this definition obeys the conditions on satisfaction of →: on one hand, if formula_9, and formula_10 is such that formula_11 and formula_12, then formula_13 by modus ponens. on the other hand, if formula_14, then formula_15 by the deduction theorem, thus the deductive closure of formula_16 is an element formula_10 such that formula_11, formula_12, and formula_20.