A small-gain-theorem-like approach to nonlinear observability via finite capacity channels * *A. Pogromsky acknowledges his partial support during his stay with the ITMO university by Government of Russian Federation grant (074-U01), Russian Federation President Grant N14.Y31.16.9281-HIII, and the Ministry of Education and Science of Russian Federation (project 14.Z50.31.0031), (Secs. 1,3,4). A. Matveev acknowledges his support by RSF 14-21-00041p and the Saint Petersburg State University (Secs. 2,5,6). A. Proskurnikov acknowledges support of RFBR, grants 17-08-00715,17-08-01728 and 17-08-01266.