Predecessor-subgraph property
I think the LHS is assumed by argument to be: $d[v_{i}]= \Sigma_{j=1}^{j=i} w(v_{j-1},v_{j})$. Since $d[v_{i-1}]= \delta(s,v_{i-1})$ = $min$$\{w(p) : s \rightsquigarrow^p v_{i-1}\} $  on the RHS, where $w(p)= \Sigma_{j=1}^{j=k}w(x_{j-1},x_{j})$ for any path $p: x_{0}=s, ... ,x_{k}=v_{i-1}$, we have the inequality $d[v_{i}]\geq d[v_{i-1}]+w(v_{i-1},v_{i})$.