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  "body": "\n## Proof\n\n1. Let $a$, $b$ and $c$ be elements of a [[group|G5S1M9]] $G$ with identity $e$, and suppose that $ab = ac$.\n2. Multiply both sides on the left by the [[inverse|0BRHKKP333]] $a^{-1}$. This gives $a^{-1}(ab) = a^{-1}(ac)$.\n3. By associativity, $(a^{-1}a)b = (a^{-1}a)c$, so $eb = ec$ and $b = c$.\n4. Now suppose that $ba = ca$. Multiply both sides on the right by $a^{-1}$.\n5. By associativity, $b(aa^{-1}) = c(aa^{-1})$, so $be = ce$ and $b = c$.\n\n## Idea\n\nMultiply both sides of the equation by the inverse of the common factor, on the same side as that factor.\n",
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        "locator": "Division",
        "url": "https://en.wikipedia.org/wiki/Group_(mathematics)"
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  "name": "Proof of the cancellation laws",
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