Are you able to read and understand it though and is it the correct use of the principle of duality?

Yes. You’re right. I should have written it like this: Let A,B,C, and K be elements of the Natural Numbers. 1. ∃K(K≤1∧∃A,B,C(A^K+B^K=C^K )) 2. ∃K(∃A,B,C(A^K+B^K=C^K )∧K≤1) 3. ∃K(∃A,B,C(A^K+B^K=C^K )→K≤1) 4. ∃K(K>1→∀A,B,C(A^K+B^K≠C^K ))

Premise 4 is trivially true for the same reason premise 3 is: choose K=1, then K>1 is false, and false implies anything.

Suppose I would pick K=2 though. What would happen then?

#3 and #4 say there exists a K for which the interior statement is true.

K=2 doesn't make the interior true for either, so choosing it doesn't tell us whether #3 and #4 are true or false.

However, if we choose K=1 then we prove #3 and #4 are true.