1. Domain: The theorem deals with the divisibility of two integers and their difference with respect to a third integer. Hypothesis\(a\) divides both \(b\) and \(c\)Conclusion: \(a\) divides \(\left(b-c\right)\).
  2. If two integers are both divisible by a third, then their difference is also divisible by the third integer.
  3. 2 divides 2 and 2 divides 4; 2 also divides (4 - 2). 3 divides 6 and 3 divides 9; 3 also divides (9 - 6). Subtracting \(c\) from \(b\) does not affect its divisibility by \(a\).
  4. By definition, \(a|b\) and \(a|c\) means \(b=ka\) and \(c=k'a\). Letting \(k''=k-k'\), we have \(b-c=k''a\). By definition, \(a|\left(b-c\right)\).
  5. This proof was still straightforward, but a tiny bit less intuitive. It took me a second to believe the theorem itself.