The importance of selected biomarkers in the clinical practice of breast cancer patients

Breast cancer is considered the most commonly diagnosed tumors. Biomarkers used for the diagnosis and treatment of breast cancer are: tissue biomarkers (PR, ER, HER2, Ki-67) and serum biomarkers (CA-15-3, CA-125, CA-27-29, CEA, cytokeratins). ECD HER2, metalloproteinases and leptin are emerging as promising biomarkers for breast cancer. There is a growing need for personalized diagnostics based on tumour genome characterization, relying on a liquid biopsy containing components such as CTC and ctDNA, cell-free RNA. Biomarkers can also be used use as a target for anti-breast cancer treatment (PGRN and sortilin, AR, PD-1/PD-L1). Another potential field of application of breast cancer biomarkers is monitoring treatment side effects, such us inflammatory biomarkers causing cardiotoxicity, thyroiditis biomarkers (TSH, FT4, TPOab TgAb) in IrAE, NF-L and MCP-1 in ICI-associated neurotoxity. It is expected that new prognostic and predictive biomarkers will be developed that can provide accurate and reliable information for clinical application. Through the re­cognition of emerging biomarkers, it is possible to identify subgroups of patients who benefit from targeted therapies and managing treatment by monitoring side effects. However, these new biomarkers need to be validated and tested for their suitability before entering clinical use

Standard and potential tumor markers in patients with cervical cancer

Oznaczanie markerów nowotworowych od wielu lat znajduje zastosowanie w ginekologii onkologicznej. Znanedotychczas markery nowotworowe wykorzystywane są przede wszystkim w prognozowaniu i monitorowaniu przebieguchoroby.Rutynowo oznaczanymi w surowicy krwi markerami nowotworowymi u chorych na płaskonabłonkowego raka szyjkimacicy są: antygen SCCAg i CYFRA 21.1, a na raka gruczołowego — CA 125 i CEA. Stężenie SCCAg przed leczeniempomocne jest w określeniu stopnia zaawansowania klinicznego i w prognozowaniu przebiegu choroby. Spadek stężeniaSCCAg w surowicy krwi po zabiegu operacyjnym świadczy o jego radykalności. Uzyskane po zakończeniu leczeniawartości SCCAg u chorych na raka płaskonabłonkowego mają wysoką dodatnią wartość predykcyjną dla wykryciawznowy lub odległych przerzutów. Komplementarne oznaczanie CYFRY 21.1 z SCCAg zwiększa czułość diagnostyczną,co ma istotne znaczenie u chorych w niskich stopniach zaawansowania. W typie gruczołowym raka szyjki macicymarkerem o najwyższej czułości jest CA 125. Stężenie jego narasta wraz ze stopniem zaawansowania klinicznego,korelując ze stanem węzłów chłonnych miednicy mniejszej i okołoaortalnych. Ponadto stężenie CA 125 w surowicykrwi jest niezależnym czynnikiem prognostycznym dla czasu wolnego od choroby. Czułość diagnostyczna CA125 w raku gruczołowym jest wyższa, w porównaniu z CEA, który jest uznanym markerem dla tego typu nowotworu.W ostatnich latach okazało się, że cytokiny mogą być klinicznie użytecznymi potencjalnymi markerami. U chorychna raka szyjki macicy istotne znaczenie może mieć oznaczanie stężeń, między innymi: naczyniowo-śródbłonkowegoczynnika wzrostu (VEGF) oraz jego rozpuszczalnych receptorów: VEGFR1 (sFLT1) i VEGFR2, interleukiny 6 (IL-6) i rozpuszczalnychreceptorów czynnika martwicy nowotworów: sTNF RI, sTNF RII. U chorych na raka szyjki macicy stężenieVEGF narasta wraz ze stopniem zaawansowania klinicznego i wielkością guza i może być klinicznie użyteczne w monitorowaniuprzebiegu choroby, a także w ocenie czasu wolnego od choroby i czasu całkowitego przeżycia. Stężeniareceptorów VEGFR1 i VEGFR2 mogą mieć znaczenie prognostyczne. Na podstawie przeglądu piśmiennictwa ostatnichlat wynika, że oznaczanie np. YKL-40, gammaglutamylotransferazy (GGT), leptyny, metaloproteinazy 9 (MMP-9) czyteż receptora HER2 u chorych na raka szyjki macicy może mieć również zastosowanie kliniczne, co stwarza szansępoprawy diagnostyki, a przez to skuteczności leczenia chorych na raka szyjki macicy.The determination of tumor markers in gynecological oncology has been useful for many years. Previously known tumor markers are used mainly when predicting and monitoring the disease. The routinely determined tumor markers in the serum of patients with squamous cell carcinoma of the cervix are: antigen SCCAg and CYFRA 21.1, and for patients with adenocarcinoma CA 125 and CEA. The concentration of SCCAg before treatment is useful for the determination of the clinical stage, as well as for the prediction of the disease. The decrease in the concentration of SCCAg in serum after surgery proves radical activity. Values of SCCAg in patients with squamous cell carcinoma after treatment, have a greater positive predictive value for the detection of a recurrence or of distant metastase. The complementary determination of CYFRA 21.1 with SCCAg increases diagnostic sensitivity, which is essential for patients in low stages of the disease. In adenocarcinoma, the tumor marker of the greatest sensitivity is CA 125. Its concentrations increase along with clinical stage, correlating with pelvic and para-aortic lymph node status. Moreover, the concentrations of CA 125 in serum are independent predictors for disease-free survival. The diagnostic sensitivity of CA 125 in adenocarcinoma is greater in comparison to CEA. During the past several years it has appeared that potentially, cytokines can be clinically useful markers. In patients with cervical cancer of significant interest is the determination of the concentrations between vascular endothelial growth factor (VEGF) and its soluble receptors: VEGFR1 (sFLT1) and VEGFR2, interleukin 6 (IL-6) and soluble receptors of the tumor necrosis factors: STNF RI, sTNF RII. In patients with cervical cancer, the concentrations of VEGF increase with the clinical stage and the tumor size. They may be clinically useful when monitoring the disease, as well as evaluating both the disease-free survival and overall survival. The concentrations of receptors VEGFR1 and VEGFR2 may play the role of a prognostic factor. On the basis of the literature review collected over the past several years, it appears that the determination of e.g., YKL-40, gamma glutamyltransferase (GGT), leptin, matrix metalloproteinase 9 (MMP-9) or HER2 receptor in patients with cervical cancer may also have a clinical application to create an opportunity to improve diagnosis and thus the effectiveness in treatment of patients with cervical cancer

