There are numerous synonyms for the word "establish", each carrying a slightly different connotation. To create or set up, you could use terms like found, initiate, start, or launch. To prove or confirm, consider terms like validate, substantiate, certify, or verify. To settle or fortify, try words like anchor, entrench, solidify, or reinforce. To establish oneself or gain recognition, you might use terms like establish one's reputation, make a name for oneself, earn one's stripes, or acquire a foothold. Ultimately, the choice of synonym depends on the context of the sentence and the precise shade of meaning that the writer wishes to convey.