 Derivative avoidance criterion for chordal restriction

Suppose a random proper simple curve from zero to infinity has the displayed avoidance probability for every admissible compact H-hull $A$, with fixed $\alpha >0$. Put ${\psi }_A=g_A-g_A(0)$. For another admissible hull $B$ in the image complex upper half-plane, map composition and the chain rule give $g_C^{\prime }(0)=g_B^{\prime }(0)g_A^{\prime }(0)$ for $C=A\cup {\psi }_A^{-1}(B)$. The conditional image curve therefore has avoidance probabilities $g_B^{\prime }(0{)}^{\alpha }$, exactly those of the original. Since avoidance probabilities determine a simple chordal curve law, the conditional image law agrees with the original. This is the chordal restriction property.