How do you prove the most basic of statements? This is the question from my homework.

I'm very confused on what this question is wanting me to do.  I mean, look at letter c... wouldn't that be like to prove that if 2 < 3 then show that 2 + 4 < 3 + 4?  I mean that's as basic as math can get... how to you prove that?  I use statements like these IN my proofs.. they are understood without proof.  So what I am supposed to do here?
 A: You're supposed to formally express ideas using the givens. Some statements might sometimes make complete intuitive sense to you, but without really showing why they're true based on something that you take to be true (whether it's an axiom or something that's already proved), you don't know whether it's true or not, since there wouldn't be proof of it being true.
This exercise wants you to think formally - based on already known things (givens, axioms, theorems).
Now, for the answers (I replaced the letters with numbers, and the ')' with a '.', for automatic aligning of lists in Math.StackExchange's text editor):


*

*You need to show that for each condition, if one of the conditions is true, then all the others are not true.
I'll do only the case that $ y>x$: $ \> y>x \Rightarrow y-x\in F^+. $ Assume also that $ y < x \Leftrightarrow x>y \Rightarrow x-y \in F^+ \stackrel{\text{distrivutivity,} +_{F} \> \text{commutativity}} \Leftrightarrow -1 \left( y-x\right) \in F^+$, which is a contradiction.
Now assume also that $y=x \Rightarrow x-y = x-x = 0 \notin F^+$, in contradiction to the fact that $x-y \in F^+$.
Again, the idea is assuming one case, and showing that it doesn't imply the others, i.e, when this case happens, the others don't, i.e that something could be of only one of those cases.

*$
    x>0 \Rightarrow x-0 \in F^+ \\
    y>0 \Rightarrow y-0 \in F^+ \\
    \text{Add both sides of the inequalities to get:} \\
    \left( x+y \right) - \left( 0+0 \right) \Rightarrow \\
    \Rightarrow x+y - 0 \in F^+
    $
And I won't finish doing the exercise - I wanted to show you the formal writing and expressing.
