Mean and expected value of normal distribution

[bibhas2]

Page: https://chrispiech.github.io/probabilityForComputerScientists/en/part2/normal/

States the following:

By definition a Normal has $E[X] = \mu$.

In fact this equality requires a proof. The proof can be seen here.

https://statproofbook.github.io/P/norm-mean.html

I do not see how this equality can be asserted from any kind of definition without making huge leaps of assumptions.

[alex-stoica]

1. This fact is well-known. I personally see no point in demonstrating every foundational result
2. You referenced quite a cumbersome demonstration. Actually, proving symmetry will suffice