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Anne Bogart Quits Trinity: Anne Bogart, artistic director of the Trinity Repertory Company in Providence, R.I., has resigned her post. Bogart and the Trinity Rep board clashed earlier this week about cuts in the proposed 1990-91 budgets. Bogart said any cuts would mean the termination of staff members and the backing off from commitments. When the board, which said the Rep’s operating costs for the past few years were more than its box office and fund-raising revenues, insisted on the cuts, Bogart resigned. No replacement has been named.