Equal sums of surfaces

Question

The inner points $X$ and $Y$ in the convex quadrilateral $ABCD$ are such that $\angle ABX=\angle CBY$, $\angle BCY=\angle DCX$, $\angle CDY=\angle ADX$ and $\angle DAY=\angle BAX$. Prove that $$S^{}_{AXB}+S^{}_{CXD}=S^{}_{AYD}+S^{}_{BYC}$$