x_{1} (li) is a bound on set x_{2} (set) in direction x_{3} (li) in ordered structure x_{4}; x_{1} bounds x_{2} from the x_{3} side in x_{4}; x_{2} is bounded from the x_{3} side by x_{1}.

x_{2} should be a subset (not necessarily proper) of the set underlying x_{4}; it may contain element which are infinite, in general. A bound on a function is really a bound on the image (set; x_{2}) of the function's domain set under the operation of the function; otherwise, it is sumti-raising. x_{3} accepts either a directional vector (normalized) / a point on the unit circle (in which case the direction from the origin to that point is the direction which is being considered), or exactly one of li ma'u and li ni'u; it is unlikely that any other input would be acceptable.
"ma'u" causes the bound to be an upper bound (a bound on the positive side or the 'right'/'above' when plotted using typical Western European conventions) - it is a number which exceeds/is greater than or equal to every element of x_{2} in the set underlying x_{4} according to the order endowing x_{4}. "ni'u" causes the bound to be an lower bound (a bound on the negative side or the 'left'/'below' when plotted using typical Western European conventions) - it is a number which is less than or equal to every element of x_{2} in the set underlying x_{4} according to the order endowing x_{4}. In order to be well-defined, all elements of x_{2} united with the singleton of x_{1} must be ordered according to the ordering endowing x_{4}; if x_{4} lacks a good order, then x_{1} is undefined. x_{4} must be an ordered structure, not simply a set. Warning: If x_{3} is explicitly filled, that is the only direction which is to be assumed. If it is not explicitly filled, only one (or, arguably, possibly zero) direction(s) is(/are) to be assumed; in this context, it is to be inferred that the set x_{2} is (possibly) bounded on at least one side/in at least one direction. Connecting arguments by 'AND' in x_{3} implies that x_{2} is the set formed by the singleton of x_{3}; it is NOT the equivalent to the English term "bound(ed)" without qualifiers/adjectives (for that meaning, use modulus/absolute value/norm on the input of x_{1}). The bound is not strict (proposal: use praperi for this sense), but must be true of all elements of x_{2} in that direction (x_{3}); it is not necessarily the extremal/best finite bound on x_{2}. x_{1} is typically finite; submitting any infinity for it is arguably bad form but would imply that no element of x_{2} is infinite on the side determined by x_{3} (where "infinite" has meaning determined also by x_{4}; again, for "finite", use modulus/absolute value/norm) - no other information (about its being bounded or unbounded) can be discerned (unless the sumti in x_{3} and/or x_{4} are incompatible). For explicit statement of unboundedness (in a more general sense, possibly including ill-definition of what "boundedness" even means in this context) in the direction determined by x_{3}, use zi'au as the sumti of x_{1} (confer earlier note about compatibility of definition and ordering of x_{4} with x_{2}). x_{4} can be more exotic than simply the field of real numbers.

- trajmaumce
- x
_{1}(li) is an extremal bound (supremum/infimum/possibly-unattained extremum (loose sense in English)/asymptote (one sense)/best possible bound (one sense)) on set x_{2}(set) in direction x_{3}(li) in ordered structure x_{4}; x_{1}bounds x_{2}tightly/maximally-strongly from the x_{3}side in x_{4}; x_{2}is bounded from the x_{3}side by x_{1}and any other bound on that side is worse than is x_{1}.