Warum muss man bei Aussagen die mit “<=>” verknüpft sind, beide Richtungen beweisen?

This double arrow just says.
Let’s take the example:
If it rains => then the soil becomes wet.
The arrow is valid only in one direction. So a follow-up. For: If the soil becomes wet, it does not necessarily have to rain. Maybe someone just pours flowers. Then the floor becomes wet. So this is enough to prove things in one direction.

In the case of an equivalence <=>, however, it is not “followed” but “almost when”.

Example of mathematics

Equivalence:

if 2x=4 <=> x=2

Conclusion:

if x=2 => x2=4

but not x2=4 => x=2 (because x could also be = -2).
I hope I could explain the difference a little.