@Preamble{"\providecommand{\calR}{\cal R} \providecommand{\calE}{\cal E} \providecommand{\rmT}{\rm T} 
\providecommand{\mbR}{\mathbb R} \providecommand{\BbbR}{\mathbb R} \providecommand{\SCM}{\bf SCM} 
\providecommand{\SCMFSA}{${\bf SCM}_{\rm FSA}$}"} \providecommand{\fxp}{$F[X]/\langle p\rangle$}
\providecommand{\fx}{$F[X]$} \providecommand{\f}{$F$}

@MISC(FF_SIEC.ABS,
      AUTHOR = {Korczy\'nski, Waldemar},
      TITLE  = {Definitions of {P}etri Net -- Part {I} -- {\tt FF\_SIEC}},
      HOWPUBLISHED = {Mizar Mathematical Library},
      YEAR = {1992})

@MISC(E_SIEC.ABS,
      AUTHOR = {Korczy\'nski, Waldemar},
      TITLE  = {Definitions of {P}etri Net -- Part {II} -- {\tt E\_SIEC}},
      HOWPUBLISHED = {Mizar Mathematical Library},
      YEAR = {1992})

@MISC(S_SIEC.ABS,
      AUTHOR = {Korczy\'nski, Waldemar},
      TITLE  = {Definitions of {P}etri Net -- Part {III} -- {\tt S\_SIEC}},
      HOWPUBLISHED = {Mizar Mathematical Library},
      YEAR = {1992})

@ARTICLE(ARYTM_0.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Introduction to Arithmetics},
      JOURNAL = {Mizar Mathematical Library})

@ARTICLE(ARYTM_3.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Arithmetic of Non-Negative Rational Numbers},
      JOURNAL = {Mizar Mathematical Library})

@ARTICLE(BINOP_2.ABS,
      AUTHOR  = {Library Committee of the Association of Mizar Users},
      TITLE   = {Binary Operations on Numbers},
      JOURNAL = {Mizar Mathematical Library})

@ARTICLE(NUMBERS.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Subsets of Complex Numbers},
      JOURNAL = {Mizar Mathematical Library})

@ARTICLE(TARSKI.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Tarski {G}rothendieck Set Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/tarski.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {9--11})

@ARTICLE(AXIOMS.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Built-in Concepts},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/axioms.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {13--15})

@ARTICLE(BOOLE.ABS,
      AUTHOR  = {Trybulec, Zinaida and {\'{S}}wi\k{e}czkowska, Halina},
      TITLE   = {Boolean Properties of Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/boole.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {17--23})

@ARTICLE(ENUMSET1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Enumerated Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/enumset1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {25--34})

@ARTICLE(REAL_1.ABS,
      AUTHOR  = {Hryniewiecki, Krzysztof},
      TITLE   = {Basic Properties  of Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/real_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {35--40})

@ARTICLE(NAT_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {The Fundamental Properties of Natural Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/nat_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {41--46})

@ARTICLE(ZFMISC_1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Some Basic Properties of Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/zfmisc_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {47--53})

@ARTICLE(FUNCT_1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Functions and Their Basic Properties},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/funct_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {55--65})

@ARTICLE(SUBSET_1.ABS,
      AUTHOR  = {Trybulec, Zinaida},
      TITLE   = {Properties of Subsets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/subset_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {67--71})

@ARTICLE(RELAT_1.ABS,
      AUTHOR  = {Woronowicz, Edmund},
      TITLE   = {Relations and Their Basic Properties},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/relat_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {73--83})

@ARTICLE(RELAT_2.ABS,
      AUTHOR  = {Woronowicz, Edmund and Zalewska, Anna},
      TITLE   = {Properties of Binary Relations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/relat_2.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {85--89})

@ARTICLE(ORDINAL1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {The Ordinal Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/ordinal1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {91--96})

@ARTICLE(MCART_1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Tuples, Projections and {C}artesian Products},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/mcart_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {97--105})

@ARTICLE(FINSEQ_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Hryniewiecki, Krzysztof},
      TITLE   = {Segments of Natural Numbers and Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/finseq_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {107--114})

@ARTICLE(DOMAIN_1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Domains and Their {C}artesian Products},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/domain_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {115--122})

@ARTICLE(WELLORD1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {The Well Ordering Relations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/wellord1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {123--129})

@ARTICLE(ZF_LANG.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {A Model of {ZF} Set Theory Language},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/zf_lang.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {131--145})

@ARTICLE(SETFAM_1.ABS,
      AUTHOR  = {Padlewska, Beata},
      TITLE   = {Families of Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/setfam_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {147--152})

@ARTICLE(FUNCT_2.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Functions from a Set to a Set},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/funct_2.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {153--164})

@ARTICLE(FINSET_1.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata},
      TITLE   = {Finite Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/finset_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {165--167})

@ARTICLE(GRFUNC_1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Graphs of Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/grfunc_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {169--173})

@ARTICLE(BINOP_1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Binary Operations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/binop_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {175--180})

@ARTICLE(RELSET_1.ABS,
      AUTHOR  = {Woronowicz, Edmund},
      TITLE   = {Relations Defined on Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/relset_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {181--186})

@ARTICLE(FINSUB_1.ABS,
      AUTHOR  = {Trybulec, Andrzej and Darmochwa{\l}, Agata},
      TITLE   = {Boolean Domains},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/finsub_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {187--190})

@ARTICLE(ZF_MODEL.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Models and Satisfiability},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/zf_model.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {191--199})

@ARTICLE(ZF_COLLA.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {The Contraction Lemma},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/zf_colla.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {201--203})

@ARTICLE(INCSP_1.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Axioms of Incidence},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/incsp_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {205--213})

@ARTICLE(LATTICES.ABS,
      AUTHOR  = {{\.{Z}}ukowski, Stanis{\l}aw},
      TITLE   = {Introduction to Lattice Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/lattices.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {215--222})

@ARTICLE(PRE_TOPC.ABS,
      AUTHOR  = {Padlewska, Beata and Darmochwa{\l}, Agata},
      TITLE   = {Topological Spaces and Continuous Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/pre_topc.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {223--230})

@ARTICLE(TOPS_1.ABS,
      AUTHOR  = {Wysocki, Miros{\l}aw and Darmochwa{\l}, Agata},
      TITLE   = {Subsets of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/tops_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {231--237})

@ARTICLE(CONNSP_1.ABS,
      AUTHOR  = {Padlewska, Beata},
      TITLE   = {Connected Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/connsp_1.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {239--244})

@ARTICLE(FUNCT_3.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Basic Functions and Operations on Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-1/funct_3.pdf},
      YEAR    = {1990},
      NUMBER  = {{\bf 1}},
      VOLUME  = 1,
      PAGES   = {245--254})

@ARTICLE(TOPS_2.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata},
      TITLE   = {Families of Subsets, Subspaces and Mappings in Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/tops_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {257--261},
      NUMBER  = {{\bf 2}})

@ARTICLE(ANAL_1.ABS,
      AUTHOR  = {Popio{\l}ek, Jan},
      TITLE   = {Some Properties of Functions Modul and Signum},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/anal_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {263--264},
      NUMBER  = {{\bf 2}})

@ARTICLE(ABSVALUE.ABS,
      AUTHOR  = {Popio{\l}ek, Jan},
      TITLE   = {Some Properties of Functions Modul and Signum},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/absvalue.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {263--264},
      NUMBER  = {{\bf 2}})

@ARTICLE(WELLORD2.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Zermelo Theorem and Axiom of Choice},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/wellord2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {265--267},
      NUMBER  = {{\bf 2}})

@ARTICLE(SEQ_1.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Real Sequences and Basic Operations on Them},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/seq_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {269--272},
      NUMBER  = {{\bf 2}})

@ARTICLE(SEQ_2.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Convergent Sequences and the Limit of Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/seq_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {273--275},
      NUMBER  = {{\bf 2}})

@ARTICLE(ZFMODEL1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Properties of {ZF} Models},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/zfmodel1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {277--280},
      NUMBER  = {{\bf 2}})

@ARTICLE(ORDINAL2.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Sequences of Ordinal Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/ordinal2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {281--290},
      NUMBER  = {{\bf 2}})

@ARTICLE(RLVECT_1.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Vectors in Real Linear Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/rlvect_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {291--296},
      NUMBER  = {{\bf 2}})

@ARTICLE(RLSUB_1.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Subspaces and Cosets of Subspaces in Real Linear Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/rlsub_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {297--301},
      NUMBER  = {{\bf 2}})

@ARTICLE(QC_LANG1.ABS,
      AUTHOR  = {Rudnicki, Piotr and Trybulec, Andrzej},
      TITLE   = {A First Order Language},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/qc_lang1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {303--311},
      NUMBER  = {{\bf 2}})

@ARTICLE(ORDERS_1.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Partially Ordered Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/orders_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {313--319},
      NUMBER  = {{\bf 2}})

@ARTICLE(RECDEF_1.ABS,
      AUTHOR  = {Hryniewiecki, Krzysztof},
      TITLE   = {Recursive Definitions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/recdef_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {321--328},
      NUMBER  = {{\bf 2}})

@ARTICLE(FUNCOP_1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Binary Operations Applied to Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/funcop_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {329--334},
      NUMBER  = {{\bf 2}})

@ARTICLE(VECTSP_1.ABS,
      AUTHOR  = {Kusak, Eugeniusz and Leo{\'n}czuk, Wojciech and Muzalewski, Micha{\l}},
      TITLE   = {Abelian Groups, Fields and Vector Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/vectsp_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {335--342},
      NUMBER  = {{\bf 2}})

@ARTICLE(PARSP_1.ABS,
      AUTHOR  = {Kusak, Eugeniusz and Leo{\'n}czuk, Wojciech and Muzalewski, Micha{\l}},
      TITLE   = {Parallelity Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/parsp_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {343--348},
      NUMBER  = {{\bf 2}})

@ARTICLE(SYMSP_1.ABS,
      AUTHOR  = {Kusak, Eugeniusz and Leo{\'n}czuk, Wojciech and Muzalewski, Micha{\l}},
      TITLE   = {Construction of a bilinear antisymmetric form in simplectic vector space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/symsp_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {349--352},
      NUMBER  = {{\bf 2}})

@ARTICLE(ORTSP_1.ABS,
      AUTHOR  = {Kusak, Eugeniusz and Leo{\'n}czuk, Wojciech and Muzalewski, Micha{\l}},
      TITLE   = {Construction of a bilinear symmetric form in orthogonal vector space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/ortsp_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {353--356},
      NUMBER  = {{\bf 2}})

@ARTICLE(PARTFUN1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Partial Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/partfun1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {357--367},
      NUMBER  = {{\bf 2}})

@ARTICLE(SETWISEO.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Semilattice Operations on Finite Subsets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/setwiseo.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {369--376},
      NUMBER  = {{\bf 2}})

@ARTICLE(CARD_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Cardinal Numbers},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {377--382},
      NUMBER  = {{\bf 2}},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/card_1.pdf})

@ARTICLE(COMPTS_1.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata},
      TITLE   = {Compact Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/compts_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {383--386},
      NUMBER  = {{\bf 2}})

@ARTICLE(ORDERS_2.ABS,
      AUTHOR  = {Trybulec, Wojciech A. and Bancerek, Grzegorz},
      TITLE   = {Kuratowski -- {Z}orn Lemma},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/orders_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {387--393},
      NUMBER  = {{\bf 2}})

@ARTICLE(RLSUB_2.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Operations on Subspaces in Real Linear Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/rlsub_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {395--399},
      NUMBER  = {{\bf 2}})

@ARTICLE(PROB_1.ABS,
      AUTHOR  = {N\k{e}dzusiak, Andrzej},
      TITLE   = { $\sigma$-Fields and Probability},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/prob_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {401--407},
      NUMBER  = {{\bf 2}})

@ARTICLE(CAT_1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Introduction to Categories and Functors},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/cat_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {409--420},
      NUMBER  = {{\bf 2}})

@ARTICLE(TREES_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Introduction to Trees},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-2/trees_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {421--427},
      NUMBER  = {{\bf 2}})

@ARTICLE(WELLSET1.ABS,
      AUTHOR  = {Nowak, Bogdan and Bia{\l}ecki, S{\l}awomir},
      TITLE   = {Zermelo's Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/wellset1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {431--432},
      NUMBER  = {{\bf 3}})

@ARTICLE(REALSET1.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef},
      TITLE   = {Group and Field Definitions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/realset1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {433--439},
      NUMBER  = {{\bf 3}})

@ARTICLE(EQREL_1.ABS,
      AUTHOR  = {Raczkowski, Konrad and Sadowski, Pawe{\l}},
      TITLE   = {Equivalence Relations and Classes of Abstraction},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/eqrel_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {441--444},
      NUMBER  = {{\bf 3}})

@ARTICLE(SQUARE_1.ABS,
      AUTHOR  = {Trybulec, Andrzej and Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Some Properties of Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/square_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {445--449},
      NUMBER  = {{\bf 3}})

@ARTICLE(QC_LANG2.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Connectives and Subformulae of the First Order Language},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/qc_lang2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {451--458},
      NUMBER  = {{\bf 3}})

@ARTICLE(QC_LANG3.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw and Bancerek, Grzegorz},
      TITLE   = {Variables in Formulae of the First Order Language},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/qc_lang3.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {459--469},
      NUMBER  = {{\bf 3}})

@ARTICLE(SEQM_3.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Monotone Real Sequences. {S}ubsequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/seqm_3.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {471--475},
      NUMBER  = {{\bf 3}})

@ARTICLE(SEQ_4.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Convergent Real Sequences. {U}pper and Lower Bound of Sets of Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/seq_4.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {477--481},
      NUMBER  = {{\bf 3}})

@ARTICLE(MIDSP_1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Midpoint algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/midsp_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {483--488},
      NUMBER  = {{\bf 3}})

@ARTICLE(QMAX_1.ABS,
      AUTHOR  = {Sadowski, Pawe{\l} and Trybulec, Andrzej and Raczkowski, Konrad},
      TITLE   = {The Fundamental Logic Structure in Quantum Mechanics},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/qmax_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {489--494},
      NUMBER  = {{\bf 3}})

@ARTICLE(FRAENKEL.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Function Domains and {F}r{\ae}nkel Operator},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/fraenkel.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {495--500},
      NUMBER  = {{\bf 3}})

@ARTICLE(INT_1.ABS,
      AUTHOR  = {Trybulec, Micha{\l} J.},
      TITLE   = {Integers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/int_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {501--505},
      NUMBER  = {{\bf 3}})

@ARTICLE(COMPLEX1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {The Complex Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/complex1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {507--513},
      NUMBER  = {{\bf 3}})

@ARTICLE(ORDINAL3.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Ordinal Arithmetics},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/ordinal3.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {515--519},
      NUMBER  = {{\bf 3}})

@ARTICLE(FUNCT_4.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {The Modification of a Function by a Function and the Iteration of the Composition of a Function},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/funct_4.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {521--527},
      NUMBER  = {{\bf 3}})

@ARTICLE(FINSEQ_2.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Finite Sequences and Tuples of Elements of a Non-empty Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/finseq_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {529--536},
      NUMBER  = {{\bf 3}})

@ARTICLE(FUNCT_5.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Curried and Uncurried Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/funct_5.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {537--541},
      NUMBER  = {{\bf 3}})

@ARTICLE(CARD_2.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Cardinal Arithmetics},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/card_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {543--547},
      NUMBER  = {{\bf 3}})

@ARTICLE(PARSP_2.ABS,
      AUTHOR  = {Kusak, Eugeniusz and Leo{\'n}czuk, Wojciech},
      TITLE   = {Fano-{D}esargues Parallelity Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/parsp_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {549--553},
      NUMBER  = {{\bf 3}})

@ARTICLE(FUNCSDOM.ABS,
      AUTHOR  = {Oryszczyszyn , Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Real Functions Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/funcsdom.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {555--561},
      NUMBER  = {{\bf 3}})

@ARTICLE(CLASSES1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Tarski's Classes and Ranks},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/classes1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {563--567},
      NUMBER  = {{\bf 3}})

