  B\'ezout's theorem, nonequivariantly, can be interpreted as a calculation of
the Euler class of a sum of line bundles over complex projective space,
expressing it in terms of the rank of the bundle and its degree. We give here a
generalization to the $C_2$-equivariant context, using the calculation of the
cohomology of a $C_2$-complex projective space from an earlier paper. We use
ordinary $C_2$-cohomology with Burnside ring coefficients and an extended
grading necessary to define the Euler class, which we express in terms of the
equivariant rank of the bundle and the degrees of the bundle and its fixed
subbundles. We do similar calculations using constant $\mathbb{Z}$ coefficients
and Borel cohomology and compare the results.
