{
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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $a, b, c \\in G$.\nIf $ab = ac$, then $b = c$.\nIf $ba = ca$, then $b = c$.\n\n## Remarks\n\nThe first law is left cancellation and the second law is right cancellation.\nTo prove the first law, multiply $ab = ac$ on the left by $a^{-1}$.\nA consequence is that in the [[Cayley table|0YB2NRP0T5]] of a finite group, no element occurs two times in one row or in one column.\nThe laws can fail in a [[monoid|0N6R902FG1]]: in $(\\mathbb{Z}, \\cdot)$, $0 \\cdot 1 = 0 \\cdot 2$ but $1 \\neq 2$.\n\n## Explanations\n\n### L1 human\n\nIn a group, you can remove the same element from both sides of an equation, if it is on the same side of each product.\nThis works because each element has an inverse that undoes it.\nTake care with the side: $ab = ca$ does not give $b = c$ in general.\n",
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