@ARTICLE(FINSEQ_3.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Non-contiguous Substrings and One-to-one Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/finseq_3.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {569--573},
      NUMBER  = {{\bf 3}})

@ARTICLE(FINSEQ_4.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Pigeon Hole Principle},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/finseq_4.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {575--579},
      NUMBER  = {{\bf 3}})

@ARTICLE(RLVECT_2.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Linear Combinations in Real Linear Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/rlvect_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {581--588},
      NUMBER  = {{\bf 3}})

@ARTICLE(CARD_3.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {K{\"o}nig's Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/card_3.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {589--593},
      NUMBER  = {{\bf 3}})

@ARTICLE(CLASSES2.ABS,
      AUTHOR  = {Nowak, Bogdan and Bancerek, Grzegorz},
      TITLE   = {Universal Classes},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/classes2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {595--600},
      NUMBER  = {{\bf 3}})

@ARTICLE(ANALOAF.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Analytical Ordered Affine Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/analoaf.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {601--605},
      NUMBER  = {{\bf 3}})

@ARTICLE(METRIC_1.ABS,
      AUTHOR  = {Kanas, Stanis{\l}awa and Lecko, Adam and Startek, Mariusz},
      TITLE   = {Metric Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/metric_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {607--610},
      NUMBER  = {{\bf 3}})

@ARTICLE(DIRAF.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Ordered Affine Spaces Defined in Terms of Directed Parallelity -- Part {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/diraf.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {611--615},
      NUMBER  = {{\bf 3}})

@ARTICLE(AFF_1.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Parallelity and Lines in Affine Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-3/aff_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {617--621},
      NUMBER  = {{\bf 3}})

@ARTICLE(AFF_2.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Classical Configurations in Affine Planes},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/aff_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {625--633},
      NUMBER  = {{\bf 4}})

@ARTICLE(AFF_3.ABS,
      AUTHOR  = {Kusak, Eugeniusz and Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Affine Localizations of {D}esargues Axiom},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/aff_3.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {635--642},
      NUMBER  = {{\bf 4}})

@ARTICLE(FINSEQOP.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Binary Operations Applied to Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/finseqop.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {643--649},
      NUMBER  = {{\bf 4}})

@ARTICLE(SETWOP_2.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Semigroup operations on finite subsets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/setwop_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {651--656},
      NUMBER  = {{\bf 4}})

@ARTICLE(COLLSP.ABS,
      AUTHOR  = {Skaba, Wojciech},
      TITLE   = {The Collinearity Structure},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/collsp.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {657--659},
      NUMBER  = {{\bf 4}})

@ARTICLE(RVSUM_1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {The Sum and Product of Finite Sequences of Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/rvsum_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {661--668},
      NUMBER  = {{\bf 4}})

@ARTICLE(CQC_LANG.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {A Classical First Order Language},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/cqc_lang.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {669--676},
      NUMBER  = {{\bf 4}})

@ARTICLE(PASCH.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof and Pra{\.z}mowska, Ma{\l}gorzata},
      TITLE   = {Classical and Non--classical {P}asch Configurations in Ordered Affine Planes},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/pasch.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {677--680},
      NUMBER  = {{\bf 4}})
      
@ARTICLE(REAL_LAT.ABS,
      AUTHOR  = {Chmur, Marek},
      TITLE   = {The Lattice of Real Numbers. {T}he Lattice of Real Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/real_lat.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {681--684},
      NUMBER  = {{\bf 4}})

@ARTICLE(TDGROUP.ABS,
      AUTHOR  = {Lewandowski, Grzegorz and Pra{\.z}mowski, Krzysztof},
      TITLE   = {A Construction of an Abstract Space of Congruence of Vectors},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/tdgroup.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {685--688},
      NUMBER  = {{\bf 4}})

@ARTICLE(CQC_THE1.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata},
      TITLE   = {A First--Order Predicate Calculus},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/cqc_the1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {689--695},
      NUMBER  = {{\bf 4}})

@ARTICLE(PARTFUN2.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Partial Functions from a Domain to a Domain},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/partfun2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {697--702},
      NUMBER  = {{\bf 4}})

@ARTICLE(RFUNCT_1.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Partial Functions from a Domain to the Set of Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/rfunct_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {703--709},
      NUMBER  = {{\bf 4}})

@ARTICLE(ORDINAL4.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Increasing and Continuous Ordinal Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/ordinal4.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {711--714},
      NUMBER  = {{\bf 4}})

@ARTICLE(TRANSGEO.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Transformations in Affine Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/transgeo.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {715--723},
      NUMBER  = {{\bf 4}})

@ARTICLE(CAT_2.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Subcategories and Products of Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/cat_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {725--732},
      NUMBER  = {{\bf 4}})

@ARTICLE(MARGREL1.ABS,
      AUTHOR  = {Woronowicz, Edmund},
      TITLE   = {Many Argument Relations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/margrel1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {733--737},
      NUMBER  = {{\bf 4}})

@ARTICLE(VALUAT_1.ABS,
      AUTHOR  = {Woronowicz, Edmund},
      TITLE   = {Interpretation and Satisfiability in the First Order Logic},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/valuat_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {739--743},
      NUMBER  = {{\bf 4}})

@ARTICLE(PROB_2.ABS,
      AUTHOR  = {N\k{e}dzusiak, Andrzej},
      TITLE   = {Probability},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/prob_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {745--749},
      NUMBER  = {{\bf 4}})

@ARTICLE(TRANSLAC.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Translations in Affine Planes},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/translac.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {751--753},
      NUMBER  = {{\bf 4}})

@ARTICLE(RPR_1.ABS,
      AUTHOR  = {Popio{\l}ek, Jan},
      TITLE   = {Introduction to Probability},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/rpr_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {755--760},
      NUMBER  = {{\bf 4}})

@ARTICLE(ANPROJ_1.ABS,
      AUTHOR  = {Leo{\'n}czuk, Wojciech and Pra{\.z}mowski, Krzysztof},
      TITLE   = {A Construction of Analytical Projective Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/anproj_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {761--766},
      NUMBER  = {{\bf 4}})

@ARTICLE(ANPROJ_2.ABS,
      AUTHOR  = {Leo{\'n}czuk, Wojciech and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Projective Spaces -- Part {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/anproj_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {767--776},
      NUMBER  = {{\bf 4}})

@ARTICLE(RCOMP_1.ABS,
      AUTHOR  = {Raczkowski, Konrad and Sadowski, Pawe{\l}},
      TITLE   = {Topological Properties of Subsets in Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/rcomp_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {777--780},
      NUMBER  = {{\bf 4}})

@ARTICLE(RFUNCT_2.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Properties of Real Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/rfunct_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {781--786},
      NUMBER  = {{\bf 4}})

@ARTICLE(FCONT_1.ABS,
      AUTHOR  = {Raczkowski, Konrad and Sadowski, Pawe{\l}},
      TITLE   = {Real Function Continuity},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/fcont_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {787--791},
      NUMBER  = {{\bf 4}})

@ARTICLE(FCONT_2.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw and Raczkowski, Konrad},
      TITLE   = {Real Function Uniform Continuity},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/fcont_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {793--795},
      NUMBER  = {{\bf 4}})

@ARTICLE(FDIFF_1.ABS,
      AUTHOR  = {Raczkowski, Konrad and Sadowski, Pawe{\l}},
      TITLE   = {Real Function Differentiability},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/fdiff_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {797--801},
      NUMBER  = {{\bf 4}})

@ARTICLE(ROLLE.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw and Raczkowski, Konrad and Sadowski, Pawe{\l}},
      TITLE   = {Average Value Theorems for Real Functions of One Variable},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-4/rolle.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {803--805},
      NUMBER  = {{\bf 4}})

@ARTICLE(REALSET2.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef},
      TITLE   = {Properties of Fields},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/realset2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {807--812},
      NUMBER  = {{\bf 5}})

@ARTICLE(FILTER_0.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Filters -- Part {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/filter_0.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {813--819},
      NUMBER  = {{\bf 5}})

@ARTICLE(GROUP_1.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Groups},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/group_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {821--827},
      NUMBER  = {{\bf 5}})

@ARTICLE(INT_2.ABS,
      AUTHOR  = {Kwiatek, Rafa{\l} and Zwara, Grzegorz},
      TITLE   = {The Divisibility of Integers and Integer Relatively Primes},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/int_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {829--832},
      NUMBER  = {{\bf 5}})

@ARTICLE(ALGSTR_1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Skaba, Wojciech},
      TITLE   = {From Loops to {A}belian Multiplicative Groups with Zero},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/algstr_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {833--840},
      NUMBER  = {{\bf 5}})

@ARTICLE(RAT_1.ABS,
      AUTHOR  = {Kondracki, Andrzej},
      TITLE   = {Basic Properties of Rational Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/rat_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {841--845},
      NUMBER  = {{\bf 5}})

@ARTICLE(RLVECT_3.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Basis of Real Linear Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/rlvect_3.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {847--850},
      NUMBER  = {{\bf 5}})

@ARTICLE(VECTSP_3.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Finite Sums of Vectors in Vector Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/vectsp_3.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {851--854},
      NUMBER  = {{\bf 5}})

@ARTICLE(GROUP_2.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Subgroup and Cosets of Subgroups},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/group_2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {855--864},
      NUMBER  = {{\bf 5}})

@ARTICLE(VECTSP_4.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Subspaces and Cosets of Subspaces in Vector Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/vectsp_4.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {865--870},
      NUMBER  = {{\bf 5}})

@ARTICLE(VECTSP_5.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Operations on Subspaces in Vector Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/vectsp_5.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {871--876},
      NUMBER  = {{\bf 5}})

@ARTICLE(VECTSP_6.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Linear Combinations in Vector Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/vectsp_6.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {877--882},
      NUMBER  = {{\bf 5}})

@ARTICLE(VECTSP_7.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Basis of Vector Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/vectsp_7.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {883--885},
      NUMBER  = {{\bf 5}})

@ARTICLE(NEWTON.ABS,
      AUTHOR  = {Kwiatek, Rafa{\l}},
      TITLE   = {Factorial and {N}ewton coefficients},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/newton.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {887--890},
      NUMBER  = {{\bf 5}})

@ARTICLE(ANALMETR.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Analytical Metric Affine Spaces and Planes},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/analmetr.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {891--899},
      NUMBER  = {{\bf 5}})

@ARTICLE(ANPROJ_3.ABS,
      AUTHOR  = {Leo{\'n}czuk, Wojciech and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Projective Spaces -- part {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/anproj_3.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {901--907},
      NUMBER  = {{\bf 5}})

@ARTICLE(ANPROJ_4.ABS,
      AUTHOR  = {Leo{\'n}czuk, Wojciech and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Projective Spaces -- part {III}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/anproj_4.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {909--918},
      NUMBER  = {{\bf 5}})

@ARTICLE(ANPROJ_5.ABS,
      AUTHOR  = {Leo{\'n}czuk, Wojciech and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Projective Spaces -- part {IV}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/anproj_5.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {919--927},
      NUMBER  = {{\bf 5}})

@ARTICLE(ANPROJ_6.ABS,
      AUTHOR  = {Leo{\'n}czuk, Wojciech and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Projective Spaces -- part {V}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/anproj_6.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {929--938},
      NUMBER  = {{\bf 5}})

@ARTICLE(ANPROJ_7.ABS,
      AUTHOR  = {Leo{\'n}czuk, Wojciech and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Projective Spaces -- part {VI}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/anproj_7.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {939--947},
      NUMBER  = {{\bf 5}})

@ARTICLE(NET_1.ABS,
      AUTHOR  = {Korczy{\'n}ski, Waldemar},
      TITLE   = {Some Elementary Notions of the Theory of {P}etri Nets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/net_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {949--953},
      NUMBER  = {{\bf 5}})

@ARTICLE(GROUP_3.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Classes of Conjugation. {N}ormal Subgroups},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/group_3.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {955--962},
      NUMBER  = {{\bf 5}})

@ARTICLE(ZF_LANG1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Replacing of Variables in Formulas of {ZF} Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/zf_lang1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {963--972},
      NUMBER  = {{\bf 5}})

@ARTICLE(ZF_REFLE.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {The Reflection Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/zf_refle.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {973--977},
      NUMBER  = {{\bf 5}})

@ARTICLE(FINSOP_1.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Binary Operations on Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/finsop_1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {979--981},
      NUMBER  = {{\bf 5}})

@ARTICLE(LATTICE2.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Finite Join and Finite Meet and Dual Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/lattice2.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {983--988},
      NUMBER  = {{\bf 5}})

@ARTICLE(ZFREFLE1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Consequences of the Reflection Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1990-1/pdf1-5/zfrefle1.pdf},
      YEAR    = {1990},
      VOLUME  = 1,
      PAGES   = {989--993},
      NUMBER  = {{\bf 5}})

@ARTICLE(VECTSP_2.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Construction of Rings and Left-, Right-, and Bi-Modules over a Ring},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/vectsp_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {3--11},
      NUMBER  = {{\bf 1}})

@ARTICLE(PROJDES1.ABS,
      AUTHOR  = {Kusak, Eugeniusz},
      TITLE   = {Desargues Theorem In Projective 3-Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/projdes1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {13--16},
      NUMBER  = {{\bf 1}})

@ARTICLE(LIMFUNC1.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {The Limit of a Real Function at Infinity},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/limfunc1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {17--28},
      NUMBER  = {{\bf 1}})

@ARTICLE(LIMFUNC2.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {One-Side Limits of a Real Function at a Point},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/limfunc2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {29--40},
      NUMBER  = {{\bf 1}})

@ARTICLE(GROUP_4.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Lattice of Subgroups of a Group. {F}rattini Subgroup},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/group_4.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {41--47},
      NUMBER  = {{\bf 1}})

@ARTICLE(REAL_2.ABS,
      AUTHOR  = {Kondracki, Andrzej},
      TITLE   = {Equalities and Inequalities in Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/real_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {49--63},
      NUMBER  = {{\bf 1}})

@ARTICLE(CARD_4.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Countable Sets and {H}essenberg's Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/card_4.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {65--69},
      NUMBER  = {{\bf 1}})

@ARTICLE(LIMFUNC3.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {The Limit of a Real Function at a Point},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/limfunc3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {71--80},
      NUMBER  = {{\bf 1}})

@ARTICLE(LIMFUNC4.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {The Limit of a Composition of Real Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/limfunc4.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {81--92},
      NUMBER  = {{\bf 1}})

@ARTICLE(CONNSP_2.ABS,
      AUTHOR  = {Padlewska, Beata},
      TITLE   = {Locally Connected Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/connsp_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {93--96},
      NUMBER  = {{\bf 1}})

@ARTICLE(ALGSEQ_1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Szczerba, Les{\l}aw W.},
      TITLE   = {Construction of Finite Sequences over Ring and Left-, Right-,  and Bi-Modules over a Ring},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/algseq_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {97--104},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOLER_1.ABS,
      AUTHOR  = {Hryniewiecki, Krzysztof},
      TITLE   = {Relations of Tolerance},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/toler_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {105--109},
      NUMBER  = {{\bf 1}})

@ARTICLE(NORMSP_1.ABS,
      AUTHOR  = {Popio{\l}ek, Jan},
      TITLE   = {Real Normed Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/normsp_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {111--115},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCHEME1.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Schemes of Existence of some Types of Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/scheme1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {117--123},
      NUMBER  = {{\bf 1}})

@ARTICLE(PREPOWER.ABS,
      AUTHOR  = {Raczkowski, Konrad},
      TITLE   = {Integer and Rational Exponents},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/prepower.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {125--130},
      NUMBER  = {{\bf 1}})

@ARTICLE(HOMOTHET.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Homotheties and Shears in Affine Planes},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/homothet.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {131--133},
      NUMBER  = {{\bf 1}})

@ARTICLE(AFVECT0.ABS,
      AUTHOR  = {Lewandowski, Grzegorz and Pra{\.z}mowski, Krzysztof and Lewandowska, Bo{\.z}ena},
      TITLE   = {Directed Geometrical Bundles and Their Analytical Representation},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/afvect0.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {135--141},
      NUMBER  = {{\bf 1}})

