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J. Let, J k be three join points and µ i =< µ i0 , µ i1 , µ i2 >, µ j =< µ j0 , µ j1 , µ j2 > and µ k =< µ k0 , µ k1 , µ k2 > the three isomorphisms associating respectively P to J i , J j and J k

F. If-e and ?. J. Belong, let e be the corresponding event in P such that e = µ i1 (e ), then according to the definition of ?, e succeeds to µ j1 (e ) and µ k1 (e ) Therefore, e and f also belong to In this way, we easily show that (J i ? J j ) ? J k = J i ? (J j ? J k ) In the same way, we also show that Finally, to prove the property (4), let us consider the two join points J i and J j and their associated morphisms µ i and µ j . Let e 2 and f 2 be two events belonging to J j and J i ? (J i ? J j ) (and consequently to J i ? J j ) but not to J i . Let us note e the event belonging to P such that e 2 = µ j1 (e ), If e 1 = µ i1 (e ), then since e 2 belongs to J i ? J j , e 2 ? e 1 , and since e 2 belongs to J i ?