Do all existence proofs proceed indirectly (a.k.a "by contradiction," "reductio ad absurdum," etc.)?
For example, can we prove directly that in a field the additive identity element and the multiplicative identity element are distinct? I have not found any proof for this that proceeds directly.
So I'm looking for some similar existence lemma that might provide hints by analogy?
If I'm on a fool's errand (direct proof is impossible), please explain or prove this.