@ARTICLE(ZFMODEL2.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Definable Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/zfmodel2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {143--145},
      NUMBER  = {{\bf 1}})

@ARTICLE(LUKASI_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz  and Darmochwa{\l}, Agata and Trybulec, Andrzej},
      TITLE   = {Propositional Calculus},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/lukasi_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {147--150},
      NUMBER  = {{\bf 1}})

@ARTICLE(COMPLSP1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw and Trybulec, Andrzej},
      TITLE   = {Complex Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/complsp1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {151--158},
      NUMBER  = {{\bf 1}})

@ARTICLE(REALSET3.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef},
      TITLE   = {Several Properties of Fields. {F}ield Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/realset3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {159--162},
      NUMBER  = {{\bf 1}})

@ARTICLE(SUPINF_1.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef},
      TITLE   = {Infimum and Supremum of the Set of Real Numbers. {M}easure Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/supinf_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {163--171},
      NUMBER  = {{\bf 1}})

@ARTICLE(SUPINF_2.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef},
      TITLE   = {Series of  Positive Real Numbers. {M}easure Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/supinf_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {173--183},
      NUMBER  = {{\bf 1}})

@ARTICLE(ALGSTR_2.ABS,
      AUTHOR  = {Skaba, Wojciech and Muzalewski, Micha{\l}},
      TITLE   = {From Double Loops to Fields},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-1/algstr_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {185--191},
      NUMBER  = {{\bf 1}})

@ARTICLE(METRIC_3.ABS,
      AUTHOR  = {Kanas, Stanis{\l}awa and Stankiewicz, Jan},
      TITLE   = {Metrics in {C}artesian Product},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/metric_3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {193--197},
      NUMBER  = {{\bf 2}})

@ARTICLE(SUB_METR.ABS,
      AUTHOR  = {Lecko, Adam and Startek, Mariusz},
      TITLE   = {Submetric Spaces -- Part {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/sub_metr.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {199--203},
      NUMBER  = {{\bf 2}})

@ARTICLE(METRIC_2.ABS,
      AUTHOR  = {Lecko, Adam and Startek, Mariusz},
      TITLE   = {On Pseudometric Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/metric_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {205--211},
      NUMBER  = {{\bf 2}})

@ARTICLE(POWER.ABS,
      AUTHOR  = {Raczkowski, Konrad and N\k{e}dzusiak, Andrzej},
      TITLE   = {Real Exponents and Logarithms},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/power.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {213--216},
      NUMBER  = {{\bf 2}})

@ARTICLE(HESSENBE.ABS,
      AUTHOR  = {Kusak, Eugeniusz and Leo{\'n}czuk, Wojciech},
      TITLE   = {Hessenberg Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/hessenbe.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {217--219},
      NUMBER  = {{\bf 2}})

@ARTICLE(MULTOP_1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Skaba, Wojciech},
      TITLE   = {Three-Argument Operations and Four-Argument Operations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/multop_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {221--224},
      NUMBER  = {{\bf 2}})

@ARTICLE(INCPROJ.ABS,
      AUTHOR  = {Leo{\'n}czuk, Wojciech and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Incidence Projective Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/incproj.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {225--232},
      NUMBER  = {{\bf 2}})

@ARTICLE(AFVECT01.ABS,
      AUTHOR  = {Konstanta, Barbara and Kowieska, Urszula and Lewandowski, Grzegorz and Pra{\.z}mowski, Krzysztof},
      TITLE   = {One-Dimensional Congruence of Segments, Basic Facts and Midpoint  Relation},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/afvect01.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {233--235},
      NUMBER  = {{\bf 2}})

@ARTICLE(NORMFORM.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Algebra of Normal Forms},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/normform.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {237--242},
      NUMBER  = {{\bf 2}})

@ARTICLE(O_RING_1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Szczerba, Les{\l}aw W.},
      TITLE   = {Ordered Rings -- Part {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/o_ring_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {243--245},
      NUMBER  = {{\bf 2}})

@ARTICLE(O_RING_2.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Szczerba, Les{\l}aw W.},
      TITLE   = {Ordered Rings -- Part {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/o_ring_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {247--249},
      NUMBER  = {{\bf 2}})

@ARTICLE(O_RING_3.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Szczerba, Les{\l}aw W.},
      TITLE   = {Ordered Rings -- Part {III}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/o_ring_3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {251--253},
      NUMBER  = {{\bf 2}})

@ARTICLE(MCART_2.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Skaba, Wojciech},
      TITLE   = {{N}-Tuples and {C}artesian Products for n$=$5},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/mcart_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {255--258},
      NUMBER  = {{\bf 2}})

@ARTICLE(ALGSTR_3.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Skaba, Wojciech},
      TITLE   = {Ternary Fields},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/algstr_3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {259--261},
      NUMBER  = {{\bf 2}})

@ARTICLE(MEASURE1.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef},
      TITLE   = {The $\sigma$-additive Measure Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/measure1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {263--270},
      NUMBER  = {{\bf 2}})

@ARTICLE(PROJRED1.ABS,
      AUTHOR  = {Kusak, Eugeniusz and Leo{\'n}czuk, Wojciech},
      TITLE   = {Incidence Projective Space (a reduction theorem in a plane)},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/projred1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {271--274},
      NUMBER  = {{\bf 2}})

@ARTICLE(MOD_1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Skaba, Wojciech},
      TITLE   = {Groups, Rings, Left- and Right-Modules},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/mod_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {275--278},
      NUMBER  = {{\bf 2}})

@ARTICLE(LMOD_1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Skaba, Wojciech},
      TITLE   = {Finite Sums of Vectors in Left Module over Associative Ring},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/lmod_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {279--282},
      NUMBER  = {{\bf 2}})

@ARTICLE(LMOD_2.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Skaba, Wojciech},
      TITLE   = {Submodules and Cosets of Submodules in Left Module over Associative Ring},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/lmod_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {283--287},
      NUMBER  = {{\bf 2}})

@ARTICLE(LMOD_3.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Skaba, Wojciech},
      TITLE   = {Operations on Submodules in Left Module over Associative Ring},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/lmod_3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {289--293},
      NUMBER  = {{\bf 2}})

@ARTICLE(LMOD_4.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Skaba, Wojciech},
      TITLE   = {Linear Combinations in Left Module over Associative Ring},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/lmod_4.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {295--300},
      NUMBER  = {{\bf 2}})

@ARTICLE(LMOD_5.ABS,
      AUTHOR  = {Muzalewski, Micha{\l} and Skaba, Wojciech},
      TITLE   = {Linear Independence in Left Module over Domain},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/lmod_5.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {301--303},
      NUMBER  = {{\bf 2}})

@ARTICLE(PROCAL_1.ABS,
      AUTHOR  = {Popio{\l}ek, Jan and Trybulec, Andrzej},
      TITLE   = {Calculus of Propositions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/procal_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {305--307},
      NUMBER  = {{\bf 2}})

@ARTICLE(CQC_THE2.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata},
      TITLE   = {Calculus of Quantifiers. {D}eduction Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-2/cqc_the2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {309--312},
      NUMBER  = {{\bf 2}})

@ARTICLE(ANALTRAP.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {A construction of analytical Ordered Trapezium Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/analtrap.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {315--322},
      NUMBER  = {{\bf 3}})

@ARTICLE(GEOMTRAP.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {A construction of analytical Ordered Trapezium Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/geomtrap.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {315--322},
      NUMBER  = {{\bf 3}})

@ARTICLE(PROJRED2.ABS,
      AUTHOR  = {Kusak, Eugeniusz and Leo{\'n}czuk, Wojciech and Pra{\.z}mowski, Krzysztof},
      TITLE   = {On Projections in Projective Planes ({P}art {II} )},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/projred2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {323--329},
      NUMBER  = {{\bf 3}})

@ARTICLE(CONAFFM.ABS,
      AUTHOR  = {{\'{S}}wierzy{\'n}ska, Jolanta and {\'{S}}wierzy{\'n}ski, Bogdan},
      TITLE   = {Metric-Affine Configurations in Metric Affine Planes -- {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/conaffm.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {331--334},
      NUMBER  = {{\bf 3}})

@ARTICLE(CONMETR.ABS,
      AUTHOR  = {{\'{S}}wierzy{\'n}ska, Jolanta and {\'{S}}wierzy{\'n}ski, Bogdan},
      TITLE   = {Metric-Affine Configurations in Metric Affine Planes -- {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/conmetr.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {335--340},
      NUMBER  = {{\bf 3}})

@ARTICLE(PAPDESAF.ABS,
      AUTHOR  = {Pra{\.z}mowski, Krzysztof},
      TITLE   = {{F}anoian, {P}appian and {D}esarguesian Affine Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/papdesaf.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {341--346},
      NUMBER  = {{\bf 3}})

@ARTICLE(PARDEPAP.ABS,
      AUTHOR  = {Pra{\.z}mowski, Krzysztof and Radziszewski, Krzysztof},
      TITLE   = {Elementary Variants of Affine Configurational Theorems},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/pardepap.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {347--348},
      NUMBER  = {{\bf 3}})

@ARTICLE(SEMI_AF1.ABS,
      AUTHOR  = {Kusak, Eugeniusz and Radziszewski, Krzysztof},
      TITLE   = {Semi\_Affine Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/semi_af1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {349--356},
      NUMBER  = {{\bf 3}})

@ARTICLE(AFF_4.ABS,
      AUTHOR  = {Leo{\'n}czuk, Wojciech and Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Planes in Affine Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/aff_4.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {357--363},
      NUMBER  = {{\bf 3}})

@ARTICLE(GRAPH_1.ABS,
      AUTHOR  = {Hryniewiecki, Krzysztof},
      TITLE   = {Graphs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/graph_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {365--370},
      NUMBER  = {{\bf 3}})

@ARTICLE(ZF_FUND1.ABS,
      AUTHOR  = {Kondracki, Andrzej},
      TITLE   = {{M}ostowski's Fundamental Operations -- {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/zf_fund1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {371--375},
      NUMBER  = {{\bf 3}})

@ARTICLE(AFPROJ.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {A Projective Closure and Projective Horizon of an Affine Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/afproj.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {377--384},
      NUMBER  = {{\bf 3}})

@ARTICLE(SCHEMS_1.ABS,
      AUTHOR  = {Czuba, Stanis{\l}aw T.},
      TITLE   = {Schemes},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/schems_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {385--391},
      NUMBER  = {{\bf 3}})

@ARTICLE(HEYTING1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Algebra of Normal Forms Is a {H}eyting Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/heyting1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {393--396},
      NUMBER  = {{\bf 3}})

@ARTICLE(TREES_2.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {K{\"o}nig's Lemma},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/trees_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {397--402},
      NUMBER  = {{\bf 3}})

@ARTICLE(FCONT_3.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Monotonic and Continuous Real Function},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/fcont_3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {403--405},
      NUMBER  = {{\bf 3}})

@ARTICLE(FDIFF_2.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw and Raczkowski, Konrad},
      TITLE   = {Real Function Differentiability -- {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/fdiff_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {407--411},
      NUMBER  = {{\bf 3}})

@ARTICLE(PRELAMB.ABS,
      AUTHOR  = {Zielonka, Wojciech},
      TITLE   = {Preliminaries to the {L}ambek calculus},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/prelamb.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {413--418},
      NUMBER  = {{\bf 3}})

@ARTICLE(OPPCAT_1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Opposite Categories and Contravariant Functors},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/oppcat_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {419--424},
      NUMBER  = {{\bf 3}})

@ARTICLE(ZF_FUND2.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Kondracki, Andrzej},
      TITLE   = {{M}ostowski's Fundamental Operations -- {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/zf_fund2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {425--427},
      NUMBER  = {{\bf 3}})

@ARTICLE(EUCLMETR.ABS,
      AUTHOR  = {Oryszczyszyn, Henryk and Pra{\.z}mowski, Krzysztof},
      TITLE   = {Fundamental Types of Metric Affine Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/euclmetr.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {429--432},
      NUMBER  = {{\bf 3}})

@ARTICLE(FILTER_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Filters -- Part {II}. {Q}uotient Lattices Modulo Filters and Direct Product of Two Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/filter_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {433--438},
      NUMBER  = {{\bf 3}})

@ARTICLE(CONMETR1.ABS,
      AUTHOR  = {{\'{S}}wierzy{\'n}ska, Jolanta and {\'{S}}wierzy{\'n}ski, Bogdan},
      TITLE   = {Shear Theorems and their role in Affine Geometry},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-3/conmetr1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {439--444},
      NUMBER  = {{\bf 3}})

@ARTICLE(SERIES_1.ABS,
      AUTHOR  = {Raczkowski, Konrad and N\k{e}dzusiak, Andrzej},
      TITLE   = {Series},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/series_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {449--452},
      NUMBER  = {{\bf 4}})

@ARTICLE(NAT_LAT.ABS,
      AUTHOR  = {Chmur, Marek},
      TITLE   = {The Lattice of Natural Numbers and The Sublattice of it. {T}he Set of Prime Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/nat_lat.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {453--459},
      NUMBER  = {{\bf 4}})

@ARTICLE(GROUP_5.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Commutator and Center of a Group},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/group_5.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {461--466},
      NUMBER  = {{\bf 4}})

@ARTICLE(NATTRA_1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Natural Transformations. {D}iscrete Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/nattra_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {467--474},
      NUMBER  = {{\bf 4}})

@ARTICLE(MATRIX_1.ABS,
      AUTHOR  = {Jankowska, Katarzyna},
      TITLE   = {Matrices. {A}belian Group of Matrices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/matrix_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {475--480},
      NUMBER  = {{\bf 4}})

@ARTICLE(PCOMPS_1.ABS,
      AUTHOR  = {Borys, Leszek},
      TITLE   = {Paracompact and Metrizable Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/pcomps_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {481--485},
      NUMBER  = {{\bf 4}})

@ARTICLE(MIDSP_2.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Atlas of {M}idpoint {A}lgebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/midsp_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {487--491},
      NUMBER  = {{\bf 4}})

@ARTICLE(MEASURE2.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef},
      TITLE   = {Several Properties of the $\sigma$-additive Measure},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/measure2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {493--497},
      NUMBER  = {{\bf 4}})

@ARTICLE(METRIC_4.ABS,
      AUTHOR  = {Kanas, Stanis{\l}awa and Lecko, Adam},
      TITLE   = {Metrics in the {C}artesian Product -- Part {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/metric_4.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {499--504},
      NUMBER  = {{\bf 4}})

@ARTICLE(ALI2.ABS,
      AUTHOR  = {de la Cruz, Alicia},
      TITLE   = {Fix Point Theorem for Compact Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/ali2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {505--506},
      NUMBER  = {{\bf 4}})

@ARTICLE(QUIN_1.ABS,
      AUTHOR  = {Popio{\l}ek, Jan},
      TITLE   = {Quadratic Inequalities},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/quin_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {507--509},
      NUMBER  = {{\bf 4}})

@ARTICLE(BHSP_1.ABS,
      AUTHOR  = {Popio{\l}ek, Jan},
      TITLE   = {Introduction to {B}anach and {H}ilbert Spaces -- Part {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/bhsp_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {511--516},
      NUMBER  = {{\bf 4}})

@ARTICLE(BHSP_2.ABS,
      AUTHOR  = {Popio{\l}ek, Jan},
      TITLE   = {Introduction to {B}anach and {H}ilbert Spaces -- Part {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/bhsp_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {517--521},
      NUMBER  = {{\bf 4}})

@ARTICLE(BHSP_3.ABS,
      AUTHOR  = {Popio{\l}ek, Jan},
      TITLE   = {Introduction to {B}anach and {H}ilbert Spaces -- Part {III}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/bhsp_3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {523--526},
      NUMBER  = {{\bf 4}})

@ARTICLE(ENS_1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Category {E}ns},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/ens_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {527--533},
      NUMBER  = {{\bf 4}})

