A proprium, for instance 'capable of laughing', implies the existence of something else, namely the existence of a certain ontologically independent entity as its appropriate subject, while it (directly) means some ontologically dependent entity (namely some quality).
This will certainly be confirmed by Classical Metaphysics.
However, according to our findings above, summarized in the diagram :
per se determinations such as 'capable of laughing' already belong to the subject itself (the carrier-only, the substrate). As such it is only a determination of the ultimate substrate or subject, namely prime matter.
'Capable of laughing' therefore (and all essential 'determinations' for that matter) is not a determination-as-a-result-of-inherence (because inherence implies the determination to be replaceable), but an essential part. It differs from that to which a difference refers by the fact that an essential 'determination' is (necessarily) caused by the Essence, and therefore resides in the phenotypic domain of the thing, while the Essence itself resides in the thing's genotypic domain and thus that to which the difference refers also resides in the genotypic domain because the difference is a genuine aspect of the Essence (it is not caused by the Essence).
As we just saw, the best distinction between that to which a difference refers and that to which a proprium refers is that the former belongs in the genotypic domain of the thing, while the latter belongs in the phenotypic domain.
And indeed, CAPABLE OF LAUGHING is embodied in certain persistent material substructures of a human body (these substructures are co-generated by the dynamical law) and is thus as such phenotypic.
RATIONAL, which is supposed to be signified by the difference, is, it is true, also phenotypic. But if we define a difference to be a part of the dynamical law, namely the over-forming of the general dynamical law ( In the case of MAN this general law is the law with respect to just [generating] ANIMAL as generic part of MAN ), then RATIONAL (as such reducible to a certain material substructure in the human body, especially the brain) is only the effect, product, ore phenotypic expression of that part of the dynamical law that is the over-forming of its generic part, and which we have called the difference.
This difference is (as is also the generic part) a non-replaceable content, residing in the genotypical domain of the given being or thing.
The dynamical law as non-replaceable genotypical content generates phenotypical generic non-replaceable features, and phenotypical specific non-replaceable features. The latter are features -- necessary determinations -- such as RATIONAL and CAPABLE OF LAUGHING (that is the material structures to which they can be reduced), and they are generated by that part of the dynamical law that we have called the difference.
Also other determinations are generated by both the generic and the difference part of the dynamical law (and thus are phenotypical), but they are not persistent, and some non-persistent determinations are only partly generated by the dynamical law (for example the being-tanned of Socrates). All these other determinations, that is all non-persistent determinations, are the replaceable determinations, and only they imply a substrate (a carrier-only). The persistent determinations (essential determinations), on the other hand, do not imply a direct substrate (only an ultimate substrate, which is prime matter). They already belong to this substrate.
So, summarizing, we can say that, with respect to MAN, the features RATIONAL and CAPABLE OF LAUGHING are ontologically equivalent. Both are generated by the dynamical law, both are permanent and so do not involve inherence, both are phenotypic essential parts, both are generated by that part of the dynamical law that we call the difference. This difference is the over-forming of the generic part of the dynamical law, and so is a genotypic essential part. In the definition of MAN we can (phenotypically) let represent this difference by either the feature RATIONAL or the feature CAPABLE OF LAUGHING.