Roughly speaking, yes.

A proof of a statement that starts with "for all", unless it is a proof by induction, must start with words like "fix arbitrary", "consider", "let," etc. In this case, you can say, "Fix arbitrary m and n" or "Let m and n be some integers." Otherwise, when you are talking about m and n later, it is not clear what those are.for all integers m and n, if m and n are odd, then m + n is even.

Further, a proof of a statement that starts with "if" must start with "assume." In this case, you say, "... and assume that m and n are odd."

This is not necessary since you are not using the letters p and q later. In general, one should not just write a proposition as a part of a proof but rather explain its status: whether the proposition is assumed, whether it's the next goal to prove, etc.p: m and n is odd

q: m+n is even(conclusion)

To show that m + n is even, consider...m + n is even

= 2(k[1] + k[2] + 1); therefore, this number is even.= 2(k[1]+k[2])+2