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a.beS addition additive inverse algorithm b-adic expansions complete lattice completely and linearly completely and weakly complex interval complex numbers computer arithmetic conditionally complete convex hull corresponding data types defined digits of base error execution exponent EXPRESSION FIGURE floating-point numbers floating-point system following properties following theorem formulas FUNCT ID greatest element IMnR infimum integer interval vectors isomorphism least element Lemma linearly ordered ringoid linearly ordered set lower bound lower resp mantissa mapping matrices monotone rounding monotone upper screen monotone upwardly directed multiplicative vectoid neutral elements obtain operands order relation ordered division ringoid ordered set outer multiplication power set programming language Proof real numbers result screen groupoid screen ringoid screen vectoid semimorphism structure subset subtraction supremum symmetric screen Theorem 4.4 underflow upper bound upwardly directed rounding VARIABLE vector vertical method weakly ordered resp weakly ordered ringoid weakly ordered vectoid zero