Is it possible to set up a resource so that the start of the lease can be chosen only within 4 days from the moment of booking, and the end of the lease is any?

Yes. Add a new work rule. The operating time, rental intervals and color of the new rule must match the old rule. In the new rule, remove the checkbox Show in colors section. Set the checkbox Ends (from the moment of booking) after 4 days.
After that, place a new work rule over the old one.
Now add a tip type Error. Specify the message that the booking is possible no later than 4 days. Remove the checkbox Apply even if there are missing rental intervals.
Include a tip only on the underlying (old) rule of work.
As a result, when booking at the intersection of the rules, the tip will not be displayed. And when booking only on the old rule (that is, for example, from the 5th to the 10th day from the current moment), an error will be displayed.
See an example in action:
Didn't find the answer to your question?
Ask it to us on