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**1 - 3**of**3**### DOI: 10.1515/forma-2015-0022 degruyter.com/view/j/forma Summable Family in a Commutative Group

"... Summary. Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters ” in Isabelle/HOL [21, 7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of limit of a family indexed by a directed set, or a sequence, in ..."

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Summary. Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters ” in Isabelle/HOL [21, 7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of limit of a family indexed by a directed set, or a sequence, in a metric space [29], a real normed linear space [28] and a linear topological space [14] with the concept of limit of image filter [?]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commu-tative group (“additive notation ” in [16]), using the notion of filters.

### Contents Formaliz. Math. 22 (2) Preface............................................................ i Pseudo-Canonical Formulae are Classical

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### Cayley’s Theorem

"... Summary. The article formalizes the Cayley’s theorem saying that every group G is isomorphic to a subgroup of the symmetric group on G. ..."

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Summary. The article formalizes the Cayley’s theorem saying that every group G is isomorphic to a subgroup of the symmetric group on G.