The coefficient ideal of a marked ideal of maximal order

Let $X$ be a smooth $K$-scheme over a field $K$, and let $(\mathcal{I},E,\mu )$ be a marked ideal of maximal order on $X$ with $\mu \ge 1$ (Smooth morphism of schemes). The coefficient ideal is the marked ideal $C(\mathcal{I},\mu ):={\sum }_{i=0}^{\mu -1}({\mathcal{D}}^i(\mathcal{I}),\mu -i),$ the sum being that of Addition and multiplication of marked ideals; explicitly $C(\mathcal{I},\mu )=({\sum }_{i=0}^{\mu -1}{\mathcal{D}}^i(\mathcal{I}{)}^{\mu !\mathrm{/}(\mu -i)},\mu !)\mathrm{.}$ Under AC (The Axiom of Choice), in every characteristic it satisfies $C(\mathcal{I},\mu )\simeq (\mathcal{I},\mu )$. Under the same AC assumption, if $K$ has characteristic zero or perfect characteristic $p>\mu$ (Field), then for every regular closed subscheme $S\subseteq X$ having SNC with $E$, $\mathrm{supp}(\mathcal{I},\mu )\cap S=\mathrm{supp}(C(\mathcal{I},\mu ){\mathrm{\mid }}_S),$ and this identity persists under multiple test blow-ups whose centers lie in the strict transforms of $S$ (proved below).