We can prove this by using Pythagoras theorem. At B we have a right angle.
In the triangle, ABE, we get: AE² = AB² + BE²
In the triangle DBC, we get : CD² = DB² + BC²
adding the above two equations. we get:
AE² + CD² = (AB² + BC²) +( DB² + BE²)
= AC² + DE²
This equation will be true for a quadrilateral with two adjacent sides being perpendicular.