We can choose $\tilde{\psi}>0$ to the following effect. Suppose $g,J,\Lambda,A$ are structures as above over $\tilde{U}$. Suppose that $\|g-g_{0}\|_{\tilde{U}},\|J-J_{0}\|_{\tilde{U}}\leq\tilde{\psi}$. Then we can find an integer $t$ with $1\leq t\leq m$ such that the data $\mu_{t}^{*}(tg),\mu_{t}^{*}(J),\mu_{t}^{*}(\Lambda^{t}),\mu_{t}^{*}(A^{\otimes t})$ over $U$ has Property(H).