Show that the primitive recursive functions are closed under definition by

cases in the following sense: if $f_1(x_1,$dots,$x_n)$ and $f_2(x_1,$dots,$x_n)$ are primitive

recursive relations and $R(x_1,$dots,$x_n)$ is a primitive recursive relation, then

the following function is primitive recursive:

$g(x_1,$dots,$x_n)=\{\begin{array}{ccc}{f}_1(x_1,& \text{d}\text{o}\text{t}\text{s}\text{,}{x}_n)ifR(x_1,\text{d}\text{o}\text{t}\text{s}\text{,}& x_n)\\ f_2(x_1,& \text{d}\text{o}\text{t}\text{s}\text{,}{x}_n)if\text{n}\text{o}\text{t}\text{R}(x_1,\text{d}\text{o}\text{t}\text{s}\text{,}& x_n)\end{array}$