 In a complete logic, a formula is contradictory if and only if it is unsatisfiable.For a set of premises formula_7 and a proposition formula_1, it is true that formula_9 (i.e., formula_7 proves formula_1) if and only if formula_12 (i.e., formula_7 and formula_14 leads to a contradiction).Therefore, a proof that formula_12 also proves that formula_1 is true under the premises formula_7.