Deduction, Daniel Bonevac, Blackwell Publishing 2003, 2nd edition, Problem 8.3(5)

The argument is very simple:

There are exactly two hemispheres. So, there are at least two hemispheres.

In terms of arguments in everyday English, it merits no attention. From a logical point of view, it is worth noting that we can't switch the sentences around. We cannot infer that there are exactly two hemispheres from the fact that there are at least two hemispheres.

We can of course negate the conclusion and derive a contradiction, which would be the stronger of the two methods of proof, but I will simply drop the universal statement which is part of the premise and do existential generalisation on what is left.

1. (∃x)(∃y){Hx • Hy • ¬ (x = y) • (z)[Hz ⊃(z = x ∨z = y)]}
2. ∴(∃x)(∃y)[Hx • Hy • ¬ (x = y)]
3. (∃y){Ha • Hy • ¬ (a = y) • (z)[Hz ⊃(z = a ∨z = y)] ......... 1 EI x/a
4. Ha • Hm • ¬ (a = m) • (z)[Hz ⊃(z = a ∨z = m) ......... 3 EI y/m
5. Ha • Hm • ¬ (a = m) ......... 4 Simp.
6. (∃y)[Ha • Hy • ¬ (a = y)] ......... 5 EG
7. (∃x)(∃y)[Hx • Hy • ¬ (x = y)] ......... 6 EG