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You respond to the exercise as a tool you have used all your life, which is very reasonable. However ‘dividing both sides’ itself requires justification. That justification being that inverses of non-zero numbers exist, multiplication on numbers being associative Funny Merry Pigmas Christmas Light Shirt. The proof you have found in the solution is exactly that. Also be aware that it is a solution manual, it has to be even more extreme in its presentation. As others have pointed out, this particular ‘exercise’ can be applied to more general structures, where division is not necessarily an option, and also the proof you found would no longer work. What even is ‘odd’ and ‘even’, exactly? Learning to think about these things is the first step to becoming a mathematician. Consider the statement ‘every integer is either odd or even’. Ok.

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Did you know that there are some operations you can perform on both sides of equality and not get the same result? The division of real numbers happens to not be one of them, but there are examples. More centrally though, the purpose of this exercise is to prove that a/a = 1. You don’t have to prove that a/a = 1 every time you want to use that fact, but proving basic concepts “from scratch” is a common pedagogical methodology to teach Funny Merry Pigmas Christmas Light Shirt. It’s much easier to prove statements you already know are true! Sure, if you respect all the rules and know what you’re doing this isn’t a big problem. But the point is that the OP isn’t doing that. The most accessible examples are from conditionally convergent series. Rearranging conditionally convergent series is extremely dangerous. I was thinking too formally rather than from the perspective of someone first approaching such things. Thank you so much. So good. So happy. Ok.

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