1. On the importance of the relation [(A,B.), (A.C.)] (A, [B,C,) (C.A.), (A,B)] between three elements of a structure ; 2. Komplexe euklidische Raume von beliebiger endliche oder transfiniter Dimensionszahl ; 3. Intrinsic topology and completion of Boolean rings ; 4. Uber die Dimension linearer Raume.

While the Dedekind axiom is identical with its dual counterpart the corresponding assertion about the equation (2) is not true. THEOREM 11. In our structure A the equation (10) is generally valid. The proof follows easily from the dual counterpart of Theorem 9. Of course Theorem 11 can also be verified simply by the table on p. 576: THEOREM 12. The assertion that in a structure the equation (2) holds for arbitrary elements, A, B, and C, and the assertion dually corresponding to this assertion are not equivalent. PROOF. See Theorem 11.