The verb establish has 8 senses (first 8 from tagged texts)
1. (50) establish, set up, found, launch -- (set up or found; "She set up a literacy program")
2. (28) establish, found, plant, constitute, institute -- (set up or lay the groundwork for; "establish a new department")
3. (24) prove, demonstrate, establish, show, shew -- (establish the validity of something, as by an example, explanation or experiment; "The experiment demonstrated the instability of the compound"; "The mathematician showed the validity of the conjecture")
4. (15) lay down, establish, make -- (institute, enact, or establish; "make laws")
5. (7) establish, give -- (bring about; "The trompe l'oeil-illusion establishes depth")
6. (2) install, instal, set up, establish -- (place; "Her manager had set her up at the Ritz")
7. (2) establish, base, ground, found -- (use as a basis for; found on; "base a claim on some observation")
8. (1) build, establish -- (build or establish something abstract; "build a reputation")