Double Diagonal Power Tower

Algebra Level 3

Let A=222 A=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}} and B=2 ⁣ ⁣2 ⁣ ⁣ ⁣ ⁣ ⁣2 B = ^{\sqrt[{}^\ddots\!\!\sqrt{2}]{2}}\!\!\!\!\!\sqrt{2}

Then it is true that:

Note: BB is not a nested tetration.

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