The model has the form (4.17): \[ \begin{equation} \log \cfrac{Pr(G = 1 | X = x)}{Pr(G = k | X =x)} = \beta_{10} + \beta_1^Tx\\ \log \cfrac{Pr(G = 2 | X = x)}{Pr(G = k | X =x)} = \beta_{20} + \beta_2^Tx\\ \vdots\\ \log \cfrac{Pr(G = K-1 | X = x)}{Pr(G = k | X =x)} = \beta_{(K-1)0} + \beta_{K-1}^Tx\\ \end{equation} \]
The model is specified in terms of K - 1 log-odds or logit transformations. A simple calculation shows that (4.18): \[ \begin{equation} Pr(G=k|X=x) = \cfrac{exp(\beta_{k0} + \beta_k^Tx)}{1+\sum_{l=1}^{K-1}exp(\beta_{l0} + \beta_l^Tx)}, k = 1,...,K - 1,\\ Pr(G=K|X=x) = \cfrac{1}{1+\sum_{l=1}^{K-1}exp(\beta_{l0} + \beta_l^Tx)} \end{equation} \]
and they clearly sum to one.