Should numeration of a list of axioms be continued? I have a definition like the following:

Definition System of sides is ... conforming to axioms:


...
...
...


Definition Monotone system of sides is a system of sides conforming to additional axiom:


...


Definition Distributive system of sides is a system of sides conforming to additional axiom:


...


My question: Should I continue numbering of axioms as above, or should the additional axioms be numbered from one (not from four or five as above)?

Also note, that every distributive system of sides is a monotone system of sides.
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:

...
...
...
...


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:

...

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

...


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;