1.(R>L)>(L>~F)
2. ~F v (R>L)
3. ~~F /~R

4. R>L         2,3, Disjunctive Syllogism  ---That's [(p v q) & ~p]>q
5. L>~F        1,4, Modus Ponens           ---That's [(p>q) & p]>q
6. R>~F        4,5, Hypothetical Syllogism ---That's [(p>q) & (q>r)]>(p>r)
7. ~R          6,3, Modus tollens          ---That's [(p>q) & ~q]>~p 

They're all common sense if you think about it and think about them in words
 using "the first", "the second" and "the third", like this:

Disjunctive Syllogism says: [(p v q) & ~p]>q

If you know that (you have the first OR the second) AND (you do NOT have the
first), then you MUST have (the second). 

I think of " > " as the same as the word "guarantees".

Modus Ponens says: [(p>q) & p]>q

If you know that (the first guarantees the second), AND you know that you have
(the first), then you MUST have (the second).

Hypothetical Syllogism says: [(p>q) & (q>r)]>(p>r)

If you know that (the first guarantees the second) AND (the second guarantees
the third), then (the first MUST guarantee the third).

Modus Tollens says: [(p>q) & ~q]>~p 

If you know that (the first guarantees the second) and (you do NOT have the
second), then (you MUST NOT have the first).

Edwin