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.