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Since we don't have N, we can't use DR1, DR2, DR3 because they were all derived from N. In system K, K was an axiom, so we can't use K either without proving it first. Here are the axioms and transformation rules for S1:

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We can't use any theorems of K because we don't have K or N. So how do we prove N here?

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I shall give a proof sketch hoping that you would find it quite appealing to intuition and steering clear of the usual tricky steps:

You have already

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and it is done for theorems of propositional logic.

You can prove its converse by reductio ad absurdum:

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Hence, all the S1-theorems with necessity prefix are constructed by theorems of propositional logic. For those, you have

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that is,

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and the substitution rules.

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