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$\Large{{1.}}$ Are proofs using strategies $P136, P143$ always easier than those using $P140$? In the former two, only one statement (either $P$ or $Q$) must be proven. In the latter, both $P$ and $Q$ must be.

$\Large{{2.}}$ If so, should I be concerned with $P140$? Why moot it at all as a proof strategy?

$\Large{{3.}}$ I remember: $P \vee Q \text{ true} \iff \text{Either $P$ or $Q$ is true}$.
Nonetheless, I'm disconcerted and agitated by the act of assuming as true either $P$ or $Q$ without any proof, before proving the other statement. I can't pinpoint why. Could someone please help?

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Related and a possible duplicate. I suggest you look at my answer since it's pretty much Velleman reworded. – Git Gud Jan 3 '14 at 9:20
In question 1. you're comparing $P\lor Q$ as a goal (P140) and as $P\lor Q$ as an hypothesis (P136) with $P\lor Q$ as a goal (P143), so there are in fact two comparisons here: P136 to P143 and P140 to P143, but the P136 to P143 doesn't seem to make much sense because in one of them $P\lor Q$ is a goal and in the other it is a given. Regarding the first part of question 2., P140 is important because it gives a different way of dealing with a $P\lor Q$ goal than that shown on P143. Regarding question 3, are you thinking of $P\lor Q$ as a goal or as an hypothesis here? – Git Gud Jan 3 '14 at 9:24

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