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\newtheorem{theorem}{Theorem}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conj}[theorem]{Conjecture}
\newtheorem*{claim}{Claim}

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\title{Lecture 1: INSERT TITLE}

\author{INSERT YOUR NAMES}

\date{January 5, 2009}
\maketitle

Insert content of lecture.

\section{Strong exchange property}

\begin{theorem}
Let $w=s_1 s_2 \cdots s_k$ be a reduced expression for $w\in W$ with
$s_i\in S$ and let $t\in T$. Then $\ell(tw)<\ell(w)$ implies that
\begin{equation}
  tw = s_1 \cdots \hat{s}_i \cdots s_k \qquad \text{for some $1\le i\le k$.}
\end{equation}
\end{theorem}

\begin{proof}
Insert proof.
\end{proof}

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