\newcommand{\ds}{\ensuremath{\displaystyle}}

\newcommand{\thm}{\noindent{{\bf \textsc{Theorem.}}~~}}

\newcommand{\prf}{\noindent{{\bf \textsc{Proof.}}~~}}

\newcommand{\prop}{\noindent{{\bf \textsc{Proposition.}}~~}}

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\newcommand{\W}{\mathbb W}

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\newcommand{\B}{\mathscr{B}}

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