The present paper introduces the notion of the Bortolotti connection in the principal fiber space $\hat H(S(\tilde M_{n,m}^{n-m}),\dot G_m)$, the notion of the pseudosurface, associated with subsurface, and the Bortolotti clothing of a pseudosurface, which generates the described connection. The paper singles out a special case of the clothing, namely, the Bortolotti clothing in the proper sense. It is demonstrated that the Bortolotti clothing in the proper sense of the pseudosurface, associated with a subsurface $\Sigma_m$, induces the Bortolotti clothing of the subsurface $\Sigma_m$ itself. The paper sets up and solves the problem of immersion of the Bortolotti connection in an $N$-dimensional projective space. It is proved that the immersion is possible, if $N\geq mn(n-m+1)+m(m-1)/2$.