Problem with a proof in probability

[DJ_JK]

Homework Statement

I have to prove that for any two events A and B
P(A and B|A) $$\geq$$ P(A and B| A or B)

Homework Equations

P(A and B) = P(A) . P(B|A)
P(A|B) = (P(A) . P(B|A))/P(B)

The Attempt at a Solution

I tried to simplify the left side with this reasoning
P(A and B|A) = P(A) . P((B|A)|A)
= P(A) . P(B|A)
= P(A and B)

My reasoning for going from step 1 to two is that condition A is already fulfilled, and asking for it a second time is needless. My friend however, disagrees with this.
I am having problems with simplifying the right side because I don't know if there is a system of priorities in probability mathematics (e.g. "condition" has a priority over "and", "or" over "condition", ...
Any help in the right direction would be greatly welcomed.

 

[statdad]

Remember that for any two events $$\mathcal{C}, \mathcal{D}$$ you have

$$\Pr(\mathcal{C} | \mathcal{D}) = \frac{\Pr(\mathcal{C} \cap \mathcal{D})}{\Pr{\mathcal{D}}}$$

Use this idea to write out both of the probabilities you need to compare. If you do it correctly you should notice something about the numerators and the denominators, and those items will allow you to argue for the conclusion you need.