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. Proof and ?. We-obtain-m-"-modp?, Also note that Modp?q ? Modpµq. We get Modp?pµ^ ?qq " Modp??q " f pModp? ? ?qq " f pttau, ta, b, euuq " N (N is closed under ^, f pN q " N holds by definition of refined operators), but Modpp?µq ^ ?q " f pMq X Modp?q " ttauu. Otherwise tau R f pMq. Since f pMq ? H and f pMq is closed under ^, by symmetry of the role played by the variables b and c, it is sufficient to examine three possibilities for f pMq: either f pMq " tta, Let ? and µ in LHorn such Modp?q

^. Modpp?µq and . ?q, thus proving that p?µq ^ ? is satisfiable, whereas ?pµ ^ ?q |ù p?µq ^ ? in LHorn

. For-?-p-t?f and ?. ?w-u,-we-have-m-"-modp?, Let us consider the possibilities for Modp?µq " f pMq. By definition of refined operators, we know that ta, bu R f pMq since ta, bu R Clmaj 3 pMq. We consider two cases: First assume tb, cu P f pMq: Let ? be such that Modp?q " ttb, cu, ta, buu " N . Clearly such a ? exists in LKrom . Besides note that Modp?q ? Modpµq. We get Modp?pµ ^ ?qq " Modp??q " f pModp? ? ?qq " ttb, cu, ta, buu " N , whereas Modpp?µq ^ ?q " ttb, cuu. Otherwise, we have tb, cu R f pMq. Since f pMq ? H and f pMq is already closed under maj 3 , by symmetry of the role played by the variables a, For L 1 " LKrom , the formulas ?, µ P LKrom with Modp?q

, It is then clear that in any case Modpp?µq ^ ?q ? H and Modp?pµ ^ ?qq ? Modpp?µq^?q, thus showing eventually that p?µq^? is satisfiable, whereas ?pµ^ ?q |ù p?µq ^ ? in LKrom

. For-l-1-"-l-affine, Observe that the set of models of µ is the set of solutions of the following equations system

?. We-have-m-"-modp? and . For-?-p-t?f,

. First and . Clearly, L affine . Also note that Modp?q ? Modpµq. We obtain on the one hand Modp?pµ ^ ?qq " Modp??q " f pModp? ? ?qq " tta, buu and on the other hand Modpp?µq ^ ?q contains tta

. Cl^ptta, If f pMq " tta, buu or f pMq " tta, b, c, euu, we consider the formula ? such that Modp?q " tta, bu, ta, b, c, euu. Clearly, such a ? exists in L affine . We obnot closed by intersection. Observe that Cl ? pModp? ? µ1qq is at distance 1 from Modp? ? µ1q, and hence Cl^pModp? ? µ1qq P FppModp? ? µ1qq. Thus, Modp? ? Prox^ µ1q, Since f pMq ? H and f pMq is closed under '3, by symmetry of the role played by the variables d and e, it is sufficient to distinguish four cases for f pMq: either f pMq " tta, buu or f pMq " tta

T. , It induces the following lexicographical order on the sets of models of FppModp? ? µ2qq: tta, bu