"The final proof requires that you isolate the virus from the bats."

In such a system each proof requires its own fresh common reference string.

But the ultimate proof, he said, required an experiment in trees in the open air.

While a proof of T13 would require induction, the intuition underlying it should be clear.

This proof requires the most basic elements of group theory.

"Does her father require proof she is still maiden?"

A proof of this requires checking four cases to demonstrate that no 3-edge-coloring exists.

We can discuss at that time what proofs you will require.

The proof of this statement requires some calculus and analytic number theory.

But some of the results are still frustratingly erratic, he added, and hard proof will require more research.