Help with theorem proof

I would appreciate if anybody could provide a clear explanation to the
theorem 12.4 on page 183 of Oliver Pretzel's book "Error Correcting
Codes and Finite Fields", OUP 1992.

I am getting stuck on the critical passage:

....'Let ord(alpha) = n and ord(beta) = m. If m does not divide n,
there must be a prime r for which the highest power x=r^k dividing m is
greater than the highest power y=r^l dividing n.' ... etc

I can follow the proof of (another version of) this theorem on Blahut's
book (Algebraic Codes ...) without trouble, but I am stuck on this one.

Apologies to the professionals, I am just an amateur.

Thank you,
ASM

.

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