@ARTICLE(BORSUK_1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {A {B}orsuk Theorem on Homotopy Types},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/borsuk_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {535--545},
      NUMBER  = {{\bf 4}})

@ARTICLE(FUNCT_6.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {{C}artesian Product of Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/funct_6.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {547--552},
      NUMBER  = {{\bf 4}})

@ARTICLE(MODAL_1.ABS,
      AUTHOR  = {de la Cruz, Alicia},
      TITLE   = {Introduction to Modal Propositional Logic},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/modal_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {553--558},
      NUMBER  = {{\bf 4}})

@ARTICLE(TBSP_1.ABS,
      AUTHOR  = {de la Cruz, Alicia},
      TITLE   = {Totally Bounded Metric Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/tbsp_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {559--562},
      NUMBER  = {{\bf 4}})

@ARTICLE(GRCAT_1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Categories of Groups},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/grcat_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {563--571},
      NUMBER  = {{\bf 4}})

@ARTICLE(GROUP_6.ABS,
      AUTHOR  = {Trybulec, Wojciech A. and Trybulec, Micha{\l} J.},
      TITLE   = {Homomorphisms and Isomorphisms of Groups. {Q}uotient Group},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/group_6.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {573--578},
      NUMBER  = {{\bf 4}})

@ARTICLE(MOD_2.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Rings and Modules -- Part {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/mod_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {579--585},
      NUMBER  = {{\bf 4}})

@ARTICLE(MOD_3.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Free Modules},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/mod_3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {587--589},
      NUMBER  = {{\bf 4}})

@ARTICLE(ANALORT.ABS,
      AUTHOR  = {Zajkowski, Jaros{\l}aw},
      TITLE   = {Oriented Metric-Affine Plane -- Part {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/analort.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {591--597},
      NUMBER  = {{\bf 4}})

@ARTICLE(EUCLID.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata},
      TITLE   = {The {E}uclidean Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/euclid.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {599--603},
      NUMBER  = {{\bf 4}})

@ARTICLE(TOPMETR.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata and Nakamura, Yatsuka},
      TITLE   = {Metric Spaces as Topological Spaces -- Fundamental Concepts},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/topmetr.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {605--608},
      NUMBER  = {{\bf 4}})

@ARTICLE(HEINE.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata and Nakamura, Yatsuka},
      TITLE   = {{H}eine--{B}orel's Covering Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/heine.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {609--610},
      NUMBER  = {{\bf 4}})

@ARTICLE(TOPMETR2.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Darmochwa{\l}, Agata},
      TITLE   = {Some Facts about Union of Two Functions and Continuity of Union of Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-4/topmetr2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {611--613},
      NUMBER  = {{\bf 4}})

@ARTICLE(TOPREAL1.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata and Nakamura, Yatsuka},
      TITLE   = {The Topological Space ${\calE}^2_{\rmT}$. {A}rcs, Line Segments and Special Polygonal Arcs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/topreal1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {617--621},
      NUMBER  = {{\bf 5}})

@ARTICLE(GR_CY_1.ABS,
      AUTHOR  = {Surowik, Dariusz},
      TITLE   = {Cyclic Groups and Some of Their Properties -- Part {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/gr_cy_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {623--627},
      NUMBER  = {{\bf 5}})

@ARTICLE(ISOCAT_1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Isomorphisms of Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/isocat_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {629--634},
      NUMBER  = {{\bf 5}})

@ARTICLE(CQC_SIM1.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata and Trybulec, Andrzej},
      TITLE   = {Similarity of Formulae},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/cqc_sim1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {635--642},
      NUMBER  = {{\bf 5}})

@ARTICLE(RINGCAT1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Category of Rings},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/ringcat1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {643--648},
      NUMBER  = {{\bf 5}})

@ARTICLE(MODCAT_1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Category of Left Modules},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/modcat_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {649--652},
      NUMBER  = {{\bf 5}})

@ARTICLE(FDIFF_3.ABS,
      AUTHOR  = {Burakowska, Ewa and Madras, Beata},
      TITLE   = {Real Function One-Side Differentiability},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/fdiff_3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {653--656},
      NUMBER  = {{\bf 5}})

@ARTICLE(METRIC_6.ABS,
      AUTHOR  = {Kanas, Stanis{\l}awa and Lecko, Adam},
      TITLE   = {Sequences in Metric Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/metric_6.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {657--661},
      NUMBER  = {{\bf 5}})

@ARTICLE(TOPREAL2.ABS,
      AUTHOR  = {Darmochwa{\l}, Agata and Nakamura, Yatsuka},
      TITLE   = {The Topological Space ${\calE}^2_{\rmT}$. {S}imple Closed Curves},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/topreal2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {663--664},
      NUMBER  = {{\bf 5}})

@ARTICLE(TSEP_1.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {Separated and Weakly Separated Subspaces of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/tsep_1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {665--674},
      NUMBER  = {{\bf 5}})

@ARTICLE(L_HOSPIT.ABS,
      AUTHOR  = {Korolkiewicz, Ma{\l}gorzata},
      TITLE   = {The de l'{H}ospital Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/l_hospit.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {675--678},
      NUMBER  = {{\bf 5}})

@ARTICLE(COMMACAT.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Darmochwa\l, Agata},
      TITLE   = {Comma Category},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/commacat.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {679--681},
      NUMBER  = {{\bf 5}})

@ARTICLE(LANG1.ABS,
      AUTHOR  = {Carlson, Patricia L. and Bancerek, Grzegorz},
      TITLE   = {Context-Free Grammar -- Part {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/lang1.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {683--687},
      NUMBER  = {{\bf 5}})

@ARTICLE(MEASURE3.ABS,
      AUTHOR  = {Bia{\l}as, J\'ozef},
      TITLE   = {Completeness of the $\sigma$-Additive Measure. {M}easure Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/measure3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {689--693},
      NUMBER  = {{\bf 5}})

@ARTICLE(BHSP_4.ABS,
      AUTHOR  = {Kraszewska, El\.zbieta and Popio{\l}ek, Jan},
      TITLE   = {Series in {B}anach and {H}ilbert {S}paces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/bhsp_4.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {695--699},
      NUMBER  = {{\bf 5}})

@ARTICLE(CAT_3.ABS,
      AUTHOR  = {Byli\'{n}ski, Czes{\l}aw},
      TITLE   = {Products and Coproducts in Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/cat_3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {701--709},
      NUMBER  = {{\bf 5}})

@ARTICLE(MATRIX_2.ABS,
      AUTHOR  = {Jankowska, Katarzyna},
      TITLE   = {Transpose Matrices and Groups of Permutations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/matrix_2.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {711--717},
      NUMBER  = {{\bf 5}})

@ARTICLE(LATTICE3.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Complete Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1991-2/pdf2-5/lattice3.pdf},
      YEAR    = {1991},
      VOLUME  = 2,
      PAGES   = {719--725},
      NUMBER  = {{\bf 5}})

@ARTICLE(TMAP_1.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {Continuity of Mappings over the Union  of Subspaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/tmap_1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {1--16},
      NUMBER  = {{\bf 1}})

@ARTICLE(SEQFUNC.ABS,
      AUTHOR  = {Perkowska, Beata},
      TITLE   = {Functional Sequence from a Domain to a Domain},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/seqfunc.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {17--21},
      NUMBER  = {{\bf 1}})

@ARTICLE(MIDSP_3.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Reper Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/midsp_3.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {23--28},
      NUMBER  = {{\bf 1}})

@ARTICLE(GR_CY_2.ABS,
      AUTHOR  = {Surowik, Dariusz},
      TITLE   = {Isomorphisms of Cyclic Groups. {S}ome Properties of Cyclic Groups},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/gr_cy_2.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {29--32},
      NUMBER  = {{\bf 1}})

@ARTICLE(ISOCAT_2.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Some Isomorphisms Between Functor Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/isocat_2.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {33--40},
      NUMBER  = {{\bf 1}})

@ARTICLE(TDLAT_1.ABS,
      AUTHOR  = {Watanabe, Toshihiko},
      TITLE   = {The Lattice of Domains of a Topological Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/tdlat_1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {41--46},
      NUMBER  = {{\bf 1}})

@ARTICLE(LMOD_6.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Submodules},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/lmod_6.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {47--51},
      NUMBER  = {{\bf 1}})

@ARTICLE(DIRORT.ABS,
      AUTHOR  = {Zajkowski, Jaros{\l}aw},
      TITLE   = {Oriented Metric-Affine Plane -- Part {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/dirort.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {53--56},
      NUMBER  = {{\bf 1}})

@ARTICLE(MOD_4.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Opposite Rings, Modules and their Morphisms},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/mod_4.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {57--65},
      NUMBER  = {{\bf 1}})

@ARTICLE(MEASURE4.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef},
      TITLE   = {Properties of {C}aratheodor's Measure},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/measure4.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {67--70},
      NUMBER  = {{\bf 1}})

@ARTICLE(TDLAT_2.ABS,
      AUTHOR  = {Karno, Zbigniew and Watanabe, Toshihiko},
      TITLE   = {Completeness of the Lattices of Domains  of a Topological Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/tdlat_2.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {71--79},
      NUMBER  = {{\bf 1}})

@ARTICLE(PCOMPS_2.ABS,
      AUTHOR  = {Borys, Leszek},
      TITLE   = {On Paracompactness of Metrizable Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/pcomps_2.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {81--84},
      NUMBER  = {{\bf 1}})

@ARTICLE(TREAL_1.ABS,
      AUTHOR  = {Watanabe, Toshihiko},
      TITLE   = {The {B}rouwer Fixed Point Theorem for Intervals},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/treal_1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {85--88},
      NUMBER  = {{\bf 1}})

@ARTICLE(CARD_5.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {On Powers of Cardinals},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/card_5.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {89--93},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPREAL3.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Kotowicz, Jaros{\l}aw},
      TITLE   = {Basic Properties of Connecting Points with Line Segments in ${\calE}^2_{\rmT}$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/topreal3.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {95--99},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPREAL4.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Kotowicz, Jaros{\l}aw},
      TITLE   = {Connectedness Conditions Using Polygonal Arcs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/topreal4.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {101--106},
      NUMBER  = {{\bf 1}})

@ARTICLE(GOBOARD1.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw and Nakamura, Yatsuka},
      TITLE   = {Introduction to {G}o-Board -- Part {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/goboard1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {107--115},
      NUMBER  = {{\bf 1}})

@ARTICLE(GOBOARD2.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw and Nakamura, Yatsuka},
      TITLE   = {Introduction to {G}o-Board -- Part {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/goboard2.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {117--121},
      NUMBER  = {{\bf 1}})

@ARTICLE(GOBOARD3.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw and Nakamura, Yatsuka},
      TITLE   = {Properties of {G}o-Board -- Part {III}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/goboard3.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {123--124},
      NUMBER  = {{\bf 1}})

@ARTICLE(GOBOARD4.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw and Nakamura, Yatsuka},
      TITLE   = {Go-Board Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/goboard4.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {125--129},
      NUMBER  = {{\bf 1}})

@ARTICLE(SYSREL.ABS,
      AUTHOR  = {Korczy\'nski, Waldemar},
      TITLE   = {Some Properties of Binary Relations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-1/sysrel.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {131--134},
      NUMBER  = {{\bf 1}})

@ARTICLE(JORDAN1.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Kotowicz, Jaros{\l}aw},
      TITLE   = {The {J}ordan's Property for Certain Subsets of the Plane},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/jordan1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {137--142},
      NUMBER  = {{\bf 2}})

@ARTICLE(TDLAT_3.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {The Lattice of Domains of an Extremally Disconnected Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/tdlat_3.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {143--149},
      NUMBER  = {{\bf 2}})

@ARTICLE(AMI_1.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {A Mathematical Model of {CPU}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/ami_1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {151--160},
      NUMBER  = {{\bf 2}})

@ARTICLE(CAT_4.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {{C}artesian Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/cat_4.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {161--169},
      NUMBER  = {{\bf 2}})

@ARTICLE(VFUNCT_1.ABS,
      AUTHOR  = {Yamazaki, Hiroshi and Shidama, Yasunari},
      TITLE   = {Algebra of Vector Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/vfunct_1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {171--175},
      NUMBER  = {{\bf 2}})

@ARTICLE(TSEP_2.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {On a Duality between Weakly Separated  Subspaces of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/tsep_2.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {177--182},
      NUMBER  = {{\bf 2}})

@ARTICLE(PETRI.ABS,
      AUTHOR  = {Kawamoto, Pauline N. and Fuwa, Yasushi and Nakamura, Yatsuka},
      TITLE   = {Basic {P}etri Net Concepts},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/petri.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {183--187},
      NUMBER  = {{\bf 2}})

@ARTICLE(FIN_TOPO.ABS,
      AUTHOR  = {Imura, Hiroshi and Eguchi, Masayoshi},
      TITLE   = {Finite Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/fin_topo.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {189--193},
      NUMBER  = {{\bf 2}})

@ARTICLE(TREES_3.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Sets and Functions of Trees and Joining Operations of Trees},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/trees_3.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {195--204},
      NUMBER  = {{\bf 2}})

@ARTICLE(FVSUM_1.ABS,
      AUTHOR  = {Zawadzka, Katarzyna},
      TITLE   = {The Sum and Product of Finite Sequences of Elements of a Field},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/fvsum_1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {205--211},
      NUMBER  = {{\bf 2}})

@ARTICLE(MONOID_0.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Monoids},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/monoid_0.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {213--225},
      NUMBER  = {{\bf 2}})

@ARTICLE(MONOID_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Monoid of Multisets and Subsets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/monoid_1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {227--233},
      NUMBER  = {{\bf 2}})

@ARTICLE(PRVECT_1.ABS,
      AUTHOR  = {Lango, Anna and Bancerek, Grzegorz},
      TITLE   = {Product of Families of Groups and Vector Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/prvect_1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {235--240},
      NUMBER  = {{\bf 2}})

@ARTICLE(AMI_2.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {On a Mathematical Model of Programs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/ami_2.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {241--250},
      NUMBER  = {{\bf 2}})

@ARTICLE(UNIALG_1.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw and Madras, Beata and Korolkiewicz, Ma{\l}gorzata},
      TITLE   = {Basic Notation of Universal Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/unialg_1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {251--253},
      NUMBER  = {{\bf 2}})

@ARTICLE(COH_SP.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw and Raczkowski, Konrad},
      TITLE   = {Coherent Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/coh_sp.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {255--261},
      NUMBER  = {{\bf 2}})

@ARTICLE(MEASURE5.ABS,
      AUTHOR  = {Bia{\l}as, J\'{o}zef},
      TITLE   = {Properties of the Intervals of Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/measure5.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {263--269},
      NUMBER  = {{\bf 2}})

@ARTICLE(RLVECT_4.ABS,
      AUTHOR  = {Trybulec, Wojciech A.},
      TITLE   = {Subspaces of Real Linear Space generated by One, Two, or Three        Vectors and Their Cosets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/rlvect_4.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {271--274},
      NUMBER  = {{\bf 2}})

@ARTICLE(RFINSEQ.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Functions and Finite Sequences of Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/rfinseq.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {275--278},
      NUMBER  = {{\bf 2}})

@ARTICLE(RFUNCT_3.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw and Sakai, Yuji},
      TITLE   = {Properties of Partial Functions from a Domain to the Set of Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/rfunct_3.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {279--288},
      NUMBER  = {{\bf 2}})

@ARTICLE(LMOD_7.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Domains of Submodules, Join and Meet of Finite Sequences of Submodules and Quotient Modules},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/lmod_7.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {289--296},
      NUMBER  = {{\bf 2}})

@ARTICLE(TOPS_3.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {Remarks on Special Subsets of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/tops_3.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {297--303},
      NUMBER  = {{\bf 2}})

@ARTICLE(TEX_1.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {On Discrete and Almost Discrete Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1992-3/pdf3-2/tex_1.pdf},
      YEAR    = {1992},
      VOLUME  = 3,
      PAGES   = {305--310},
      NUMBER  = {{\bf 2}})

