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\begin{document}
\begin{equation*}
\aligned
  y(t) &= H(t-t_0) u(t_0) + H(t-t_1) \bigl( u(t_1) - u(t_0) \bigr) + \cdots \\
    &= \bigl( H(t-t_0) - H(t-t_1) \bigr) u(t_0) + 
       \bigl( H(t-t_1) - H(t-t_2) \bigr) u(t_1) + \cdots \\
    &= \sum_{n=0}^{\boldmath{t_n < t}} \bigl(H(t-t_n) - H(t-t_{n+1}) \bigr) u(t_n) \\
    &= \sum_{n=0}^{\boldmath{t_n < t}} \frac{H(t-t_n) - H(t-t_{n+1})}{t_{n+1}-t_{n}}
       u(t_n) \bigl( t_{n+1}-t_{n} \bigr).
\endaligned
\end{equation*}
\end{document}