Measurement of HE4 six months after first-line treatment as optimal time in identifying patients at high risk of progression advanced ovarian cancer

Objectives: The objective of the study was to assess the usefulness of determining HE4 and CA125 in ovarian cancer patients, to indicate which of the measurements may be optimal in the prognosis, depending on the treatment scheme. Material end methods: The concentrations of CA125 and HE4 were performed in 70 patients with advanced ovarian cancer during I-line therapy and after treatment. The subjects were divided based on the treatment scheme: group I - primary surgery and adjuvant chemotherapy, II- neoadjuvant therapy, and surgery. Results: Multivariate analysis showed that HE4 levels six months after treatment was significantly higher in patients with disease progression. ROC analysis in the group of patients treated with neoadjuvant therapy showed that the cut-off values indicating relapse for HE4 and CA125 after six months of follow up, were > 90.4 pmol/L, > 25.6 IU/mL, respectively. In the group of patients not treated with neoadjuvant therapy, the cut-off points differentiating patients with progression were: HE4 > 79.1 pmol/L, CA125 > 30.7 IU/mL. We demonstrated significantly higher HE4 and CA125 at both 6- and 12-months follow-up in patients treated with neoadjuvant therapy. In both groups of patients, the cut-off points were lower than those proposed by the manufacturer of the kits. Conclusions: Measurement of HE4 six months after treatment may be useful in identifying patients at high risk of progression, especially when CA125 levels may be non-specifically elevated. The cut-off values indicating relapse for HE4 and CA125 after six months of follow up may be lower than the normal range

Seed dispersal in six species of terrestrial orchids in Biebrza National Park (NE Poland)

Knowledge about seed dispersal is required to explain problems in ecology, phylogeography, and conservation biology. Even though seed dispersal is a fundamental mechanism to understand problems at different levels of biological organization (individual, population, species, landscape), it remains one of the least recognized processes. Similar to other groups of plants, very little is known regarding patterns and distances of seed dispersal in orchids. Orchid seeds are generally assumed to be widely dispersed by wind because of their small size and low weight. Between 2006 and 2008, we conducted a field study of the distances at which orchid seeds are dispersed, and determined factors affecting dispersal. Investigations included 13 populations of six terrestrial orchid species – Cypripedium calceolus, Cephalanthera rubra, Epipactis helleborine, Goodyera repens, Neottia ovata, and Platanthera bifolia. To evaluate seed dispersal in orchid populations, 8.5-cm Petri dishes (traps) with self-adhesive paper were placed along transects, starting from a group of fruiting plants, which were considered to be the dispersal source. Seeds of the investigated orchid species were dispersed over relatively short distances. There were statistically significant negative correlations between seed density and distance from the fruiting plants. Seeds of species with taller fruiting shoots were dispersed farther than those with shorter ones (R = 0.68, p < 0.05). We discuss the causes and consequences of the dispersal patterns of orchid seeds

