What to use as numeraire in pricing the derivative?

Question

Assume that there exists derivative which payoff is given by $C_T = 1(S_{1,T}>S_{2,T})$. We have given two underlying stock price processes such as: $$dS_{1,t} = μ_1S_{1,t}dt + σ_1S_{1,t}dW_t$$ $$dS_{2,t} = μ_2S_{2,t}dt + σ_2S_{2,t}dW_t$$ From the first fundemental theorem of asset pricing we have that $$\frac{C_0}{N_0} = Ε_Q\left[\frac{C_T}{N_T}\right]$$Can we use two assets as a numeraire such as $N_t = \frac{S_{2,t}}{S_{1,t}}$, from where we obtain $$C_0 = \frac{S_{2,0}}{S_{1,0}}*Ε_Q\left[\frac{S_{1,T}}{S_{2,T}}*1\left(\frac{S_{1,T}}{S_{2,T}}>1\right)\right]$$From equivalent martingale measure we have $$\frac{S_{1,T}}{S_{2,T}} = \frac{S_{1,0}}{S_{2,0}}*e^{(σ_1 + σ_2)W_T^Q - \frac{1}{2}(σ_1 + σ_2)^2T}$$ Now we have $$C_0 = e^{-\frac{1}{2}(σ_1 + σ_2)^2T}*Ε_Q\left[e^{(σ_1 + σ_2)W_T^Q}*Ρ\left(Z<d_1\right)\right]$$ $$d_1 = \frac{ln\left(\frac{S_{1,0}}{S_{2,0}}\right) - \frac{1}{2}(σ_1 + σ_2)^2T}{(σ_1 + σ_2)\sqrt{T}}$$Finally, solution is $$C_0 = P\left(Z<d1 + (σ_1 + σ_2)\sqrt{T}\right)$$ Is this approach correct?