups is solvable.\n\nExample: in $\\Z^2 = \\langle a, b \\mid a b a^{-1} b^{-1} \\rangle$, the theorem says that $a$ generates an infinite cyclic subgroup, a free group of rank $1$.\nIn the [[Baumslag–Solitar group|429JJ5ZNBY]] $\\langle a, b \\mid b a b^{-1} a^{-2} \\rangle$, both $a$ and $b$ have infinite order.\n\nThe word $r$ must be cyclically reduced.\nFor $r = b a b^{-1}$, which contains $b$, the group $\\langle a, b \\mid b a b^{-1} \\rangle$ is infinite cyclic generated by $b$, and $a = 1$ in it.\nSo $\\set{a}$ omits $b$, but $a$ does not generate a free group of rank $1$.\nThe cyclic reduction of $r$ is $a$, which does not contain $b$.\n\n## Explanations\n\n### L1 human\n\nFreiheitssatz is German for \"freeness theorem\".\nA one-relator group is a free group in which one rule is imposed.\nThe theorem says that the rule cannot affect any set of generators that leaves out a letter of the rule.\nThose generators stay as free as they were before.\n\n### L2 human\n\nThe theorem is the starting point of the theory of one-relator groups.\nMagnus's method of writing a one-relator group as an HNN extension of a group with a shorter relator is the main tool of the area.\nSuppose that $r$ is cyclically reduced and that every generator in $X$ occurs in $r$.\nThen every proper subset of $X$ is a free basis of a free subgroup of $G$.\n",
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