Let us define for a sufficiently small $\delta$ a linear operator $D^{\prime}(Y)\to\Omega^{\ast}(Y)$ by the formula $\Pi_{\delta}(\alpha)=\sum_{x\in Cr(f)}(\int_{Y}\alpha\wedge R_{\delta}([S_{x}]))\cdot R_{\delta}([U_{x}]).$

Then

1) $\Pi_{\delta}^{2}(\alpha)=\Pi_{\delta}(\alpha)$ if $\alpha\in\Omega^{\ast}(Y)$, and $\Pi_{\delta}d=d\Pi_{\delta}.$

2) The image $\Pi_{\delta}(M^{\ast}(Y,f))$ is a subcomplex in $\Omega^{\ast}(Y)$ which is canonically isomorphic to the Morse complex $M^{\ast}(Y,f)$.