On the direct image of intersections in exact homological categories

Given a regular epimorphismf:X?Y in an exact homological categoryC, 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. WhenC 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 mappingδ:Y/f(R∩S)?Y/(f(R)∩f(S)) has an abelian kernel relation. This last result actually holds true in any exact Mal'cev category. SettingY=X/T, this result says that the difference mapping determined by the inclusionT∪(R∩S)?(T∪R)∩(T∪S) has an abelian kernel relation, which casts a new light on the congruence distributive property.