elementary number theory - Prove that if $pmid ab$ where $a$ and $b$ are positive integers and $alt p$ then $ple b$

I have found an old textbook called "Real Variables by Claude W. Burrill and John R. Knudsen" in the first chapter this textbook uses 15 axioms to derive much of the well known and basic facts about the integers, i have been reading and solving all the exercise and so far so good until exercise 1-27 which asks the following: "Prove that if $p$ is prime and divides $ab$ where $a$ and $b$ are positive and $a\lt p$, then $p\le b$." this would be very easy if we assume Euclid's lemma but it hasn't been proven and the very next exercise asks for its proof so i believe that there is a way to prove it without Euclid's lemma but how? Is there even a way to prove this without Euclid's lemma? I also believe i'm not allowed to use Bézout's identity because its proof is exercise 1-29

I have been thinking about this problem since yesterday and i searched online for exercise solutions for this textbook but there was no results.

As another question:does the theorem above imply Euclid's lemma in a straightforward way?