if \{ U_ i \to U\} _{i \in I} is a \tau -covering, then there exist
a \tau '-covering \{ V_ j \to U\} _{j \in J},
a \tau -covering \{ f_ j : W_ j \to V_ j\} consisting of a single f_ j \in \mathcal{P}, and
a \tau '-covering \{ W_{jk} \to W_ j\} _{k \in K_ j}
such that \{ W_{jk} \to U\} _{j \in J, k \in K_ j} is a refinement of \{ U_ i \to U\} _{i \in I}.
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