@ARTICLE(MATRIX_3.ABS,
      AUTHOR  = {Zawadzka, Katarzyna},
      TITLE   = {The Product and the Determinant of Matrices with Entries in a Field},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/matrix_3.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {1--8},
      NUMBER  = {{\bf 1}})

@ARTICLE(REARRAN1.ABS,
      AUTHOR  = {Sakai, Yuji and Kotowicz, Jaros{\l}aw},
      TITLE   = {Introduction to Theory of Rearrangement},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/rearran1.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {9--13},
      NUMBER  = {{\bf 1}})

@ARTICLE(PBOOLE.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Many sorted Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/pboole.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {15--22},
      NUMBER  = {{\bf 1}})

@ARTICLE(UNIALG_2.ABS,
      AUTHOR  = {Burakowska, Ewa},
      TITLE   = {Subalgebras of the Universal Algebra. {L}attices of Subalgebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/unialg_2.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {23--27},
      NUMBER  = {{\bf 1}})

@ARTICLE(HAHNBAN.ABS,
      AUTHOR  = {Nowak, Bogdan and Trybulec, Andrzej},
      TITLE   = {Hahn-{B}anach Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/hahnban.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {29--34},
      NUMBER  = {{\bf 1}})

@ARTICLE(LATTICE4.ABS,
      AUTHOR  = {Kamie{\'n}ska, Jolanta and Walijewski, Jaros{\l}aw Stanis{\l}aw},
      TITLE   = {Homomorphisms of Lattices, Finite Join and Finite Meet},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/lattice4.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {35--40},
      NUMBER  = {{\bf 1}})

@ARTICLE(OPENLATT.ABS,
      AUTHOR  = {Kamie\'nska, Jolanta},
      TITLE   = {Representation Theorem for {H}eyting Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/openlatt.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {41--45},
      NUMBER  = {{\bf 1}})

@ARTICLE(LOPCLSET.ABS,
      AUTHOR  = {Walijewski, Jaros{\l}aw Stanis{\l}aw},
      TITLE   = {Representation Theorem for {B}oolean Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/lopclset.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {45--50},
      NUMBER  = {{\bf 1}})

@ARTICLE(AMI_3.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {Some Remarks on the Simple Concrete Model of Computer},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/ami_3.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {51--56},
      NUMBER  = {{\bf 1}})

@ARTICLE(AMI_4.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {{E}uclid's Algorithm},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/ami_4.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {57--60},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCM_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Rudnicki, Piotr},
      TITLE   = {Development of Terminology for {\SCM}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/scm_1.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {61--67},
      NUMBER  = {{\bf 1}})

@ARTICLE(PRE_FF.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Rudnicki, Piotr},
      TITLE   = {Two Programs for {\SCM}. {P}art {I} -- Preliminaries},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/pre_ff.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {69--72},
      NUMBER  = {{\bf 1}})

@ARTICLE(FIB_FUSC.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Rudnicki, Piotr},
      TITLE   = {Two Programs for {\SCM}. {P}art {II} -- Programs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/fib_fusc.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {73--75},
      NUMBER  = {{\bf 1}})

@ARTICLE(TREES_4.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Joining of Decorated Trees},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/trees_4.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {77--82},
      NUMBER  = {{\bf 1}})

@ARTICLE(BINARITH.ABS,
      AUTHOR  = {Nishiyama, Takaya and Mizuhara, Yasuho},
      TITLE   = {Binary Arithmetics},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/binarith.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {83--86},
      NUMBER  = {{\bf 1}})

@ARTICLE(BOOLMARK.ABS,
      AUTHOR  = {Kawamoto, Pauline N. and Fuwa, Yasushi and Nakamura, Yatsuka},
      TITLE   = {Basic Concepts for {P}etri Nets with {B}oolean Markings},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/boolmark.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {87--90},
      NUMBER  = {{\bf 1}})

@ARTICLE(DTCONSTR.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Rudnicki, Piotr},
      TITLE   = {On Defining Functions on Trees},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/dtconstr.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {91--101},
      NUMBER  = {{\bf 1}})

@ARTICLE(PRALG_1.ABS,
      AUTHOR  = {Madras, Beata},
      TITLE   = {Product of Family of Universal Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/pralg_1.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {103--108},
      NUMBER  = {{\bf 1}})

@ARTICLE(ALG_1.ABS,
      AUTHOR  = {Korolkiewicz, Ma{\l}gorzata},
      TITLE   = {Homomorphisms of Algebras. {Q}uotient Universal Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/alg_1.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {109--113},
      NUMBER  = {{\bf 1}})

@ARTICLE(FREEALG.ABS,
      AUTHOR  = {Perkowska, Beata},
      TITLE   = {Free Universal Algebra Construction},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/freealg.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {115--120},
      NUMBER  = {{\bf 1}})

@ARTICLE(COMSEQ_1.ABS,
      AUTHOR  = {Banachowicz, Agnieszka and Winnicka, Anna},
      TITLE   = {Complex Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/comseq_1.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {121--124},
      NUMBER  = {{\bf 1}})

@ARTICLE(TEX_2.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {Maximal Discrete Subspaces of Almost Discrete Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/tex_2.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {125--135},
      NUMBER  = {{\bf 1}})

@ARTICLE(TEX_3.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {On Nowhere and Everywhere Dense Subspaces of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1993-4/pdf4-1/tex_3.pdf},
      YEAR    = {1993},
      VOLUME  = 4,
      PAGES   = {137--146},
      NUMBER  = {{\bf 1}})

@ARTICLE(AMI_5.ABS,
      AUTHOR  = {Tanaka, Yasushi},
      TITLE   = {On the Decomposition of the States of {SCM}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/ami_5.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {1--8},
      NUMBER  = {{\bf 1}})

@ARTICLE(BINTREE1.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Rudnicki, Piotr},
      TITLE   = {On Defining Functions on Binary Trees},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/bintree1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {9--13},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCM_COMP.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Rudnicki, Piotr},
      TITLE   = {A Compiler of Arithmetic Expressions for {SCM}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/scm_comp.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {15--20},
      NUMBER  = {{\bf 1}})

@ARTICLE(MEASURE6.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef},
      TITLE   = {Some Properties of the Intervals},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/measure6.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {21--26},
      NUMBER  = {{\bf 1}})

@ARTICLE(BINARI_2.ABS,
      AUTHOR  = {Mizuhara, Yasuho and Nishiyama, Takaya},
      TITLE   = {Binary Arithmetics, Addition and Subtraction of Integers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/binari_2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {27--29},
      NUMBER  = {{\bf 1}})

@ARTICLE(BOOLEALG.ABS,
      AUTHOR  = {Marasik, Agnieszka Julia},
      TITLE   = {Boolean Properties of Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/boolealg.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {31--35},
      NUMBER  = {{\bf 1}})

@ARTICLE(MSUALG_1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Many Sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/msualg_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {37--42},
      NUMBER  = {{\bf 1}})

@ARTICLE(AUTGROUP.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Group of Inner Automorphisms},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/autgroup.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {43--45},
      NUMBER  = {{\bf 1}})

@ARTICLE(MSUALG_2.ABS,
      AUTHOR  = {Burakowska, Ewa},
      TITLE   = {Subalgebras of Many Sorted Algebra. {L}attice of Subalgebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/msualg_2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {47--54},
      NUMBER  = {{\bf 1}})

@ARTICLE(PRALG_2.ABS,
      AUTHOR  = {Madras, Beata},
      TITLE   = {Products of Many Sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/pralg_2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {55--60},
      NUMBER  = {{\bf 1}})

@ARTICLE(MSUALG_3.ABS,
      AUTHOR  = {Korolkiewicz, Ma{\l}gorzata},
      TITLE   = {Homomorphisms of Many Sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/msualg_3.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {61--65},
      NUMBER  = {{\bf 1}})

@ARTICLE(MSAFREE.ABS,
      AUTHOR  = {Perkowska, Beata},
      TITLE   = {Free Many Sorted Universal Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/msafree.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {67--74},
      NUMBER  = {{\bf 1}})

@ARTICLE(T_0TOPSP.ABS,
      AUTHOR  = {\.Zynel, Mariusz and Guzowski, Adam},
      TITLE   = { \Tzero\ Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/t_0topsp.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {75--77},
      NUMBER  = {{\bf 1}})

@ARTICLE(MSUALG_4.ABS,
      AUTHOR  = {Korolkiewicz, Ma{\l}gorzata},
      TITLE   = {Many Sorted Quotient Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/msualg_4.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {79--84},
      NUMBER  = {{\bf 1}})

@ARTICLE(QUANTAL1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Quantales},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/quantal1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {85--91},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPRNS_1.ABS,
      AUTHOR  = {Sakowicz, Agnieszka and Gryko, Jaros{\l}aw and Grabowski, Adam},
      TITLE   = {Sequences in ${\calE}^{N}_{\rmT}$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/toprns_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {93--96},
      NUMBER  = {{\bf 1}})

@ARTICLE(SPPOL_1.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Byli\'nski, Czes\l{}aw},
      TITLE   = {Extremal Properties of Vertices on Special Polygons. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/sppol_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {97--102},
      NUMBER  = {{\bf 1}})

@ARTICLE(RELOC.ABS,
      AUTHOR  = {Tanaka, Yasushi},
      TITLE   = {Relocatability},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/reloc.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {103--108},
      NUMBER  = {{\bf 1}})

@ARTICLE(TEX_4.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {Maximal Anti-Discrete Subspaces of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/tex_4.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {109--118},
      NUMBER  = {{\bf 1}})

@ARTICLE(TSP_1.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {On {K}olmogorov Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/tsp_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {119--124},
      NUMBER  = {{\bf 1}})

@ARTICLE(TSP_2.ABS,
      AUTHOR  = {Karno, Zbigniew},
      TITLE   = {Maximal {K}olmogorov Subspaces of a Topological Space as {S}tone Retracts of the Ambient Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/tsp_2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {125--130},
      NUMBER  = {{\bf 1}})

@ARTICLE(PROJPL_1.ABS,
      AUTHOR  = {Muzalewski, Micha{\l}},
      TITLE   = {Projective Planes},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/projpl_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {131--136},
      NUMBER  = {{\bf 1}})

@ARTICLE(SGRAPH1.ABS,
      AUTHOR  = {Toda, Yozo},
      TITLE   = {The Formalization of Simple Graphs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/sgraph1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {137--144},
      NUMBER  = {{\bf 1}})

@ARTICLE(GRSOLV_1.ABS,
      AUTHOR  = {Zawadzka, Katarzyna},
      TITLE   = {Solvable Groups},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-1/grsolv_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {145--147},
      NUMBER  = {{\bf 1}})

@ARTICLE(FILTER_2.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Ideals},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/filter_2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {149--156},
      NUMBER  = {{\bf 2}})

@ARTICLE(CAT_5.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Categorial Categories and Slice Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/cat_5.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {157--165},
      NUMBER  = {{\bf 2}})

@ARTICLE(PRE_CIRC.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Rudnicki, Piotr and Trybulec, Andrzej and Kawamoto, Pauline N.},
      TITLE   = {Preliminaries to Circuits, {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/pre_circ.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {167--172},
      NUMBER  = {{\bf 2}})

@ARTICLE(FSM_1.ABS,
      AUTHOR  = {Kaloper , Miroslava and Rudnicki, Piotr},
      TITLE   = {Minimization of finite state machines},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/fsm_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {173--184},
      NUMBER  = {{\bf 2}})

@ARTICLE(TREES_9.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Subtrees},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/trees_9.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {185--190},
      NUMBER  = {{\bf 2}})

@ARTICLE(MSATERM.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Terms Over Many Sorted Universal Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/msaterm.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {191--198},
      NUMBER  = {{\bf 2}})

@ARTICLE(DECOMP_1.ABS,
      AUTHOR  = {Przemski, Marian},
      TITLE   = {On the Decomposition of the Continuity},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/decomp_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {199--204},
      NUMBER  = {{\bf 2}})

@ARTICLE(MSAFREE1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {A Scheme for Extensions of Homomorphisms of Many sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/msafree1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {205--209},
      NUMBER  = {{\bf 2}})

@ARTICLE(MSUHOM_1.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {The Correspondence Between Homomorphisms of Universal Algebra   \& Many Sorted Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/msuhom_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {211--214},
      NUMBER  = {{\bf 2}})

@ARTICLE(MSAFREE2.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Rudnicki, Piotr and Trybulec, Andrzej and Kawamoto, Pauline N.},
      TITLE   = {Preliminaries to Circuits, {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/msafree2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {215--220},
      NUMBER  = {{\bf 2}})

@ARTICLE(AUTALG_1.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Group of Automorphisms of Universal Algebra \&         Many Sorted Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/autalg_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {221--226},
      NUMBER  = {{\bf 2}})

@ARTICLE(CIRCUIT1.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Rudnicki, Piotr and Trybulec, Andrzej and Kawamoto, Pauline N.},
      TITLE   = {Introduction to Circuits, {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/circuit1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {227--232},
      NUMBER  = {{\bf 2}})

@ARTICLE(CANTOR_1.ABS,
      AUTHOR  = {Shibakov, Alexander Yu. and Trybulec, Andrzej},
      TITLE   = {The {C}antor Set},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/cantor_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {233--236},
      NUMBER  = {{\bf 2}})

@ARTICLE(CQC_THE3.ABS,
      AUTHOR  = {Okhotnikov, Oleg},
      TITLE   = {Logical Equivalence of Formulae},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/cqc_the3.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {237--240},
      NUMBER  = {{\bf 2}})

@ARTICLE(FINSEQ_5.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Some Properties of Restrictions of Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/finseq_5.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {241--245},
      NUMBER  = {{\bf 2}})

@ARTICLE(SPPOL_2.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw and Nakamura, Yatsuka},
      TITLE   = {Special Polygons},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/sppol_2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {247--252},
      NUMBER  = {{\bf 2}})

@ARTICLE(MEASURE7.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef},
      TITLE   = {The One-Dimensional {L}ebesgue Measure},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/measure7.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {253--258},
      NUMBER  = {{\bf 2}})

@ARTICLE(ALTCAT_1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Categories without Uniqueness of {\bf cod} and {\bf dom}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/altcat_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {259--267},
      NUMBER  = {{\bf 2}})

@ARTICLE(EXTENS_1.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Extensions of Mappings on Generator Set},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/extens_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {269--272},
      NUMBER  = {{\bf 2}})

@ARTICLE(CIRCUIT2.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Rudnicki, Piotr and Trybulec, Andrzej and Kawamoto, Pauline N.},
      TITLE   = {Introduction to Circuits, {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/circuit2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {273--278},
      NUMBER  = {{\bf 2}})

@ARTICLE(MBOOLEAN.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Definitions and Basic Properties of {B}oolean \& Union of Many Sorted Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/mboolean.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {279--281},
      NUMBER  = {{\bf 2}})

@ARTICLE(CIRCCOMB.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Bancerek, Grzegorz},
      TITLE   = {Combining of Circuits},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-2/circcomb.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {283--295},
      NUMBER  = {{\bf 2}})

@ARTICLE(GRAPH_2.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Rudnicki, Piotr},
      TITLE   = {Vertex Sequences Induced by Chains},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/graph_2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {297--304},
      NUMBER  = {{\bf 3}})

@ARTICLE(VECTSP_8.ABS,
      AUTHOR  = {Iwaniuk, Andrzej},
      TITLE   = {On the Lattice of Subspaces of a Vector Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/vectsp_8.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {305--308},
      NUMBER  = {{\bf 3}})

@ARTICLE(LATSUBGR.ABS,
      AUTHOR  = {Ganczarski, Janusz},
      TITLE   = {On the Lattice of Subgroups of a Group},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/latsubgr.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {309--312},
      NUMBER  = {{\bf 3}})

@ARTICLE(UNIALG_3.ABS,
      AUTHOR  = {Paszek, Miros{\l}aw Jan},
      TITLE   = {On the Lattice of Subalgebras of a Universal Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/unialg_3.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {313--316},
      NUMBER  = {{\bf 3}})

