Double Diagonal Power Tower

Algebra Level 3

Let \[A=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}\] \[B=^\sqrt[{}^\ddots\!\!\sqrt{2}]{2}\!\!\!\!\!\sqrt{2}\] Then it is true that:

Note: \(B\) is not a nested tetration.