~te wa ～ては

In Japanese, ~te wa ～ては is the the te-form of a verb or adjective plus the wa は particle. This can have two different functions.

Auxiliary Focus

The te-form of verbs can connect to an auxiliary verb or auxiliary adjective.

kare wa tabete-iru?
彼は食べている？
Him, eating is?
He is eating?
Is he eating?

Above, we tabete connects to the auxiliary verb iru いる, forming its te-iru form.

The wa は particle can be inserted in the middle to mark as topic and focus the phrases tabete and iru.

tabete wa iru kedo
食べてはいるけど
Eating, [he] is, but... (he isn't doing something else.)
- This is a contrastive wa は.

Another example:

ganbatte hoshii
頑張ってほしい
[I] want [you] to try your best. To try hard. To put effort.
- hoshii
  ほしい
  [I] want [you] to do something. (auxiliary adjective.)
ganbatte wa hoshii kedo muri wa shinai de
頑張ってはほしいけど無理はしないで
[I] want [you] your best but please don't do [something] unreasonable.
- muri shinai de means, for example, that the speaker doesn't want the listener to get sick from working too much, or ending up dying from trying too hard, etc.

ai shite-iru kedo ai sarete wa inai
愛しているけど愛されてはいない
[I] am loving but being loved [I] am not.
- —Excel Saga Opening Song.

Condition

A second function of ~te wa ～ては is creating a condition for which the following statement holds true. For example:

wasurete wa komaru
忘れては困る
Forgetting: be troubled.
[I'll] be troubled if [you] forget [about it].
- komaru
  困る
  To be trouble [by something]. To be inconvenienced.

Normally, the marked verb is in reference to something that did happen, or might happen, or someone talked about recently.

For example, if an assassin is tasked with protecting a guy from other assassins, and the guy is having a nervous breakdown screaming he's going to get killed, the assassin may say:

korosarete wa komaru
殺されては困る
Be killed: be troubled.
[I'll] be trouble if [you] are killed.

After all, he was hired to keep the guy alive.

Obviously, you can use this te wa with all sorts of verbs besides komaru, it just tends to be used with komaru a lot. If you wanted to, you could say:

korosarete wa iki-kaeru
殺されては生き返る
Be killed: living-return.
If [you] are killed [you'll] return to life.
[You'll] revive in the case [you] are killed.
- Is this re:zero?

Although this function is notably used with conditions and hypothetical cases—if, in case of, etc.—it's actually not much different from a simple topic-comment structure talking about something regarding the verb.

For example, this can be noticed in phrases like:

tsuduite wa
続いては
[Regarding] the continuation.
Continuing...
- Phrase used before continuing talking about something.
sore ni tsuite wa
それについては
[Regarding] attached to that.
About that...
- sore ni tsuite means literally "attached to that," but less literally in respect to information "attached" to that, about that.

The ~te wa ～ては is also commonly used with dame 駄目, "no good," ikenai いけない, "can't go," and naranai ならない, "won't be." In any of these cases, it means that the verb shouldn't be done. For example:

hito wo koroshite wa dame
人を殺してはダメ
Killing people: no good.
Killing people is bad. (don't do this.)
hito wo koroshite wa ikenai
人を殺してはいけない
(same meaning, but more serious.)
hito wo koroshite wa naranai
人を殺してはならない
(same meaning, but even more serious.)

In some cases, the part that comes after the wa は is omitted.

watashi mo ganbarakute wa!
私も頑張らなくては！
If I, too, don't work hard... [it's no good]!
I, too, should work hard!
I gotta try my best, too!
- Phrase used when a character sees someone else working hard and gets fired up.

This "no good" usage is often seen with negative phrases. For example:

soitsu wo taosanakute wa naranai
そいつを倒さなくてはならない
Not defeating that guy: won't be.
"Not defeat that guy" won't be.
Must not "not defeat that guy."
Must defeat that guy.

In this case you have a double negative. You're saying that "not doing something is bad." Thus, you must do that something, otherwise it's going to be bad.

Contractions

There are a few contractions associated with ~te wa ～ては.

~cha ～ちゃ
From ~te wa ～ては.
- ganbaranakucha
  頑張らなくちゃ
  Gotta try [my] best! (again!)
-ja ～じゃ
From -de wa ～では, because the te-form of the da だ copula is de で, and because the te-form of verbs undergoing renjoudaku 連声濁 also end in de で. The de で particle marked by wa は also contracts to this.
- kirei da
  綺麗だ
  [It] is pretty.
- kirei de wa aru ga
  綺麗ではあるが
  Pretty, [it] is, but.
- kirei ja aru ga
  綺麗じゃあるが
  (same meaning.)
- shinde wa dame!
  死んではダメ！
  Dying: no good!
  It'd be no good if [you] died!
  It'd be bad if [you] died!
  Don't die!
- shinja dame!
  死んじゃダメ！
  (same meaning.)
- koko de hataraku
  ここで働く
  To work here.
- koko de wa hatarakanai
  ここでは働かない
  Here, [it] won't work.
- koko ja hatarakanai!
  ここじゃ働かない！
  (same meaning.)
~kya ～きゃ
From ~kute wa ～くては, the -te form of i-adjectives.
- ganbaranakya
  頑張らなきゃ
  Gotta try [my] best! (one more time!)
- takakute wa kaenai
  高くては買えない
  Expensive: can't buy.
  [I] can't buy [it] if [it's] expensive.
- takakya kaenai
  高きゃ買えない
  (same meaning)