@ARTICLE(FINSEQ_6.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {On the Decomposition of Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/finseq_6.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {317--322},
      NUMBER  = {{\bf 3}})

@ARTICLE(GOBOARD5.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {Decomposing a {G}o-Board into Cells},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/goboard5.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {323--328},
      NUMBER  = {{\bf 3}})

@ARTICLE(INDEX_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Indexed Category},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/index_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {329--337},
      NUMBER  = {{\bf 3}})

@ARTICLE(MATRLIN.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Associated Matrix of Linear Map},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/matrlin.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {339--345},
      NUMBER  = {{\bf 3}})

@ARTICLE(GOBOARD6.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {On the Geometry of a {G}o-{B}oard},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/goboard6.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {347--352},
      NUMBER  = {{\bf 3}})

@ARTICLE(WEIERSTR.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef and Nakamura, Yatsuka},
      TITLE   = {The Theorem of {W}eierstrass},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/weierstr.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {353--359},
      NUMBER  = {{\bf 3}})

@ARTICLE(URYSOHN1.ABS,
      AUTHOR  = {Bia{\l}as, J{\'o}zef and Nakamura, Yatsuka},
      TITLE   = {Dyadic Numbers and {T}${}_4$ Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/urysohn1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {361--366},
      NUMBER  = {{\bf 3}})

@ARTICLE(FACIRC_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Nakamura, Yatsuka},
      TITLE   = {Full Adder Circuit.  {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/facirc_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {367--380},
      NUMBER  = {{\bf 3}})

@ARTICLE(COHSP_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Continuous, Stable, and Linear Maps of Coherence Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/cohsp_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {381--393},
      NUMBER  = {{\bf 3}})

@ARTICLE(PZFMISC1.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Some Basic Properties of Many Sorted Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/pzfmisc1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {395--399},
      NUMBER  = {{\bf 3}})

@ARTICLE(TREES_A.ABS,
      AUTHOR  = {Okhotnikov, Oleg},

      TITLE   = {Replacement of Subtrees in a Tree},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/trees_a.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {401--403},
      NUMBER  = {{\bf 3}})

@ARTICLE(PUA2MSS1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Minimal Signature for Partial Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/pua2mss1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {405--414},
      NUMBER  = {{\bf 3}})

@ARTICLE(QC_LANG4.ABS,
      AUTHOR  = {Okhotnikov, Oleg},
      TITLE   = {The Subformula Tree of a Formula of the First Order Language},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/qc_lang4.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {415--422},
      NUMBER  = {{\bf 3}})

@ARTICLE(VECTSP_9.ABS,
      AUTHOR  = {{\.Z}ynel, Mariusz},
      TITLE   = {The {S}teinitz Theorem and the Dimension of a Vector Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/vectsp_9.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {423--428},
      NUMBER  = {{\bf 3}})

@ARTICLE(GOBOARD7.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {On the {G}o-{B}oard of a Standard Special Circular Sequence},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/goboard7.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {429--438},
      NUMBER  = {{\bf 3}})

@ARTICLE(ENDALG.ABS,
      AUTHOR  = {Gryko, Jaros{\l}aw},
      TITLE   = {On the Monoid of Endomorphisms of Universal Algebra and Many Sorted Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/endalg.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {439--442},
      NUMBER  = {{\bf 3}})

@ARTICLE(GOBOARD8.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {More on Segments on a {G}o-{B}oard},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/goboard8.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {443--450},
      NUMBER  = {{\bf 3}})

@ARTICLE(MSSUBFAM.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Certain Facts about Families of Subsets of Many Sorted Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/mssubfam.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {451--456},
      NUMBER  = {{\bf 3}})

@ARTICLE(TRIANG_1.ABS,
      AUTHOR  = {Madras, Beata},
      TITLE   = {On the Concept of the Triangulation},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-3/triang_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {457--462},
      NUMBER  = {{\bf 3}})

@ARTICLE(GOBOARD9.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Left and Right Component of the Complement of a Special Closed Curve},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/goboard9.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {465--468},
      NUMBER  = {{\bf 4}})

@ARTICLE(REWRITE1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Reduction Relations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/rewrite1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {469--478},
      NUMBER  = {{\bf 4}})

@ARTICLE(MSUALG_5.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Lattice of Congruences in Many Sorted Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/msualg_5.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {479--483},
      NUMBER  = {{\bf 4}})

@ARTICLE(FUNCT_7.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Trybulec, Andrzej},
      TITLE   = {Miscellaneous Facts about Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/funct_7.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {485--492},
      NUMBER  = {{\bf 4}})

@ARTICLE(ALTCAT_2.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Examples of Category Structures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/altcat_2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {493--500},
      NUMBER  = {{\bf 4}})

@ARTICLE(ORDERS_3.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {On the Category of Posets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/orders_3.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {501--505},
      NUMBER  = {{\bf 4}})

@ARTICLE(SCMFSA_1.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka and Rudnicki, Piotr},
      TITLE   = {An Extension of {\SCM}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/scmfsa_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {507--512},
      NUMBER  = {{\bf 4}})

@ARTICLE(CONNSP_3.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {Components and Unions of Components},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/connsp_3.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {513--517},
      NUMBER  = {{\bf 4}})

@ARTICLE(SCMFSA_2.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka and Rudnicki, Piotr},
      TITLE   = {The {\SCMFSA} Computer},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/scmfsa_2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {519--528},
      NUMBER  = {{\bf 4}})

@ARTICLE(CLOSURE1.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Many Sorted Closure Operator and the Many Sorted Closure System},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/closure1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {529--536},
      NUMBER  = {{\bf 4}})

@ARTICLE(SCMFSA_3.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {Computation in {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/scmfsa_3.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {537--542},
      NUMBER  = {{\bf 4}})

@ARTICLE(CLOSURE2.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Closure Operator and the Closure System of Many Sorted Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/closure2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {543--551},
      NUMBER  = {{\bf 4}})

@ARTICLE(MSUALG_6.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Translations, Endomorphisms, and Stable Equational Theories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/msualg_6.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {553--564},
      NUMBER  = {{\bf 4}})

@ARTICLE(MSUALG_7.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {More on the Lattice of Many Sorted Equivalence Relations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/msualg_7.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {565--569},
      NUMBER  = {{\bf 4}})

@ARTICLE(SCMFSA_4.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {Modifying Addresses of Instructions of {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/scmfsa_4.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {571--576},
      NUMBER  = {{\bf 4}})

@ARTICLE(MSSCYC_1.ABS,
      AUTHOR  = {Byli\'nski, Czes{\l}aw and Rudnicki, Piotr},
      TITLE   = {The Correspondence Between Monotonic Many Sorted Signatures and Well-Founded Graphs. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/msscyc_1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {577--582},
      NUMBER  = {{\bf 4}})

@ARTICLE(SCMFSA_5.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {Relocability for {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/scmfsa_5.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {583--586},
      NUMBER  = {{\bf 4}})

@ARTICLE(MSUALG_8.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {More on the Lattice of Congruences in Many Sorted Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/msualg_8.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {587--590},
      NUMBER  = {{\bf 4}})

@ARTICLE(MSSCYC_2.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw and Rudnicki, Piotr},
      TITLE   = {The Correspondence Between Monotonic Many Sorted Signatures and Well-Founded Graphs. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/msscyc_2.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {591--593},
      NUMBER  = {{\bf 4}})

@ARTICLE(FUNCTOR0.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Functors for Alternative Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/functor0.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {595--608},
      NUMBER  = {{\bf 4}})

@ARTICLE(FUNCTOR1.ABS,
      AUTHOR  = {Zinn, Claus and Jaksch, Wolfgang},
      TITLE   = {Basic Properties of Functor Structures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/functor1.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {609--613},
      NUMBER  = {{\bf 4}})

@ARTICLE(SCMFSA_7.ABS,
      AUTHOR  = {Asamoto, Noriko},
      TITLE   = {Some Multi Instructions Defined by Sequence of Instructions of {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/scmfsa_7.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {615--619},
      NUMBER  = {{\bf 4}})

@ARTICLE(PRALG_3.ABS,
      AUTHOR  = {Giero, Mariusz},
      TITLE   = {More on Products of Many Sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1996-5/pdf5-4/pralg_3.pdf},
      YEAR    = {1996},
      VOLUME  = 5,
      PAGES   = {621--626},
      NUMBER  = {{\bf 4}})

@ARTICLE(GOBRD10.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {Adjacency Concept for Pairs of Natural Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/gobrd10.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {1--3},
      NUMBER  = {{\bf 1}})

@ARTICLE(MSALIMIT.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {Inverse Limits of Many Sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/msalimit.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {5--8},
      NUMBER  = {{\bf 1}})

@ARTICLE(MSUALG_9.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Trivial Many Sorted Algebras and Many Sorted Congruences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/msualg_9.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {9--15},
      NUMBER  = {{\bf 1}})

@ARTICLE(MSINST_1.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {Examples of Category Structures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/msinst_1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {17--20},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMFSA6A.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka and Asamoto, Noriko},
      TITLE   = {On the Compositions of Macro Instructions. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/scmfsa6a.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {21--27},
      NUMBER  = {{\bf 1}})

@ARTICLE(SF_MASTR.ABS,
      AUTHOR  = {Rudnicki, Piotr and Trybulec, Andrzej},
      TITLE   = {Memory Handling for {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/sf_mastr.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {29--36},
      NUMBER  = {{\bf 1}})

@ARTICLE(GOBRD11.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {Some Topological Properties of Cells in ${\calE}^2_{\rmT}$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/gobrd11.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {37--40},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMFSA6B.ABS,
      AUTHOR  = {Asamoto, Noriko and Nakamura, Yatsuka and Rudnicki, Piotr and Trybulec, Andrzej},
      TITLE   = {On the Composition of Macro Instructions. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/scmfsa6b.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {41--47},
      NUMBER  = {{\bf 1}})

@ARTICLE(GOBRD12.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {The First Part of {J}ordan's Theorem for Special Polygons},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/gobrd12.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {49--51},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMFSA6C.ABS,
      AUTHOR  = {Asamoto, Noriko and Nakamura, Yatsuka and Rudnicki, Piotr and Trybulec, Andrzej},
      TITLE   = {On the Composition of Macro Instructions. {P}art {III}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/scmfsa6c.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {53--57},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMFSA7B.ABS,
      AUTHOR  = {Asamoto, Noriko},
      TITLE   = {Constant Assignment Macro Instructions of {\SCMFSA}. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/scmfsa7b.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {59--63},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMFSA8A.ABS,
      AUTHOR  = {Asamoto, Noriko},
      TITLE   = {Conditional Branch Macro Instructions of {\SCMFSA}. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/scmfsa8a.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {65--72},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMFSA8B.ABS,
      AUTHOR  = {Asamoto, Noriko},
      TITLE   = {Conditional Branch Macro Instructions of {\SCMFSA}. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/scmfsa8b.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {73--80},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW_0.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Bounds in Posets and Relational Substructures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/yellow_0.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {81--91},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL_0.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Directed Sets, Nets, Ideals, Filters, and Maps},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/waybel_0.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {93--107},
      NUMBER  = {{\bf 1}})

@ARTICLE(KNASTER.ABS,
      AUTHOR  = {Rudnicki, Piotr and Trybulec, Andrzej},
      TITLE   = {Fixpoints in Complete Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/knaster.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {109--115},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW_1.ABS,
      AUTHOR  = {Grabowski, Adam and Milewski, Robert},
      TITLE   = {Boolean Posets, Posets under Inclusion and Products of Relational Structures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/yellow_1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {117--121},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW_2.ABS,
      AUTHOR  = {{\.Z}ynel, Mariusz and Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Properties of Relational Structures, Posets, Lattices and Maps},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/yellow_2.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {123--130},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL_1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Galois Connections},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/waybel_1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {131--143},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW_3.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Cartesian Products of Relations and Relational Structures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/yellow_3.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {145--152},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW_4.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Definitions and Properties of the Join and Meet of Subsets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/yellow_4.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {153--158},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL_2.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Meet--Continuous Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/waybel_2.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {159--167},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL_3.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {The ``Way-Below'' Relation},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-1/waybel_3.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {169--176},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL_4.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {Auxiliary and Approximating Relations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/waybel_4.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {179--188},
      NUMBER  = {{\bf 2}})

@ARTICLE(TWOSCOMP.ABS,
      AUTHOR  = {Wasaki, Katsumi and Kawamoto, Pauline N.},
      TITLE   = {2's Complement Circuit},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/twoscomp.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {189--197},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL_5.ABS,
      AUTHOR  = {{\.Z}ynel, Mariusz},
      TITLE   = {The Equational Characterization of Continuous Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/waybel_5.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {199--205},
      NUMBER  = {{\bf 2}})

@ARTICLE(YELLOW_5.ABS,
      AUTHOR  = {Marasik, Agnieszka Julia},
      TITLE   = {Miscellaneous Facts about Relation Structure},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/yellow_5.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {207--211},
      NUMBER  = {{\bf 2}})

@ARTICLE(YELLOW_6.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {{M}oore-{S}mith Convergence},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/yellow_6.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {213--225},
      NUMBER  = {{\bf 2}})

@ARTICLE(YELLOW_7.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Duality in Relation Structures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/yellow_7.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {227--232},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL_6.ABS,
      AUTHOR  = {Madras, Beata},
      TITLE   = {Irreducible and Prime Elements},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/waybel_6.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {233--239},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL_7.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Prime Ideals and Filters},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/waybel_7.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {241--247},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL_8.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Algebraic Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/waybel_8.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {249--254},
      NUMBER  = {{\bf 2}})

@ARTICLE(JORDAN3.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Matuszewski, Roman},
      TITLE   = {Reconstructions of Special Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/jordan3.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {255--263},
      NUMBER  = {{\bf 2}})

@ARTICLE(COMSEQ_2.ABS,
      AUTHOR  = {Naumowicz, Adam},
      TITLE   = {Conjugate Sequences, Bounded Complex Sequences and    Convergent Complex Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/comseq_2.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {265--268},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL_9.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Topological Properties of Meet-Continuous Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/waybel_9.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {269--277},
      NUMBER  = {{\bf 2}})

@ARTICLE(INSTALG1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Institution of Many Sorted Algebras. {P}art {I}: Signature Reduct of an Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/instalg1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {279--287},
      NUMBER  = {{\bf 2}})

@ARTICLE(YELLOW_8.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {{B}aire Spaces, {S}ober Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/yellow_8.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {289--294},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL10.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Closure Operators and Subalgebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/waybel10.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {295--301},
      NUMBER  = {{\bf 2}})

@ARTICLE(CATALG_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Algebra of Morphisms},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/catalg_1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {303--310},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL11.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {{S}cott Topology},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/waybel11.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {311--319},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL12.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the {B}aire Category Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-2/waybel12.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {321--327},
      NUMBER  = {{\bf 2}})

@ARTICLE(ALTCAT_3.ABS,
      AUTHOR  = {Madras, Beata},
      TITLE   = {Basic Properties of Objects and Morphisms},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/altcat_3.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {329--334},
      NUMBER  = {{\bf 3}})

@ARTICLE(ABIAN.ABS,
      AUTHOR  = {Rudnicki, Piotr and Trybulec, Andrzej},
      TITLE   = {{A}bian's Fixed Point Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/abian.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {335--338},
      NUMBER  = {{\bf 3}})

@ARTICLE(WELLFND1.ABS,
      AUTHOR  = {Rudnicki, Piotr and Trybulec, Andrzej},
      TITLE   = {On Same Equivalents of Well-foundedness},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/wellfnd1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {339--343},
      NUMBER  = {{\bf 3}})

@ARTICLE(WAYBEL13.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Algebraic and Arithmetic Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/waybel13.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {345--349},
      NUMBER  = {{\bf 3}})

@ARTICLE(JORDAN4.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Matuszewski, Roman and Grabowski, Adam},
      TITLE   = {Subsequences of Standard Special Circular Sequences in ${\calE}^2_{\rmT}$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/jordan4.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {351--358},
      NUMBER  = {{\bf 3}})

