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...e article we show two schemes of cardinal induction. Some definitions are based on [9] and [10]. MML Identifier: CARD_1. WWW: http://mizar.org/JFM/Vol1/card_1.html The articles [11], [7], [13], [12], =-=[1]-=-, [14], [6], [4], [2], [3], [5], and [8] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: A, B denote ordinal numbers, X, X1, Y , Y1, Z, x, y denote s...

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... n . Also some basic facts about tuples of a non-empty set are proved. MML Identifier: FINSEQ_2. WWW: http://mizar.org/JFM/Vol2/finseq_2.html The articles [13], [12], [9], [16], [2], [3], [14], [11], =-=[1]-=-, [15], [17], [6], [8], [7], [4], [5], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following convention: i, j, l are natural numbers, k is a natural numb...

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...al linear space, operations on vectors: addition, multiplication by real number, inverse vector, subtraction. The sum of finite sequence of the vectors is also defined. Theorems that belong rather to =-=[1]-=- or [4] are proved. MML Identifier: RLVECT_1. WWW: http://mizar.org/JFM/Vol1/rlvect_1.html The articles [11], [8], [13], [2], [3], [12], [10], [14], [6], [7], [5], [9], [4], and [1] provide the notati...

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...range. At the end of the article we prove the pigeon hole principle. MML Identifier: FINSEQ_4. WWW: http://mizar.org/JFM/Vol2/finseq_4.html The articles [7], [10], [8], [3], [11], [4], [1], [5], [6], =-=[2]-=-, and [9] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: f is a function, p, q are finite sequences, A, B, x, y, z are sets, and i, k, n are natural...

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...rties of real numbers and set representations of binary operations on real numbers. MML Identifier: RVSUM_1. WWW: http://mizar.org/JFM/Vol2/rvsum_1.html The articles [17], [21], [8], [2], [18], [12], =-=[1]-=-, [19], [22], [5], [7], [6], [4], [15], [14], [16], [3], [10], [20], [13], and [9] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: i, j, k denote nat...

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...on increasing sequences, the definition of a subsequence and their basic properties. MML Identifier: SEQM_3. WWW: http://mizar.org/JFM/Vol1/seqm_3.html The articles [7], [2], [8], [9], [3], [5], [4], =-=[1]-=-, and [6] provide the notation and terminology for this paper. We follow the rules: n, m, k are natural numbers, r is a real number, and f , s1, s2, s3 are sequences of real numbers. Let f be a partia...

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...try to approach it. 9 How to learn Mizar? The Mizar language, its processor, and the organization of the Mizar library evolve, and therefore there is not much in the way of written documentation, see =-=[1]-=-. In the face of documentation shortages the best way to learn Mizar is to spend approximately four weeks in Bia lstok 6 and co-author a Mizar article with a native user of the system. However, numero...