Double Diagonal Power Tower

Algebra Level 3

Let $A=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}$ and $B = ^{\sqrt[{}^\ddots\!\!\sqrt{2}]{2}}\!\!\!\!\!\sqrt{2}$

Then it is true that:

Note: $B$ is not a nested tetration.