Prove $A\cap (B\cup C) \subseteq (A\cap B) \cup C$ Prove $$\Large{A\ {\cap}\ (B\cup C}) \subseteq (A\ \cap \ B) \ \cup \ C$$
I begin my proof by writing that:

*

*Let $x\in A$ and $x\in (B\cup C)$. then I jump to my next train of thought.

*Let $x\in A$ and $x\in B$ or $x\in A$ and $x\in C$. Here I get a bit confused...

I know that all proofs follow the if.... then methodology.
So then I would write that If Line 1., Line 2, some intermediate Line, then $x \in (A\cap B)$ or $x \in C$.
How would I gain access to those intermediate steps? I know after Line2 I could possibly reorganize it but I don't quite know how the distributive property would work.
 A: Short answer: Case distinction.
A bit longer answer: In your second line you notice that one of two things is true: You might have $x \in A$ and $x \in B$ or you might have $x \in A$ and $x \in C$ (or you might have both). Since you don't know which of the two is the case for your generic $x$, you have to show that your desired conclusion $x \in (A \cap B) \cup C$ follows in both cases.
Formally, this is done by a case distinction, i.e.:
Case 1: If $x \in A$ and $x \in B$, then ... intermediate steps ... which shows $x \in (A \cap B) \cup C$
Case 2: If $x \in A$ and $x \in C$, then ... (different) intermediate steps ... which shows $x \in (A \cap B) \cup C$

An additional, stylistic, remark: In mathematical proofs "let" is used for a very specific purpose -- namely, introducing new objects. So, using "let" in your first line of thought is correct. In your second line, however, you shouldn't use "let" since you aren't introducing a new $x$ but rather deducing a fact about the x you introduced prior. So, I would just say "Then $x \in A$ ..." here.
A: First of all, I would start with saying "let $x\in A\cap(B\cup C)$". Your line 1 follows easily, but it's important that your proof follow the form $x\in X\to x\in Y$ to show that $X\subseteq Y$.
Next, you have the right idea for line 2, but you've already defined $x$. Get rid of the "let" in that statement, and replace it with something like "it must be the case that". You're showing a logical step, not a new definition.
Now you have a disjunction (aka OR statment), and you can finish the proof by cases. You know $x\in A$. Show that if $x\in B$, then $x\in(A\cap B)\to x\in(A\cap B) \cup C$. Show that if $x\in C$, then $x\in(A\cap B) \cup C$.