@ARTICLE(SUBSTLAT.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {Lattice of Substitutions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/substlat.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {359--361},
      NUMBER  = {{\bf 3}})

@ARTICLE(EQUATION.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Equations in Many Sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/equation.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {363--369},
      NUMBER  = {{\bf 3}})

@ARTICLE(FUNCTOR2.ABS,
      AUTHOR  = {Nieszczerzewski, Robert},
      TITLE   = {Category of Functors Between Alternative Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/functor2.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {371--375},
      NUMBER  = {{\bf 3}})

@ARTICLE(YONEDA_1.ABS,
      AUTHOR  = {Wojciechowski, Miros{\l}aw},
      TITLE   = {Yoneda Embedding},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/yoneda_1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {377--379},
      NUMBER  = {{\bf 3}})

@ARTICLE(GCD_1.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {The Correctness of the Generic Algorithms of {B}rown and {H}enrici Concerning Addition and Multiplication in Fraction Fields},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/gcd_1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {381--388},
      NUMBER  = {{\bf 3}})

@ARTICLE(BIRKHOFF.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {{B}irkhoff Theorem for Many Sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/birkhoff.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {389--395},
      NUMBER  = {{\bf 3}})

@ARTICLE(CLOSURE3.ABS,
      AUTHOR  = {Marasik, Agnieszka Julia},
      TITLE   = {Algebraic Operation on Subsets of Many Sorted Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/closure3.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {397--401},
      NUMBER  = {{\bf 3}})

@ARTICLE(COMSEQ_3.ABS,
      AUTHOR  = {Shidama, Yasunari and Korni{\l}owicz, Artur},
      TITLE   = {Convergence and the Limit of Complex Sequences. {S}eries},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/comseq_3.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {403--410},
      NUMBER  = {{\bf 3}})

@ARTICLE(RLVECT_5.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {The {S}teinitz Theorem and the Dimension of a Real Linear Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/rlvect_5.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {411--415},
      NUMBER  = {{\bf 3}})

@ARTICLE(GRAPH_3.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Rudnicki, Piotr},
      TITLE   = {{E}uler Circuits and Paths},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/graph_3.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {417--425},
      NUMBER  = {{\bf 3}})

@ARTICLE(PSCOMP_1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw and Rudnicki, Piotr},
      TITLE   = {Bounding Boxes for Compact Sets in ${\calE}^2$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/pscomp_1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {427--440},
      NUMBER  = {{\bf 3}})

@ARTICLE(WAYBEL14.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw and Rudnicki, Piotr},
      TITLE   = {The {S}cott Topology. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-3/waybel14.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {441--446},
      NUMBER  = {{\bf 3}})

@ARTICLE(BORSUK_2.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {Introduction to the Homotopy Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/borsuk_2.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {449--454},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN5A.ABS,
      AUTHOR  = {Grabowski, Adam and Nakamura, Yatsuka},
      TITLE   = {Some Properties of Real Maps},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/jordan5a.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {455--459},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN5B.ABS,
      AUTHOR  = {Grabowski, Adam and Nakamura, Yatsuka},
      TITLE   = {The Ordering of Points on a Curve. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/jordan5b.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {461--465},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN5C.ABS,
      AUTHOR  = {Grabowski, Adam and Nakamura, Yatsuka},
      TITLE   = {The Ordering of Points on a Curve. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/jordan5c.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {467--473},
      NUMBER  = {{\bf 4}})

@ARTICLE(ALTCAT_4.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Categories Without Uniqueness of { \bf cod} and { \bf dom}. {S}ome Properties of the Morphisms and the Functors},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/altcat_4.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {475--481},
      NUMBER  = {{\bf 4}})

@ARTICLE(SCMFSA8C.ABS,
      AUTHOR  = {Asamoto, Noriko},
      TITLE   = {The {\tt loop} and {\tt Times} Macroinstruction for {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/scmfsa8c.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {483--497},
      NUMBER  = {{\bf 4}})

@ARTICLE(WAYBEL15.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Algebraic and Arithmetic Lattices. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/waybel15.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {499--503},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN2B.ABS,
      AUTHOR  = {Matuszewski, Roman and Nakamura, Yatsuka},
      TITLE   = {Projections in $n$-Dimensional {E}uclidean Space to Each Coordinates},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/jordan2b.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {505--509},
      NUMBER  = {{\bf 4}})

@ARTICLE(TOPREAL5.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {Intermediate Value Theorem and Thickness of Simple Closed Curves},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/topreal5.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {511--514},
      NUMBER  = {{\bf 4}})

@ARTICLE(LATTICE5.ABS,
      AUTHOR  = {Gryko, Jaros{\l}aw},
      TITLE   = {The {J}\'onson's Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/lattice5.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {515--524},
      NUMBER  = {{\bf 4}})

@ARTICLE(UNIFORM1.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {{L}ebesgue's Covering Lemma, Uniform Continuity and Segmentation of Arcs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/uniform1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {525--529},
      NUMBER  = {{\bf 4}})

@ARTICLE(SPRECT_1.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {On the Rectangular Finite Sequences of the Points of the Plane},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/sprect_1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {531--539},
      NUMBER  = {{\bf 4}})

@ARTICLE(SPRECT_2.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {On the Order on a Special Polygon},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/sprect_2.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {541--548},
      NUMBER  = {{\bf 4}})

@ARTICLE(EULER_1.ABS,
      AUTHOR  = {Fujisawa, Yoshinori and Fuwa, Yasushi},
      TITLE   = {The {E}uler's Function},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/euler_1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {549--551},
      NUMBER  = {{\bf 4}})

@ARTICLE(SCMFSA_9.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {While Macro Instructions of {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/scmfsa_9.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {553--561},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN6.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {A Decomposition of a Simple Closed Curves and the Order of Their Points},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/jordan6.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {563--572},
      NUMBER  = {{\bf 4}})

@ARTICLE(WSIERP_1.ABS,
      AUTHOR  = {Kondracki, Andrzej},
      TITLE   = {The {C}hinese {R}emainder {T}heorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1997-6/pdf6-4/wsierp_1.pdf},
      YEAR    = {1997},
      VOLUME  = 6,
      PAGES   = {573--577},
      NUMBER  = {{\bf 4}})

@ARTICLE(FUNCTOR3.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Composition of Functors and Transformations in Alternative   Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/functor3.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {1--7},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL16.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Completely-Irreducible Elements},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/waybel16.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {9--12},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL17.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {{S}cott-Continuous Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/waybel17.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {13--18},
      NUMBER  = {{\bf 1}})

@ARTICLE(NAT_2.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Natural Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/nat_2.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {19--22},
      NUMBER  = {{\bf 1}})

@ARTICLE(BINARI_3.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Binary Arithmetics. {B}inary Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/binari_3.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {23--26},
      NUMBER  = {{\bf 1}})

@ARTICLE(BINTREE2.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Full Trees},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/bintree2.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {27--30},
      NUMBER  = {{\bf 1}})

@ARTICLE(T_1TOPSP.ABS,
      AUTHOR  = {{N}aumowicz, Adam and {\L}api{\'n}ski, Mariusz},
      TITLE   = {On \Tone\ Reflex of Topological Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/t_1topsp.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {31--34},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW_9.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Bases and Refinements of Topologies},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/yellow_9.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {35--43},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW10.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Properties of Product of Relational Structures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/yellow10.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {45--52},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW11.ABS,
      AUTHOR  = {Naumowicz, Adam},
      TITLE   = {On the Characterization of Modular and Distributive Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/yellow11.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {53--55},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL18.ABS,
      AUTHOR  = {Gryko, Jaros{\l}aw},
      TITLE   = {Injective Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/waybel18.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {57--62},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW12.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Characterization of {H}ausdorff Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/yellow12.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {63--68},
      NUMBER  = {{\bf 1}})

@ARTICLE(QUOFIELD.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {The Field of Quotients Over an Integral Domain},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/quofield.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {69--79},
      NUMBER  = {{\bf 1}})

@ARTICLE(FRECHET.ABS,
      AUTHOR  = {Skorulski, Bart{\l}omiej},
      TITLE   = {First-countable, Sequential, and {F}rechet Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/frechet.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {81--86},
      NUMBER  = {{\bf 1}})

@ARTICLE(SFMASTR1.ABS,
      AUTHOR  = {Rudnicki, Piotr},
      TITLE   = {On the Composition of Non-parahalting Macro Instructions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/sfmastr1.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {87--92},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMFSA9A.ABS,
      AUTHOR  = {Rudnicki, Piotr},
      TITLE   = {The {\tt while} Macro Instructions of {\SCMFSA}. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/scmfsa9a.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {93--100},
      NUMBER  = {{\bf 1}})

@ARTICLE(SFMASTR2.ABS,
      AUTHOR  = {Rudnicki, Piotr},
      TITLE   = {Another {\tt times} Macro Instruction},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/sfmastr2.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {101--105},
      NUMBER  = {{\bf 1}})

@ARTICLE(SFMASTR3.ABS,
      AUTHOR  = {Rudnicki, Piotr},
      TITLE   = {The {\tt for} (going up) Macro Instruction},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/sfmastr3.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {107--114},
      NUMBER  = {{\bf 1}})

@ARTICLE(JORDAN5D.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Grabowski, Adam},
      TITLE   = {Bounding Boxes for Special Sequences in ${\calE}^2$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/jordan5d.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {115--121},
      NUMBER  = {{\bf 1}})

@ARTICLE(EULER_2.ABS,
      AUTHOR  = {Fujisawa, Yoshinori and Fuwa, Yasushi and Shimizu, Hidetaka},
      TITLE   = {{E}uler's {T}heorem and Small {F}ermat's {T}heorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/euler_2.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {123--126},
      NUMBER  = {{\bf 1}})

@ARTICLE(GROUP_7.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Product of the Families of the Groups},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/group_7.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {127--134},
      NUMBER  = {{\bf 1}})

@ARTICLE(JORDAN7.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {On the Dividing Function of the Simple Closed Curve into Segments},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/jordan7.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {135--138},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCM_HALT.ABS,
      AUTHOR  = {Chen, Jing-Chao and Nakamura, Yatsuka},
      TITLE   = {Initialization Halting Concepts and Their Basic Properties of {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/scm_halt.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {139--151},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMBSORT.ABS,
      AUTHOR  = {Chen, Jing-Chao and Nakamura, Yatsuka},
      TITLE   = {Bubble Sort on {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-1/scmbsort.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {153--161},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL19.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {The {L}awson Topology},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/waybel19.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {163--168},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL20.ABS,
      AUTHOR  = {Rudnicki, Piotr},
      TITLE   = {Kernel Projections and Quotient Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/waybel20.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {169--175},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL21.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {{L}awson Topology in Continuous Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/waybel21.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {177--184},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL22.ABS,
      AUTHOR  = {Rudnicki, Piotr},
      TITLE   = {Representation Theorem for Free Continuous Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/waybel22.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {185--188},
      NUMBER  = {{\bf 2}})

@ARTICLE(GRAPH_4.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Rudnicki, Piotr},
      TITLE   = {Oriented Chains},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/graph_4.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {189--192},
      NUMBER  = {{\bf 2}})

@ARTICLE(JGRAPH_1.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {Graph Theoretical Properties of Arcs in the Plane and {F}ashoda {M}eet {T}heorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/jgraph_1.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {193--201},
      NUMBER  = {{\bf 2}})

@ARTICLE(IDEA_1.ABS,
      AUTHOR  = {Fuwa, Yasushi and Fujisawa, Yoshinori},
      TITLE   = {Algebraic Group on Fixed-length Bit Integer and its Adaptation to {IDEA} Cryptography},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/idea_1.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {203--215},
      NUMBER  = {{\bf 2}})

@ARTICLE(TOPGRP_1.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Definition and Basic Properties of Topological Groups},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/topgrp_1.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {217--225},
      NUMBER  = {{\bf 2}})

@ARTICLE(MSSUBLAT.ABS,
      AUTHOR  = {Naumowicz, Adam and Marasik, Agnieszka Julia},
      TITLE   = {The Correspondence Between Lattices of Subalgebras of  Universal Algebras and Many Sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/mssublat.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {227--231},
      NUMBER  = {{\bf 2}})

@ARTICLE(CONLAT_1.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {Introduction to Concept Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/conlat_1.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {233--241},
      NUMBER  = {{\bf 2}})

@ARTICLE(PARTIT1.ABS,
      AUTHOR  = {Kobayashi, Shunichi and Jia, Kui},
      TITLE   = {A Theory of Partitions. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/partit1.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {243--247},
      NUMBER  = {{\bf 2}})

@ARTICLE(BVFUNC_1.ABS,
      AUTHOR  = {Kobayashi, Shunichi and Jia, Kui},
      TITLE   = {A Theory of {B}oolean Valued Functions and Partitions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/bvfunc_1.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {249--254},
      NUMBER  = {{\bf 2}})

@ARTICLE(SIN_COS.ABS,
      AUTHOR  = {Yang, Yuguang and Shidama, Yasunari},
      TITLE   = {Trigonometric Functions and Existence of Circle Ratio},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/sin_cos.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {255--263},
      NUMBER  = {{\bf 2}})

@ARTICLE(SPRECT_3.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {Some Properties of Special Polygonal Curves},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/sprect_3.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {265--272},
      NUMBER  = {{\bf 2}})

@ARTICLE(VECTMETR.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Real Linear-Metric Space and Isometric Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/vectmetr.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {273--277},
      NUMBER  = {{\bf 2}})

@ARTICLE(YELLOW13.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Introduction to Meet-Continuous Topological Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/yellow13.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {279--283},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL23.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Bases of Continuous Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/waybel23.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {285--294},
      NUMBER  = {{\bf 2}})

@ARTICLE(SCMRING1.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Construction of {\SCM} over Ring},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/scmring1.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {295--300},
      NUMBER  = {{\bf 2}})

@ARTICLE(SCMRING2.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Basic Properties of {\SCM} over Ring},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/scmring2.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {301--305},
      NUMBER  = {{\bf 2}})

@ARTICLE(BVFUNC_2.ABS,
      AUTHOR  = {Kobayashi, Shunichi and Nakamura, Yatsuka},
      TITLE   = {A Theory of {B}oolean Valued Functions and Quantifiers with Respect to Partitions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/bvfunc_2.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {307--312},
      NUMBER  = {{\bf 2}})

@ARTICLE(BVFUNC_3.ABS,
      AUTHOR  = {Kobayashi, Shunichi and Nakamura, Yatsuka},
      TITLE   = {Predicate Calculus for {B}oolean Valued Functions. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/bvfunc_3.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {313--315},
      NUMBER  = {{\bf 2}})

@ARTICLE(PEPIN.ABS,
      AUTHOR  = {Fujisawa, Yoshinori and Fuwa, Yasushi and Shimizu, Hidetaka},
      TITLE   = {Public-Key Cryptography and {P}epin's Test for the Primality of {F}ermat Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/pepin.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {317--321},
      NUMBER  = {{\bf 2}})

@ARTICLE(HEYTING2.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {Lattice of Substitutions is a {H}eyting Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1998-7/pdf7-2/heyting2.pdf},
      YEAR    = {1998},
      VOLUME  = 7,
      PAGES   = {323--327},
      NUMBER  = {{\bf 2}})

@ARTICLE(JORDAN2C.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej and Byli\'nski, Czes{\l}aw},
      TITLE   = {Bounded Domains and Unbounded Domains},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/jordan2c.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {1--13},
      NUMBER  = {{\bf 1}})

@ARTICLE(REVROT_1.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Rotating and Reversing},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/revrot_1.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {15--20},
      NUMBER  = {{\bf 1}})

@ARTICLE(SPRECT_4.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {On the Components of the Complement of a Special Polygonal Curve},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/sprect_4.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {21--23},
      NUMBER  = {{\bf 1}})

@ARTICLE(JORDAN8.ABS,
      AUTHOR  = {Byli\'nski, Czes\l{}aw},
      TITLE   = {Gauges},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/jordan8.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {25--27},
      NUMBER  = {{\bf 1}})

