  We propose an optimal Fermi-averaging for an elementary
$\pi^+$+$n$$\to$$K^+$+$\Lambda$ t-matrix under the on-energy-shell condition,
in order to describe ($\pi^+$,$K^+$) reactions on a nuclear target in the
framework of a distorted-wave impulse approximation. We apply it to calculate
$\Lambda$ quasi-free spectra from $^{12}$C($\pi^+$,$K^+$) reactions at $p_\pi$=
1.20 GeV/c and 1.05 GeV/c, and compare them with experimental data. The results
show that the calculated spectra are in excellent agreement with the data,
because the energy-dependence originates from the nature of the optimal
Fermi-averaging t-matrix.
