Proposition 32

Any number is either prime or is measured by some prime number.

Let A be a number.

I say that A either is prime or is measured by some prime number.

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If now A is prime, then that which was proposed is done.

VII.31

But if it is composite, then some prime number measures it.

Therefore any number either is prime or is measured by some prime number.

Therefore, any number is either prime or is measured by some prime number.

Q.E.D.

Guide

After the previous proposition, this one really doesn’t need to be stated at all.