In a space $L_N$ of asymmetric affine connection one observes a submanifold, defined in local coordinates. Because of the asymmetry of the connection in the space, the connection of submanifolds is generally asymmetric. Based on this, it follows that 4 kinds of covariant derivatives and 4 kinds of derivational equations are possible. In the present paper is proved that by applying the $3^{rd}$, or the $4^{th}$ kind of covariant derivative, it follows that the induced connection is symmetric (Theorem 1.2.). In the pseudonormal submanifold are defined 2 connections (2.4) and 4 kinds of covariant derivative. It is proved that by applying the $3^{rd}$ or the $4^{th}$ kind of derivative one concludes that the induced connections in this case is unique (Theorem 2.2). In $\S 3$ are examined some properties of coefficients of derivational equations and induced connection in pseudonormal subspace.