A homotopy principle for maps with prescribed Thom-Boardman singularities

Let $N$ and $P$ be smooth manifolds of dimensions $n$ and $p$ ( $n\geq p\geq2$) respectively. Let $\Omega^{I}(N,P)$ denote an open subspace of $J^{\infty }(N,P)$ which consists of all Boardman submanifolds $\Sigma^{J}(N,P)$ of symbols $J$ with $J\leq I$. An $\Omega^{I}$-regular map $f:N\rightarrow P$ refers to a smooth map such that $j^{\infty}f(N)\subset\Omega^{I}(N,P)$. We will prove what is called the homotopy principle for $\Omega^{I}$-regular maps on the existence level. Namely, a continuous section $s$ of $\Omega^{I}(N,P)$ over $N$ has an $\Omega^{I}$-regular map $f$ such that $s$ and $j^{\infty}f$ are homotopic as sections.