Hey Amysky,

This is a rare version of the COMMAND subjunctive with the idiom "on condition." First, let's look at some more common subjunctives:

I demand that you buy dinner.
I request that you eat lunch.
I require that you play golf.

In these three examples, the verbs instigate a subjunctive structure. We always begin with THAT, then the object of the verb, then the subjunctive form of the verb in question (buy, eat, play). That's formed by taking the infinitive and removing the "to": to buy = buy, to eat = eat, to play = play.

If a verb needs to be passive, we would remove the "to" from the passive infinitive: to be bought = be bought, to be helped = be helped.

In this case, "on condition" signals our structure. We follow it with THAT, then the object (they), then the subjunctive.

Hope that helps!

-t