How to prove this property for repeated squaring?

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This is from https://courses.cs.washington.edu/courses/cse311/14au/slides/lecture12-filled.pdf

This property is used for modular exponentiation, that is to do a problem like this (from slide 6) enter image description here

Here is the property(slide 7) enter image description here

How would you go about proving this?

The first thing I did was by definition of a mod m is congruent to a modulo m,

a mod m - a = km for some integer k

What step would you take next to prove this property?