Theorem of reduction in the problem of reconstruction of submanifolds in Euclidean space by a given Grassmann image

A necessary condition for the Grassmann image of submanifolds in the Euclidean space is proved. It is shown that the reconstruction of a submanifold $F^n\subset E^{n+m}$ with the constant dimension $l$ of the first normal space by a given $k$-dimensional Grassmann image $\Gamma$ is equivalent to the reconstruction of some submanifold $\tilde F^k\subset E^{k+l}$ with the constant dimension I of the first normal space by a given fe-dimensional Grassmann image $\tilde\Gamma$, where $\tilde\Gamma$ is connected with $\Gamma$ in a special way.