All 3 Uses

establish
in
Around the World in 80 Days
(Edited)

But he must go to the consul's in person, so as to establish his identity.
Chpt 6 *

establish = prove
…founded the church of the "Latter Day Saints," which, adopted not only in America, but in England, Norway and Sweden, and Germany, counts many artisans, as well as men engaged in the liberal professions, among its members; how a colony was established in Ohio, a temple erected there at a cost of two hundred thousand dollars, and a town built at Kirkland; how Smith became an enterprising banker, and received from a simple mummy showman a papyrus scroll written by Abraham and several…
Chpt 27

established = settled
A great crowd was collected in Pall Mall and the neighbouring streets on Saturday evening; it seemed like a multitude of brokers permanently established around the Reform Club.
Chpt 36 *

Definitions:

(1) (establish as in: establish a positive tone)
create, start, or set in [a] place
(2) (establish as in: establish that there is a need)
show or determine (cause to be recognized or figure out)
(3) (meaning too rare to warrant focus)