Under the caveat/understanding that... 怎麼說？

[anonymoose]

How would one translate "under the caveat that..." or "under the understanding that..." into Chinese?

Some examples:

Under the understanding that this experiment is a tentative first attempt, we can glean much information concerning the methods by which cloned cells can be regenerated in the laboratory.

Concerning the EU, under the caveat that the number of studies is smaller, similar conflicting results are found. [*]

This program is free for use by any individual for any purpose, under the understanding that the author of this program is not liable for any damages incurred from the use or mis-use of this program. [*]

Under the caveat that such practices not damage the body, which we honor as a gift of God’s creation, it is reasonable to believe that some people, in some circumstances at some times in their lives, do receive the charism to accept such mortifications as wholesome and profitable both to their own souls and for the building up of the Church. [*]

[semantic nuance]

you may consider '在...前提下'.