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From eudict.com the simple translation of discipules(discipulus) est prioris posterior dies is:
each day succeeding is the student of the one preceding (Publilius Syrus)
Breaking it down:
discipulus --> student
est --> is
prioris --> of the former (or previous or prior)
posterior --> latter or next or subsequent
dies --> day
You already have the Latin, so . . .
The next day is the student of the previous.
. . . would be the literal English translation.