# K-Path sampling overall transfer function

I'm trying to derive the overall Z-transform for the K-path sampling system shown below. I numbered some of the equations that I'll reference.

I don't think I understand going from equation 3 to equation 4. I understand that $$\frac{V_{o,i}(z)}{V_{in,i}(z)} = H(z)$$, but why is $$\frac{V_{o,i}(z^{1/k})}{V_{in,i}(z^{1/k})} = H(z)$$? Does this follow from equation 1?