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A multi-set design of order \(v\), \({MB}_t(v, k, \lambda)\), first defined by Assaf, Hartman, and Mendelsohn, is an ordered pair \((V, B)\), where \(V\) is a set of cardinality \(v\) and \(B\) is a collection of multi-subsets of \(V\) of size \(k\) (called blocks), with the property that every multi-subset of \(V\) of size \(t\) is contained a total of \(\lambda\) times in the blocks of \(B\). (For example, the multi-set \(\{a,b,b\}\) is contained \(\binom{2}{1}\binom{4}{2} = 12\) times in the multi-set \(\{a,a,b,b,b,b,c\}\) and not at all in the multi-set \(\{a,a,b,c\}\).) Previously, the first author had pointed out that any \(t\)-multi-set design is a \(1\)-design. Here, we show the pleasant yet not obvious fact that any \(t\)-multi-set design is a \(t’\)-multi-set design for any positive integer \(t’ \leq t\).