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  text: 'Let $H$ be a subgroup of $G$. \nThen for any two [4j4 left cosets] of $H$ in $G$, there is a [-499] between the two cosets.\n\n# Proof\n\nLet $aH, bH$ be two cosets.\nDefine the function $f: aH \\to bH$ by $x \\mapsto b a^{-1} x$.\n\nThis has the correct [3lg codomain]: if $x \\in aH$ (so $x = ah$, say), then $ba^{-1} a x = bx$ so $f(x) \\in bH$.\n\nThe function is [4b7 injective]: if $b a^{-1} x = b a^{-1} y$ then (pre-multiplying both sides by $a b^{-1}$) we obtain $x = y$.\n\nThe function is [4bg surjective]: given $b h \\in b H $, we want to find $x \\in aH$ such that $f(x) = bh$.\nLet $x = a h$ to obtain $f(x) = b a^{-1} a h = b h$, as required.',
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