1. Domain: The theorem deals with the modular congruence of an integer with respect to two others. Hypothesis\(n\) and 7 are congruent modulo 2. Conclusion: \(n\) and 3 are congruent modulo 2.
  2. If an integer and 7 are congruent modulo 2, then the integer and 3 are also congruent modulo 2.
  3. 7 and 7 are congruent modulo 2, and 7 and 3 are also congruent modulo 2, since 2 divides (7 - 3). 9 and 7 are congruent modulo 2, since 2 divides (9 - 7), and 9 and 3 are also congruent modulo 2, since 2 divides (9 - 3). Adding 4 to the difference does not affect its divisibility by 2.
  4. By definition, \(n\equiv7\) (mod 2) means 2 divides \(n-7\) and \(n-7=2k\). Letting \(k'=k+2\), we have \(n-3=2k'\), and, since \(k'\) is an integer, by definition, 2 divides \(n-3\) and n and 3 are congruent modulo 2.
  5. I don't feel like I worded this proof clearly. I have a lot to learn about organizing the thought and logic process behind proofs.