A = (√2)
B = (√(2-√2))
C = (√(2-√(2+√2)))
D = (√(2-√(2+√(2+√2))))
E = (√(2-√(2+√(2+√(2+√2)))))
F = (√(2+√(2+√(2+√(2+√2)))))

A = sqrt(2) ===> A^2 = 2

B = sqrt(2 - sqrt(2)) 

A/B = sqrt(2)/(sqrt(2 - sqrt(2))) = sqrt(2)*[sqrt(2 + sqrt(2))]/[sqrt(4 - 2)] = sqrt(4 + 2*sqrt(2))/sqrt(2) = sqrt(4 + 2*A)/A 

B^2 = 2 - sqrt(2) = 2 - A

A * B = sqrt(2) * sqrt(2 - sqrt(2)) = sqrt(4 - 2*sqrt(2)) = sqrt (4 - 2*A)

B = B * (sqrt(2 + sqrt(2))/(sqrt(2 + sqrt(2))

B = sqrt(4 - 2)/(sqrt(2 + sqrt(2)) = sqrt(2)/(sqrt(2 + sqrt(2))

B = A / (sqrt(2 + sqrt(2))

sqrt(2 + sqrt(2)) = A / B.

C = sqrt(2 - A/B)

E * F = D (already showed in previous comment)

D * E * F = D^2 = 2 - sqrt(2 + A/B)

P = A * B * sqrt(2 - A/B) * [2 - sqrt(2 + A/B)]

P = [sqrt(4 - 2*A) * sqrt(2 - A/B)] * [2 - sqrt(2 + A/B)]

P = [sqrt(8 - 4*A/B - 4*A + 2*A^2/B)] * [2 - sqrt(2 + A/B)]

P = [sqrt(8 - 4*A/B - 4*A + 4/B] * [2 - sqrt(2 + A/B)]

P = 2 * sqrt(8 - 4*A/B - 4*A + 4/B) - sqrt(16 + 8*A/B - 8*A/B - 4*A^2/B^2 - 8*A - 4*A^2/B + 8/B + 4*A/B^2)

A^2/B^2 = 2 + sqrt(2) = 2 + A

A/(B^2) = 1 + A

P = 2 * sqrt(8 - 4*A/B - 4*A + 4/B) - sqrt(16 - 8 - 4*A - 8*A - 8/B + 8/B + 4 + 4*A)

P = 2 * sqrt(8 - 4*A/B - 4*A + 4/B) - sqrt(12 - 8*A)

Until here is correct...P = 0,01623...

I'm tired today. I"ll continue tomorrow !