Should numeration of a list of axioms be continued?
I would recommend thinking about this presentation problem in the following way: if you were inclined to be verbose, it would be perfectly correct to say:

Definition Monotone system of sides is a system of sides conforming to the axioms:

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As such, it is not so much that you wish to continue a list of axioms, as it is that you wish to take advantage of the similarity of the two lists of axioms to present this information in a way that is more streamlined and easier to understand.
As such, I would recommend making the elision explicit, as in:

Definition A monotone system of sides conforms to the same axioms 1-3 for a system of sides, plus an additional axiom:

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Definition A distributive system of sides conforms to the same axioms 1-4 for a monotone system of sides, plus an additional axiom:

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Notice also that I have extended the italics to the whole statement &quot;X system of sides,&quot; rather than just the adjective, since you are not defining e.g., &quot;monotone&quot; but rather &quot;monotone system of sides.&quot;