The total cross section of the process $\gamma e\to eWW$ at energies up to 2 TeV has been calculated. The contributions to the process come from the three- and fourboson vertices ($\gamma WW$, $ZWW$, $\gamma \gamma WW$ and $\gamma ZWW$). The large magnitude of the cross section suggests that this will be one of the most interesting processes in studying the structure of the electroweak sector of the Standard Model using the linear colliders that are now being designed.
$^1$Institute of Mathematics, Siberian Division of Russian Academy of Sciences, Novosibirsk
$^2$Research Institute of Nuclear Physics, Moscow State University, Leninskie Gory, Moscow, 119992, Russia
$^3$Novosibirsk State University
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