Let $S$ be a complex algebraic surface and let $D_{\lambda}$ be a pencil of curves without fixed part and with an irreducible and reduced divisor in it I would like to see an analytic (or just related to complex geometry not more general schematic fact) proof of the following fact (which I've seen called as the Bertini extended theorem):
The generic element of $D_{\lambda}$ is an irreducible divisor.
I tried to emulate the proof of the Bertini theorem about smoothness but I did not make up anything.