@MISC{Sakowicz_mmlidentifier:, author = {Agnieszka Sakowicz and Sequences In E N T and Adam Grabowski}, title = {MML Identifier: TOPRNS 1.}, year = {} }

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Abstract

notation and terminology for this paper. For simplicity we adopt the following rules: f denotes a function, N, n, m denote natural numbers, q, r, r1, r2 denote real numbers, x is arbitrary, and w, w1, w2, g denote points of EN T. Let us consider N. A sequence in E N T E N T. In the sequel s1, s2, s3, s4, s ′ 1 are sequences in EN T. is a function from ¦ into the carrier of Next we state two propositions: (1) f is a sequence in EN T if and only if domf ¦ = and for every x such that x ¦ ∈ holds f(x) is a point of EN T. (2) f is a sequence in EN T iff dom f ¦ = and for every n holds f(n) is a point of EN T. Let us consider N, s1, n. Then s1(n) is a point of EN T. Let us consider N. A sequence in EN T is non-zero if: (Def.1) rng it ⊆ (the carrier of EN T) \ {0EN T We now state several propositions: (3) s1 is non-zero iff for every x such that x ¦ ∈ holds s1(x) � = 0EN. T (4) s1 is non-zero iff for every n holds s1(n) � = 0 E N T (5) For all N, s1, s2 such that for every x such that x ¦ ∈ holds s1(x) = s2(x) holds s1 = s2. (6) For all N, s1, s2 such that for every n holds s1(n) = s2(n) holds s1 = s2. (7) For every point w of EN T there exists s1 such that rng s1 = {w}. The scheme ExTopRealNSeq deals with a natural number A and a unary functor F yielding a point of E A T, and states that: