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The auxiliary properties of Remark 3.4 below have been used in the proof of Lemma 3.3.
Considering the encoding functions f1 and f2, defined in Definition 1, and the above definitions for auxiliary RVs, we remark that (X1,i, U i ) → T i → (X2,i, V i ) forms a Markov chain.
(Shapiro remarks that this auxiliary assumption seems best to capture the sense of multiple realization stressed by proponents of the standard argument).
Remark Let us now give certain auxiliary functions.
Remarks (i) In both standard DPC [18] and GDPC, the auxiliary random variable is given by.
The system NK has no logical axioms and provides two introduction-elimination rules for each logical connective: A few remarks: First, the expression represents the fact that α is an auxiliary assumption in the proof of γ that eventually gets discharged, i.e. discarded.
Remark 14 As in Theorem 13, by changing the property of the auxiliary function, we get various results (see, e.g., [6 9] and related references therein).
Remark 3.2 Notice that if ( u, u ) is a solution of the auxiliary system (1.3), then u is a solution of the given problem (1.1) under the assumption f ( t, u, w, u ) ≡ g ( t, u, w ).
