On the direct image of intersections in exact homological categories

Given a regular epimorphism f:Xtwo headed rightarrowY in an exact homological category image, and a pair (U,V) of kernel subobjects of X, we show that the quotient (f(U)∩f(V))/f(U∩V) is always abelian. When image is nonpointed, i.e. only exact protomodular, the translation of the previous result is that, given any pair (R,S) of equivalence relations on X, the difference mapping δ:Y/f(R∩S)two headed rightarrowY/(f(R)∩f(S)) has an abelian kernel relation. This last result actually holds true in any exact Malʹcev category. Setting Y=X/T, this result says that the difference mapping determined by the inclusion Tunion or logical sum(R∩S)less-than-or-equals, slant(Tunion or logical sumR)∩(Tunion or logical sumS) has an abelian kernel relation, which casts a new light on the congruence distributive property.