if we are to prove

X-(X-A) = A,

what we normally do is for an arbitrary x suppose ,

x E X-(X-A)

--> proof

--> x E A

and then

x E A

--->proof

---> x E X-(X-A)

this tells us X-(X-A) = A,

its perfect and theres no wrong with it.

but my question is what happens if I use biconditional statement as

x E X-(X-A) <---> x E X AND x ~E X -A

<---> some proof here

<---> x E A

then we dont want to split it into two parts as did earlier.

but why dont we normally prove like this???