praperi fu'ivla

x1 is a strict/proper sub-x2 [structure] in/of x3; x2 is a structure and x1 and x3 are both examples of that structure x2 such that x1 is entirely contained within x3 (where containment is defined according to the standard/characteristics/definition of x2; but in any case, no member/part/element that belongs to x1 does not also belong to x3), but there is some member/part/element of x3 that does not belong to x1 in the same way.

If x2 is a (sub)set, then x1 is a proper subset of x3; if x2 is a mathematical/algebraic (sub)group, then x1 is a proper subgroup of x3; etc. Can also be used for describing proper sub-lakes (such as Lake Michigan), proper super-selma'o, and other non-mathmetical usages. x3 is a proper super-x2 of/with x1. Biological taxa, if comparable, are usually/hypothetically proper. See also: klesi, enklesi, cletu, cmeta.

In notes:

enklesi
x1 is an (arbitrary) x2-set (li) of superset x3; x1 is subset/subgroup/subcategory/subclass/vel sim. of x3 with cardinality/size x2.
trajmaumce
x1 (li) is an extremal bound (supremum/infimum/possibly-unattained extremum (loose sense in English)/asymptote (one sense)/best possible bound (one sense)) on set x2 (set) in direction x3 (li) in ordered structure x4; x1 bounds x2 tightly/maximally-strongly from the x3 side in x4; x2 is bounded from the x3 side by x1 and any other bound on that side is worse than is x1.
tsekane
x1 (linear manifold, vector, etc.) is/lies secant to x2 (object, surface, curve, manifold, etc.) passing through or toward points/loci x3 (set of intersected points), by standard/definition/in system x4.
zmaumce
x1 (li) is a bound on set x2 (set) in direction x3 (li) in ordered structure x4; x1 bounds x2 from the x3 side in x4; x2 is bounded from the x3 side by x1.