Determining the rate of stratospheric ozone destruction
Dioxygen is split in the stratosphere by UV light into oxygen atoms. This is the main source of free oxygen atoms. Their production rate depends on the intensity of the UV light.

In steady-state we can assume that the concentration of oxygen atoms is constant.

$$\frac{d[\ce{O}]}{dt} = 0  \tag{1}$$

For the above destruction mechanism we can formulate the rate laws

$$\begin{align}
\frac{d[\ce{O3}]}{dt} &amp;= -k_1 [\ce{O3}] [\ce{M}] + k_{-1} [\ce{O}] [\ce{O2}] [\ce{M}] - k_2 [\ce{O}] [\ce{O3}] \tag{2}\\
\\
\frac{d[\ce{O}]}{dt} &amp;= k_1 [\ce{O3}] [\ce{M}] - k_{-1} [\ce{O}] [\ce{O2}] [\ce{M}] - k_2 [\ce{O}] [\ce{O3}]  \tag{3}
\end{align}$$

Adding equation $(2)$ and $(3)$ which by $(1)$ is $0$ we get

$$\frac{d[\ce{O3}]}{dt} = -2~k_2 [\ce{O}] [\ce{O3}] \tag{4}$$

With equations $(1)$ and $(3)$ we can express the concentration of oxygen atoms as

$$[\ce{O}] = \frac{k_1 [\ce{O3}] [\ce{M}]}{k_{-1} [\ce{O2}] [\ce{M}] + k_2 [\ce{O3}]} \tag{5}$$

Substituting $(5)$ into equation $(4)$ we get

$$\frac{d[\ce{O3}]}{dt} = \frac{-2~k_1~k_2 [\ce{O3}]^2 [\ce{M}]}{k_{-1} [\ce{O2}] [\ce{M}] + k_2 [\ce{O3}]}  \tag{6}$$

So your friend seems to be right.