sets theory-Realation proof question Let $A,B,C$ and $R,S,T$ be sets. 
And assume that

$$ R \subseteq A \times B, ~~ S \subseteq B \times C, ~~
T \subseteq B \times C.$$

Then, I want to show that 
$$
\begin{equation}
(S \circ R) \cap (T \circ R) \subseteq (S\cap T) \circ R, \\
(S \circ R) \cup (T \circ R) = (S \cup T) \circ R.
\end{equation}
$$

I tried solving this for hours.
I have no idea to even handle this.
I would love any kind of help or assistence.
Thank you.
If a(R o (S $\cap$ T))c, then exists
b in B with aRb and b(S $\cap$ T)c.
Thus aRb, bSc and bTc;  a(RoS)c, a(RoT)c.
In conclusion, R o (S $\cap$ T) subset RoS $\cap$ RoT.  

a(R o (S $\cup$ T))c iff exists b in B with
aRb, b(S $\cup$ T)c iff aRb and (bSc or bTc)
iff (aRb and bSc) or (aRb and bTc)
iff a(RoS)c or a(RoT)c.
In conclusion R o (S $\cup$ T) = RoS $\cup$ RoT.  

Note that I'm using the conventions aRb for (a,b) in R and
for composition RoS ={ (a,c) : exists b with (aRb and bSc) }
If a(R o (S $\cap$ T))c, then exists
b in B with aRb and b(S $\cap$ T)c.
Thus aRb, bSc and bTc;  a(RoS)c, a(RoT)c.
In conclusion, R o (S $\cap$ T) subset RoS $\cap$ RoT.  

a(R o (S $\cup$ T))c iff exists b in B with
aRb, b(S $\cup$ T)c iff aRb and (bSc or bTc)
iff (aRb and bSc) or (aRb and bTc)
iff a(RoS)c or a(RoT)c.
In conclusion R o (S $\cup$ T) = RoS $\cup$ RoT.  

Note that I'm using the conventions aRb for (a,b) in R and
for composition RoS ={ (a,c) : exists b with (aRb and bSc) }