Semantics of MML Query - Ordering

In this paper X denotes a set, R, R1, R2 denote binary relations, x, y, z denote sets, and n, m, k denote natural numbers. Let us consider a binary relation R on X. Now we state the propositions: (1) fieldR ⊆ X. (2) If x, y ∈ R, then x, y ∈ X. Now we state the propositions: (3) Let us consider sets X, Y . Then (idX)Y = X ∩ Y . (4) 〈x, y〉 ∈ R |2 X if and only if x, y ∈ X and 〈x, y〉 ∈ R. (5) dom(X R) ⊆ domR. (6) Let us consider a total reflexive binary relation R on X and a subset S of X. Then R |2 S is a total reflexive binary relation on S. The theorem is a consequence of (4). Proof: Set Q = R |2 S. domQ = S. (7) Let us consider transfinite sequences f , g. Then rng(fag) = rng f∪rng g. Let us consider R. Let us note that R is transitive if and only if the condition (Def. 1) is satisfied.