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\begin{frame}
  \small
  
  \vspace{-1ex}
  
  \begin{definitions}
  \begin{itemize}
  \item
  An equation $s \approx t$ is \alert{valid} in the $\Sigma$-algebra
  $\AA$ \quad(\alert{$\AA \vDash s \approx t$})\quad %, \alert{$s =_\AA t$}) 
  if
  \[
  [s,\alpha]_\AA = [t,\alpha]_\AA
  \]
  for all assignments $\alpha$.
  \item<2->
  $\FF$-algebra $\AA$ is \alert{model} of ES $\ES$ if
  $\AA \vDash s \approx t$ for all equations $s \approx t \in \EE$.
  \end{itemize}
  \end{definitions}
  
  \smallskip
  
  \begin{example}<3->
  \begin{itemize}
  \item
  $\Aa = ( \NN, [\cdot] )$
  with \quad $[0] = 0$, \quad $[s](x) = x+1$, \quad $[+](x,y) = x+y$
  \item
  $\Bb = ( \NN, [\cdot] )$
  with \quad $[0] = 1$, \quad $[s](x) = x+1$, \quad $[+](x,y) = 2x+y$
  \item
  ES $\EE$
  \[
  \GREEN{\begin{array}{r@{~}c@{~}l}
  \m{0}+y & \approx & y \\[.5ex]
  \m{s}(x)+y & \approx & \m{s}(x+y) 
  \end{array}}
  \]
  \end{itemize}
  \onslide<4->
  \begin{center}
  $\AA$ is model of $\EE$ \qquad
  \onslide<5->
  $\BB$ is no model of $\EE$
  \end{center}
  \vspace{-2ex}
  \end{example}
\end{frame}