Using the basic natural deduction rules for predicate logic, prove the validity of the following sequent:

$($i$)$x$($P$($x$)$ Q$($x$)),$yS$($y$)$z$($P$($z$)$ S$($z$))$

$($ii$)$x $($P$($x$)($Q$($x$)$ R$($x$))),$x $($P$($x$)$ R$($x$))$x $($P$($x$)$ Q$($x$))$