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Church 1956). , the relation is antisymmetric. , the relation is reflexive. Thus, is a partial order in Ro. Lemma is proved. This allows us to define: – – – – The set Rþþ o of all positive real hypernumbers The set Rþ o of all non-negative real hypernumbers The set RÀ o of all non-positive real hypernumbers The set RÀÀ o of all negative real hypernumbers It is interesting that although real numbers are isomorphically included into the set of all real hypernumbers, some concepts change their meaning.

Proof If a hypernumber a is defined by a bounded sequence l, then l either converges or has, at least, two subsequences that converge to two different points. In the first case, a is a real number. 9. Proposition is proved. 1 Any finite increasing (decreasing) real hypernumber is a real number. 10 imply the following result. 5 Any infinite real hypernumber is either an infinite increasing hypernumber or an infinite decreasing hypernumber or an oscillating real hypernumber. Proof If a hypernumber a is defined by a sequence l, then being unbounded, l is either unbounded only above (case 1) or unbounded only below (case 2) or unbounded below and above (case 3).

M3 d(x, y) d(x, z) + d(z, y) for all x, y, z 2 X. (b) A set X with a hypermetric d is called a hypermetric space. (c) The real hypernumber d(x, y) is called the distance between x and y in the hypermetric space X. Note that the distance between two elements in a hypermetric space can be a real number, finite hypernumber or infinite hypernumber. 2 A hypernorm in a vector space L induces a hypermetric in this space. Indeed, if q : X ! Rþ o is a hypernorm in L and x and y are elements from L, then we can define dðx; yÞ ¼ qðx À yÞ.