Can you clarify the exact statement of Open Induction and why it implies the Fan Theorem and is implied by LPO + Zorn's Lemma? (Is this the sequential or analytic LPO?) It might also be worth while, for the completeness of MathOverflow, to copy the statement of Zorn's lemma from Johnstone, because I assume there are dozens of natural statements classically equivalent to each other that might be called “Zorn's lemma” and which are not constructively equivalent. Generally speaking, in constructive math, I think it is wiser to always write down every statement explicitly.
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