Taking formula_71 gives formula_72, so the supremum is always of a nonempty bounded set and thus well-defined. For a general t-norm there remains the possibility that formula_73 has a jump discontinuity at formula_74, in which case formula_75 could come out strictly larger than formula_28 even though formula_77 is defined as the least upper bound of formula_78s satisfying formula_79; to prevent that and have the construction work as expected, we require that the t-norm formula_7 is left-continuous. The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic.