  Normal monomorphisms in the sense of Bourn describe the equivalence classes
of an internal equivalence relation. Although the definition is given in the
fairly general setting of a category with finite limits, later investigations
on this subject often focus on protomodular settings, where normality becomes a
property. This paper clarifies the connections between internal equivalence
relations and Bourn-normal monomorphisms in regular Mal'tesv categories with
pushouts of split monomorphisms along arbitrary morphisms, whereas a full
description is achieved for quasi-pointed regular Mal'tsev categories with
pushouts of split monomorphisms along arbitrary morphisms.
