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\begin{document}\Century{1100}

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\Dimostrazione {Siano\m

$A=\Matrice{a & b\\c & d}$ \qquad e \qquad
$B=\Matrice{\alpha & \beta\\\gamma & \delta}$

\NI Allora\m

$AB=\Matrice{a\alpha+b\gamma & a\beta+b\delta\\
       c\alpha+d\gamma & c\beta+d\delta}$

\NI mentre per $z\in\CC$ abbiamo

\Align {f_A\circ f_B (z) &= f_A(\Frac{\alpha z+\beta}{\gamma z + \delta})
= \Frac{a\Frac{\alpha z+\beta}{\gamma z + \delta}+b}
  {c\Frac{\alpha z+\beta}{\gamma z + \delta}+d}\\
&=\Frac{a(\alpha z+\beta) + b(\gamma z+\delta)}
     {c(\alpha z+\beta) +d(\gamma z+\delta)}
=\Frac{(a\alpha +b\gamma)z+a\beta+b\delta}
  {(c\alpha+d\gamma)z+c\beta+d\delta} = f_{AB}(z)}}


\end{document}