He mathematically proved that a manager should not have more than 4-5 subordinates.

He tried to mathematically prove to himself that he would not run from a battle.

He proved mathematically and empirically the effectiveness of the design.

We proved mathematically that there is absolutely no effective way to protect against computer viruses in software.

The former is mathematically proven, subject to the practical availability of random numbers in sufficient quantity.

This is a consequence of Noether's theorem, which can be proven mathematically.

This can be proven mathematically using the Euler characteristic and other mathematical concepts.

However, an active attacker can break the system using a chosen ciphertext attack, as has been mathematically proven.

Process of elimination does not automatically disregard what has not yet been mathematically proven.

This calculation of achievable combinations has not been mathematically proven and can only be considered an upper bound.