v[n] = x[n]-w[n] => V(z) = X(z) - W(z)
Let the output of the H(z) block be y[n]. Then
w[n] = e[n] + y[n] => W(z) = E(z) + Y(z) = E(z) + H(z) V(z)
Now eliminate V(z) between the above two equations:
W(z) = E(z) + H(z)(X(z)-W(z))
Thus,
H_{1}(z) = H(z)/(1+H(z)) and
H_{2}(z) = 1/(1+H(z)).