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I was writing a proof where I wanted to show a statement y is true and ended up with two different cases to consider which were essentially: $x = 1$ and $x \neq 1$. I showed the result I wanted is true in both cases and I was just wondering how I would phrase the resulting conclusion. Essentially I want to say "We have $x = 1$ or $x \neq 1$. Since y is true if $x = 1$ and and y is true if $x \neq 1$, y must be true." What would be the best way of phrasing this so my intention is clear?

Henry Brown
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  • In such a stark case, I would just remark something like "Of course, we must have $x=1$ or $x\neq 1$. Case I: $x=1$....". It's tougher when it isn't obvious that the cases exhaust all the possibilities. – lulu Dec 23 '20 at 11:00

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I'd probably write it like this: "At this point of the proof, we have two different cases: $x=1$ and $x \neq 1$. First, if $x=1$, then ..., and $y$ is true. Otherwise, we have ... and $y$ is also true. Since it is true in both cases, $y$ must be true."

Flo Spich'r
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