Proof of Lower Bound for Deterministic Distinct Elements Algorithm
Consider the following argument:


  If $0 &gt; 1$ then $1 &gt; 2$.


Proof: Starting with $0 &gt; 1$, add $1$ to both sides.

Since it is not true that $1 &gt; 2$, we can conclude that $0 &gt; 1$ is also false.

The same is true in your example. Starting with some assumption A, you prove property B. However, property B is clearly false, so assumption A must have been false as well. Don't forget that you can derive everything from False.