The degree of a sum of two polynomials (proof question) 
Let $R$ be a ring and let $f = (a_0,a_1,a_2,...)$ and $g = (b_0,b_1,b_2,...)$ be arbitrary polynomials and let $\deg f = m$, $\deg g = n$.
  Then $\deg(f+g) = \max\{m,n\}$, if $m\neq n$ or $\deg(f+g) \leq m$, in case $m = n$.

The proof itself is quite easy, but it's a little detail I don't quite understand.
The author proves three cases: $m < n$, $n < m$ and $m = n$. At the end of the proof he asks why one could not have just dropped the second case $n < m$ and just regarded the first two cases as one $m < n$ without loss of generality.
This is exactly something I would have done and that's why I have no clue what he's referring to. :)
Can somebody help?
 A: Shortcuts like these can be hard to understand, but once you do, they make life easier.
The original set of cases are: $m < n$, $m > n$, and $m=n$.  But notice that the proposition's statement is symmetric in $f$ and $g$. That is, switching the role of $f$ and $g$ in the statement creates an equivalent statement.  Basically, $\max\{m,n\} = \max\{n,m\}$, and if $m=n$, $\deg(f+g) \leq m \iff \deg(f+g)\leq n$.
Which means there are really only two cases: either $m=n$ or $m\neq n$.  If $m=n$, use the same argument as in the “trichotomous” (three-case) proof.  If $m \neq n$, well, one of them must be less than the other.  Let's call the lesser one $m$ and the greater one $n$.  The symmetry of the proposition says that it's OK to do this.  Now we use the $m<n$ case from the trichotomous proof.
Another way to see the symmetry in the proposition is to rewrite it without naming the variables:

  
*
  
*If two polynomials have the same degree, the degree of the sum is at most this common degree.
  
*If two polynomials have different degrees, the degree of the sum is the maximum of the degrees of each polynomial.
  

But of course, you need to name them to do anything.  To prove case 1, you need to name two generic polynomials of the same degree.  Call them $f$ and $g$ and their degree $n$.  To prove case 2, you need to name two generic polynomials of different degree.  Call the one with lesser degree $f$ (and its degree $m$), and the one with greater degree $g$ (and its degree $n$).   
When using simplications like this in proofs, we write the idiom “without loss of generality”.  It means that this extra assumption does not actually skip cases of the proof.
After posting, I noticed @mfl's comment to the same effect.  His additional remark is also worth fleshing out.  The lecturer was recognizing that this may be the first time you've seen a WLOG argument like this.  But like I said at the top, it may take a couple of explanations before this sinks in.
