Write a formal proof (numbered propositions, justifications, etc.) of the following theorem: If a and b are odd, then a+b is even. Do not use modular arithmetic, "properties of odds" or similar shortcuts: use propositional logic as we've done in class. You may use the definitions of "even" and "odd," as well as "closure of integers" (adding, subtracting and multiplying integers gives you an integer) and "Algebra" (which includes things like x+x=2x (and other algebraic facts) and 1+1=2 (and other arithmetic facts))