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They should proc independently of one another; that is, each has a 10% chance to proc when the cooldown is up, but their proc chance and cooldowns are not shared or linked in any way.
Also note that it's not a 10% chance to proc every 45 seconds; it's a 10% chance to proc on any crit as long as it's not on cooldown, meaning that, depending on your crit rate, it probably procs in 10 or 15 seconds; once it does, there is then 45 seconds where it can't proc at all, at which point it's back to a 10% chance on any crit until it procs again.
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