Product theorem for Euclidean Ramsey sets

If finite point sets $X$ and $Y$ are Euclidean Ramsey, then their orthogonal Cartesian product $X\times Y$ is Euclidean Ramsey. The proof first chooses a finite Ramsey witness for $X$, colors a witness for $Y$ by the complete color pattern it induces on the first witness, and then applies the two Ramsey properties in succession.