Proof of New Implicational Relationships between Generalized Symmetries (appendix for journal paper)

This note provides proof of some new implicational relationships between generalized symmetries. These relationships are formulated in terms of twelve symmetry types. Six of these symmetries are denoted T_n^x_i,x_j where the index n?[1,6] indicates that a specific co-factor equivalence property holds between the variables x_i and x_j. The other six symmetries are denoted neg T_n^x_i,x_j, and indicate that one co-factor is equivalent to the negation of the other. The relationships that are specified take the form, if T_p^x_i,x_j and T_q^x_j,x_k hold then T_r^x_i,x_j holds where T_p,T_q and T_r denote one of these twelve symmetry types.