Let the mapping from level 1 to level 2 be defined by the
narrowest mapping possible. That is, of all the objects
that exist in your representation that represent supersets
of a given level 1 interval, let the mapping be to some
object which is a subset of all of them. There may be
many such mappings. And the are all, of course, equal in
the above sense.
Now, with mappings back & forth we can define arithmetic.
Let any operation be defined by mapping its operand(s) to
level 1, applying that operation there, finding the
contiguous hull of the result there & mapping that result
back into your format in the tightest sense as defined
above.
That gets us most of it. Doesn't it?