Sard on the nonflat critical strata

Fix $x\in K$. [L1, given, choose] Because $x\mathrm{\notin }{C}_{j+1}$, some partial derivative of order $j+1$ of some component of $f$ is nonzero at $x$. After reordering coordinates and components, choose a multi-index $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }=j$ and a component $f^{\mu }$ such that, for $g:=D^{\alpha }{f}^{\mu },$ one has $\mathrm{\partial }g\mathrm{/}\mathrm{\partial }{x}^1(x)\mathrm{\ne }0$. Since $g$ is $C^{r-j}$, [L1] applied to ${\mathrm{\Phi }}_x(y):=(g(y),y^2,\dots ,y^m)$ gives a neighbourhood $W_x$ of $x$ and a $C^{r-j}$ diffeomorphism from $W_x$ onto an open set $I_x\times {\mathrm{\Omega }}_x\subseteq \mathbb{R}\times {\mathbb{R}}^{m-1}$.

Every point of $K\cap W_x$ lies in $C_j$, so $g$ vanishes there by definition of $C_j$. Hence ${\mathrm{\Phi }}_x(K\cap W_x)\subseteq \{0\}\times {\mathrm{\Omega }}_x\mathrm{.}$ If $m=1$, then ${\mathrm{\Omega }}_x\subseteq {\mathbb{R}}^0$ has at most one point. Thus $K\cap W_x$ has at most one point, and its image is null in ${\mathbb{R}}^n$. Assume $m\ge 2$. Define the $C^{r-j}$ graph parametrization $b_x:{\mathrm{\Omega }}_x\to U,b_x(u):={\mathrm{\Phi }}_x^{-1}(0,u),$ and the relatively closed set $A_x^{\ast }:=b_x^{-1}(C_j\cap W_x)$ in ${\mathrm{\Omega }}_x$. The map $f$ is $j$-flat on $C_j$, and $b_x(A_x^{\ast })\subseteq C_j$. By [L2] there is a $C^r$ map $H_x:{\mathrm{\Omega }}_x\to {\mathbb{R}}^n$ agreeing with $f\circ b_x$ on $A_x^{\ast }$ and satisfying $D{H}_x=0$ there. Thus every $u\in b_x^{-1}(K\cap W_x)$ is a critical point of the $C^r$ map $H_x$, and $f(K\cap W_x)=H_x(b_x^{-1}(K\cap W_x))\subseteq H_x(\mathrm{Crit}{H}_x)\mathrm{.}$ The induction hypothesis, at the unchanged differentiability order $r$, gives nullity of this image.

Finitely many neighbourhoods $W_x$ cover the compact set $K$, so $f(K)$ is a finite union of null sets and therefore null.