On Lp Space Formed by Real-Valued Partial Functions

This article is the continuation of [31]. We define the set of Lp integrable functions - the set of all partial functions whose absolute value raised to the p-th power is integrable. We show that Lp integrable functions form the Lp space. We also prove Minkowski's inequality, Hölder's inequality and that Lp space is Banach space ([15], [27]).Watase Yasushige - Graduate School of Science and Technology, Shinshu University, Nagano, JapanEndou Noboru - Gifu National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanJonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Józef Białas. The s-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006, doi:10.2478/v10037-006-0008-x.Noboru Endou, Yasunari Shidama, and Keiko Narita. Egoroff's theorem. Formalized Mathematics, 16(1):57-63, 2008, doi:10.2478/v10037-008-0009-z.P. R. Halmos. Measure Theory. Springer-Verlag, 1987.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Keiko Narita, Noboru Endou, and Yasunari Shidama. Integral of complex-valued measurable function. Formalized Mathematics, 16(4):319-324, 2008, doi:10.2478/v10037-008-0039-6.Keiko Narita, Noboru Endou, and Yasunari Shidama. Lebesgue's convergence theorem of complex-valued function. Formalized Mathematics, 17(2):137-145, 2009, doi: 10.2478/v10037-009-0015-9.Andrzej Nędzusiak. s-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.Walter Rudin. Real and Complex Analysis. Mc Graw-Hill, Inc., 1974.Yasunari Shidama and Noboru Endou. Integral of real-valued measurable function. Formalized Mathematics, 14(4):143-152, 2006, doi:10.2478/v10037-006-0018-8.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Yasushige Watase, Noboru Endou, and Yasunari Shidama. On L1 space formed by real-valued partial functions. Formalized Mathematics, 16(4):361-369, 2008, doi:10.2478/v10037-008-0044-9.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Riemann Integral of Functions from R into Rⁿ

In this article, we define the Riemann Integral of functions from R into Rⁿ, and prove the linearity of this operator. The presented method is based on [21].Miyajima Keiichi - Ibaraki University, Hitachi, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

The Geometric Interior in Real Linear Spaces

We introduce the notions of the geometric interior and the centre of mass for subsets of real linear spaces. We prove a number of theorems concerning these notions which are used in the theory of abstract simplicial complexes.Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Karol Pąk. Affine independence in vector spaces. Formalized Mathematics, 18(1):87-93, 2010, doi: 10.2478/v10037-010-0012-z.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313-319, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Riemann Integral of Functions from R into Real Normed Space

In this article, we define the Riemann integral on functions from R into real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to a wider range of functions. The proof method follows the [16].Miyajima Keiichi - Faculty of Engineering, Ibaraki University, Hitachi, JapanKato Takahiro - Faculty of Engineering, Graduate School of Ibaraki University, Hitachi, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux's theorem. Formalized Mathematics, 9(1):197-200, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992

Planes and Spheres as Topological Manifolds. Stereographic Projection

The goal of this article is to show some examples of topological manifolds: planes and spheres in Euclidean space. In doing it, the article introduces the stereographic projection [25].Via del Pero 102, 54038 Montignoso, ItalyGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213-225, 1992.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475-480, 1991.Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in En/T Formalized Mathematics, 12(3):301-306, 2004.Artur Korniłowicz and Yasunari Shidama. Some properties of circles on the plane. Formalized Mathematics, 13(1):117-124, 2005.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.John M. Lee. Introduction to Topological Manifolds. Springer-Verlag, New York Berlin Heidelberg, 2000.Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.Yatsuka Nakamura, Artur Korniłowicz, Nagato Oya, and Yasunari Shidama. The real vector spaces of finite sequences are finite dimensional. Formalized Mathematics, 17(1):1-9, 2009, doi:10.2478/v10037-009-0001-2.Henryk Oryszczyszyn and Krzysztof Prażmowski. Real functions spaces. Formalized Mathematics, 1(3):555-561, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93-96, 1991.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol Pąk. Basic properties of metrizable topological spaces. Formalized Mathematics, 17(3):201-205, 2009, doi: 10.2478/v10037-009-0024-8.Marco Riccardi. The definition of topological manifolds. Formalized Mathematics, 19(1):41-44, 2011, doi: 10.2478/v10037-011-0007-4.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297-301, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Mariusz Żynel and Adam Guzowski. T0 topological spaces. Formalized Mathematics, 5(1):75-77, 1996

Differentiable Functions into Real Normed Spaces

In this article, we formalize the differentiability of functions from the set of real numbers into a normed vector space [14].Okazaki Hiroyuki - Shinshu University, Nagano, JapanEndou Noboru - Nagano National College of Technology, Nagano, JapanNarita Keiko - Hirosaki-city, Aomori, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55- 65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011, doi: 10.2478/v10037-011-0008-3.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Laurent Schwartz. Cours d'analyse, vol. 1. Hermann Paris, 1967. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992