Theorems on intervals of ordered sets

An interval of the totally ordered set (S, Oforu, v @? I, @?"v"@?"IA"v O 2.Theorem 2. Let k be a positive integer and "X(N) =< kfor |N| = k + 1. Then"X(I) =<. 3.Theorem 3. There is J @? Iwith @?"v"@?"JA"v = @?"v"@?"IA"vand"X(J) finite if and only if there is a well-order of I in which, for every x@e @?"v"@?"IA"v, theset {v: x @? A"v} has a last element.