How to prove that sentence using constructive proof?

Question

Ok, I admit it's homework, I am supposed to prove some sentences using natural logic, I can do some, but I'm stuck on this one:

¬E→(E→(E→F))

I can prove it using contradiction, but I'm supposed to do it using intuitionistic logic only (no double negation, law of excluded middle, or proof by contradiction).

How to do it?

Answer 1

1. $\neg E$ (Assumption)
2. $E$ (Assumption)
3. $\perp$ ($\rightarrow$E 1,2)
4. $E \rightarrow F$ ($\bot$E / Ex falso quodlibet 3)

and now you discharge the second and first assumption (that is you use $\rightarrow$I). Note, that $\bot$E is NOT the same thing as "proof by contradiction" and a valid rule in intuitionistic logic.