@ARTICLE(INT_3.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {The Ring of Integers, {E}uclidean Rings and Modulo Integers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/int_3.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {29--34},
      NUMBER  = {{\bf 1}})

@ARTICLE(GATE_1.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {Logic Gates and Logical Equivalence of Adders},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/gate_1.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {35--45},
      NUMBER  = {{\bf 1}})

@ARTICLE(FRECHET2.ABS,
      AUTHOR  = {Skorulski, Bart{\l}omiej},
      TITLE   = {The Sequential Closure Operator in Sequential and {F}rechet Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/frechet2.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {47--54},
      NUMBER  = {{\bf 1}})

@ARTICLE(BORSUK_3.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {Properties of the Product of Compact Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/borsuk_3.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {55--59},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPREAL6.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Compactness of the Bounded Closed Subsets of ${\calE}^2_{\rmT}$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/topreal6.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {61--68},
      NUMBER  = {{\bf 1}})

@ARTICLE(HILBERT1.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {{H}ilbert Positive Propositional Calculus},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/hilbert1.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {69--72},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPREAL7.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Homeomorphism between [:${\calE}^i_{\rmT}, {\calE}^j_{\rmT}$:] and ${\calE}^{i+j}_{\rmT}$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/topreal7.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {73--76},
      NUMBER  = {{\bf 1}})

@ARTICLE(FSCIRC_1.ABS,
      AUTHOR  = {Wasaki, Katsumi and Endou, Noboru},
      TITLE   = {Full Subtracter Circuit. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/fscirc_1.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {77--81},
      NUMBER  = {{\bf 1}})

@ARTICLE(GATE_2.ABS,
      AUTHOR  = {Yang, Yuguang and Wasaki, Katsumi and Fuwa, Yasushi and Nakamura, Yatsuka},
      TITLE   = {Correctness of Binary Counter Circuits},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/gate_2.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {83--85},
      NUMBER  = {{\bf 1}})

@ARTICLE(GATE_3.ABS,
      AUTHOR  = {Yang, Yuguang and Wasaki, Katsumi and Fuwa, Yasushi and Nakamura, Yatsuka},
      TITLE   = {Correctness of {J}ohnson Counter Circuits},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/gate_3.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {87--91},
      NUMBER  = {{\bf 1}})

@ARTICLE(INTEGRA1.ABS,
      AUTHOR  = {Endou, Noboru and Korni{\l}owicz, Artur},
      TITLE   = {The Definition of the {R}iemann Definite Integral and some Related Lemmas},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/integra1.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {93--102},
      NUMBER  = {{\bf 1}})

@ARTICLE(SIN_COS2.ABS,
      AUTHOR  = {Mitsuishi, Takashi and Yang, Yuguang},
      TITLE   = {Properties of the Trigonometric Function},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/sin_cos2.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {103--106},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC_4.ABS,
      AUTHOR  = {Kobayashi, Shunichi and Nakamura, Yatsuka},
      TITLE   = {Predicate Calculus for {B}oolean Valued Functions. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/bvfunc_4.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {107--109},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC_5.ABS,
      AUTHOR  = {Kobayashi, Shunichi and Nakamura, Yatsuka},
      TITLE   = {Propositional Calculus for Boolean Valued Functions. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/bvfunc_5.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {111--113},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC_6.ABS,
      AUTHOR  = {Kobayashi, Shunichi and Nakamura, Yatsuka},
      TITLE   = {Propositional Calculus for Boolean Valued Functions. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/bvfunc_6.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {115--117},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMISORT.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {Insert Sort on {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/scmisort.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {119--127},
      NUMBER  = {{\bf 1}})

@ARTICLE(GATE_4.ABS,
      AUTHOR  = {Yang, Yuguang and Wasaki, Katsumi and Fuwa, Yasushi and Nakamura, Yatsuka},
      TITLE   = {Correctness of a Cyclic Redundancy Check Code Generator},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/gate_4.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {129--132},
      NUMBER  = {{\bf 1}})

@ARTICLE(HILBERT2.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Defining by Structural Induction in the Positive Propositional Language},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/hilbert2.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {133--137},
      NUMBER  = {{\bf 1}})

@ARTICLE(GOBRD13.ABS,
      AUTHOR  = {Byli\'nski, Czes{\l}aw},
      TITLE   = {Some Properties of Cells on {G}o-Board},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/gobrd13.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {139--146},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC_7.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Propositional Calculus for Boolean Valued Functions. {P}art {III}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/bvfunc_7.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {147--148},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC_8.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Propositional Calculus for Boolean Valued Functions. {P}art {IV}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/bvfunc_8.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {149--150},
      NUMBER  = {{\bf 1}})

@ARTICLE(GENEALG1.ABS,
      AUTHOR  = {Uchibori, Akihiko and Endou, Noboru},
      TITLE   = {Basic Properties of Genetic Algorithm},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/genealg1.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {151--160},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC_9.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Propositional Calculus for Boolean Valued Functions. {P}art {V}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/bvfunc_9.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {161--162},
      NUMBER  = {{\bf 1}})

@ARTICLE(GOBRD14.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Properties of Left and Right Components},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/gobrd14.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {163--168},
      NUMBER  = {{\bf 1}})

@ARTICLE(LATTICE6.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {Noetherian Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/lattice6.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {169--174},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMPDS_1.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {A Small Computer Model with Push-Down Stack},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/scmpds_1.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {175--182},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMPDS_2.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {The {SCMPDS} Computer and the Basic Semantics of its Instructions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/scmpds_2.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {183--191},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMPDS_3.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {Computation and Program Shift in the {SCMPDS} Computer},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/scmpds_3.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {193--199},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMPDS_4.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {The Construction and Shiftability of Program Blocks for {SCMPDS}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/scmpds_4.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {201--210},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMPDS_5.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {Computation of Two Consecutive Program Blocks for {SCMPDS}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/scmpds_5.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {211--217},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMPDS_6.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {The Construction and Computation of Conditional Statements for {SCMPDS}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/1999-8/pdf8-1/scmpds_6.pdf},
      YEAR    = {1999},
      VOLUME  = 8,
      PAGES   = {219--234},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMP_GCD.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {Recursive {E}uclide Algorithm},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/scmp_gcd.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {1--4},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL24.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {{S}cott-Continuous Functions. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/waybel24.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {5--11},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW14.ABS,
      AUTHOR  = {Gryko, Jaros{\l}aw and Korni{\l}owicz, Artur},
      TITLE   = {Some Properties of Isomorphism between Relational Structures. {O}n the Product of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/yellow14.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {13--18},
      NUMBER  = {{\bf 1}})

@ARTICLE(JORDAN9.ABS,
      AUTHOR  = {Byli\'nski, Czes{\l}aw and \.Zynel, Mariusz},
      TITLE   = {Cages -- the External Approximation of {J}ordan's Curve},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/jordan9.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {19--24},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW15.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Components and Basis of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/yellow15.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {25--29},
      NUMBER  = {{\bf 1}})

@ARTICLE(JORDAN10.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Properties of the External Approximation of {J}ordan's Curve},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/jordan10.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {31--34},
      NUMBER  = {{\bf 1}})

@ARTICLE(IRRAT_1.ABS,
      AUTHOR  = {Wiedijk, Freek},
      TITLE   = {Irrationality of $e$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/irrat_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {35--39},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL25.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Gryko, Jaros{\l}aw},
      TITLE   = {Injective Spaces. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/waybel25.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {41--47},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC10.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Propositional Calculus for Boolean Valued Functions. {P}art {VI}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc10.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {49--50},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC11.ABS,
      AUTHOR  = {Kobayashi, Shunichi and Nakamura, Yatsuka},
      TITLE   = {Predicate Calculus for Boolean Valued Functions. {P}art {III}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc11.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {51--53},
      NUMBER  = {{\bf 1}})

@ARTICLE(CONLAT_2.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {A Characterization of Concept Lattices. {D}ual Concept Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/conlat_2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {55--59},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC12.ABS,
      AUTHOR  = {Kobayashi, Shunichi and Nakamura, Yatsuka},
      TITLE   = {Predicate Calculus for Boolean Valued Functions. {P}art {IV}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc12.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {61--63},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC13.ABS,
      AUTHOR  = {Kobayashi, Shunichi and Nakamura, Yatsuka},
      TITLE   = {Predicate Calculus for Boolean Valued Functions. {P}art {V}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc13.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {65--70},
      NUMBER  = {{\bf 1}})

@ARTICLE(RADIX_1.ABS,
      AUTHOR  = {Fujisawa, Yoshinori and Fuwa, Yasushi},
      TITLE   = {Definitions of Radix-$2^k$ Signed-Digit Number and its Adder Algorithm},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/radix_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {71--75},
      NUMBER  = {{\bf 1}})

@ARTICLE(YELLOW16.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Retracts and Inheritance},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/yellow16.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {77--85},
      NUMBER  = {{\bf 1}})

@ARTICLE(ALGSPEC1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Technical Preliminaries to Algebraic Specifications},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/algspec1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {87--93},
      NUMBER  = {{\bf 1}})

@ARTICLE(POLYNOM1.ABS,
      AUTHOR  = {Rudnicki, Piotr and Trybulec, Andrzej},
      TITLE   = {Multivariate Polynomials  with Arbitrary Number of Variables},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/polynom1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {95--110},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL26.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Continuous Lattices of Maps between {T}$_0$ Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/waybel26.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {111--117},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC14.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Predicate Calculus for Boolean Valued Functions. {P}art {VI}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc14.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {119--121},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC15.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Predicate Calculus for Boolean Valued Functions. {P}art {VII}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc15.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {123--125},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC16.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Predicate Calculus for Boolean Valued Functions. {P}art {VIII}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc16.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {127--129},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC17.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Predicate Calculus for Boolean Valued Functions. {P}art {IX}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc17.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {131--133},
      NUMBER  = {{\bf 1}})

@ARTICLE(ASYMPT_0.ABS,
      AUTHOR  = {Krueger, Richard and Rudnicki, Piotr and Shelley, Paul},
      TITLE   = {Asymptotic Notation. {P}art {I}: {T}heory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/asympt_0.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {135--142},
      NUMBER  = {{\bf 1}})

@ARTICLE(ASYMPT_1.ABS,
      AUTHOR  = {Krueger, Richard and Rudnicki, Piotr and Shelley, Paul},
      TITLE   = {Asymptotic Notation. {P}art {II}: {E}xamples and Problems},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/asympt_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {143--154},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC18.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Predicate Calculus for Boolean Valued Functions. {P}art {X}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc18.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {155--156},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC19.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Predicate Calculus for Boolean Valued Functions. {P}art {XI}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc19.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {157--159},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC20.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Four Variable Predicate Calculus for Boolean Valued Functions. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc20.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {161--165},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC21.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Four Variable Predicate Calculus for Boolean Valued Functions. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc21.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {167--170},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL27.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Naumowicz, Adam},
      TITLE   = {Function Spaces in the Category of Directed Suprema Preserving Maps},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/waybel27.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {171--177},
      NUMBER  = {{\bf 1}})

@ARTICLE(CFUNCT_1.ABS,
      AUTHOR  = {Mitsuishi, Takashi and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Property of Complex Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/cfunct_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {179--184},
      NUMBER  = {{\bf 1}})

@ARTICLE(CFCONT_1.ABS,
      AUTHOR  = {Mitsuishi, Takashi and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Property of Complex Sequence and Continuity of Complex Function},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/cfcont_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {185--190},
      NUMBER  = {{\bf 1}})

@ARTICLE(INTEGRA2.ABS,
      AUTHOR  = {Endou, Noboru and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Scalar Multiple of {R}iemann Definite Integral},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/integra2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {191--196},
      NUMBER  = {{\bf 1}})

@ARTICLE(INTEGRA3.ABS,
      AUTHOR  = {Endou, Noboru and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {{D}arboux's Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/integra3.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {197--200},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC22.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Five Variable Predicate Calculus for Boolean Valued Functions. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc22.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {201--204},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC23.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Six Variable Predicate Calculus for Boolean Valued Functions. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc23.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {205--208},
      NUMBER  = {{\bf 1}})

@ARTICLE(SCMPDS_7.ABS,
      AUTHOR  = {Chen, Jing-Chao and Rudnicki, Piotr},
      TITLE   = {The Construction and Computation of for-loop Programs for {SCMPDS}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/scmpds_7.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {209--219},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC24.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Predicate Calculus for Boolean Valued Functions. {P}art {XII}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-1/bvfunc24.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {221--235},
      NUMBER  = {{\bf 1}})

@ARTICLE(WAYBEL28.ABS,
      AUTHOR  = {Skorulski, Bart{\l}omiej},
      TITLE   = {Lim-Inf Convergence},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/waybel28.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {237--240},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL29.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Naumowicz, Adam},
      TITLE   = {The Characterization of the Continuity of Topologies},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/waybel29.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {241--247},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL30.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Meet Continuous Lattices Revisited},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/waybel30.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {249--254},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL31.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Weights of Continuous Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/waybel31.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {255--259},
      NUMBER  = {{\bf 2}})

@ARTICLE(LATTICE7.ABS,
      AUTHOR  = {Dudzicz, Marek},
      TITLE   = {Representation Theorem for Finite Distributive Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/lattice7.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {261--264},
      NUMBER  = {{\bf 2}})

@ARTICLE(COMPLFLD.ABS,
      AUTHOR  = {Milewska, Anna Justyna},
      TITLE   = {The Field of Complex Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/complfld.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {265--269},
      NUMBER  = {{\bf 2}})

@ARTICLE(INTEGRA4.ABS,
      AUTHOR  = {Endou, Noboru and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Integrability of Bounded Total Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/integra4.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {271--274},
      NUMBER  = {{\bf 2}})

@ARTICLE(RADIX_2.ABS,
      AUTHOR  = {Fuwa, Yasushi and Fujisawa, Yoshinori},
      TITLE   = {High-Speed Algorithms for {RSA} Cryptograms},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/radix_2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {275--279},
      NUMBER  = {{\bf 2}})

@ARTICLE(INTEGRA5.ABS,
      AUTHOR  = {Endou, Noboru and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Definition of Integrability for Partial Functions from {$\mathbb{R}$} to {$\mathbb{R}$} and Integrability for Continuous Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/integra5.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {281--284},
      NUMBER  = {{\bf 2}})

@ARTICLE(RFUNCT_4.ABS,
      AUTHOR  = {Endou, Noboru and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Introduction to Several Concepts of Convexity and Semicontinuity for Function from {$\mathbb{R}$} to {$\mathbb{R}$}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/rfunct_4.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {285--289},
      NUMBER  = {{\bf 2}})

@ARTICLE(AMISTD_1.ABS,
      AUTHOR  = {Trybulec, Andrzej and Rudnicki, Piotr and Korni{\l}owicz, Artur},
      TITLE   = {Standard Ordering of Instruction Locations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/amistd_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {291--301},
      NUMBER  = {{\bf 2}})

@ARTICLE(AMISTD_2.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Composition of Macro Instructions of Standard Computers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/amistd_2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {303--316},
      NUMBER  = {{\bf 2}})

@ARTICLE(SCMRING3.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Properties of Instructions of {SCM} over Ring},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/scmring3.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {317--322},
      NUMBER  = {{\bf 2}})

@ARTICLE(CARD_FIL.ABS,
      AUTHOR  = {Urban, Josef},
      TITLE   = {Basic Facts about Inaccessible and Measurable Cardinals},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/card_fil.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {323--329},
      NUMBER  = {{\bf 2}})

@ARTICLE(POLYNOM2.ABS,
      AUTHOR  = {Schwarzweller, Christoph and Trybulec, Andrzej},
      TITLE   = {The Evaluation of Multivariate Polynomials},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/polynom2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {331--338},
      NUMBER  = {{\bf 2}})

@ARTICLE(POLYNOM3.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {The Ring of Polynomials},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/polynom3.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {339--346},
      NUMBER  = {{\bf 2}})

@ARTICLE(POLYEQ_1.ABS,
      AUTHOR  = {Liang, Xiquan},
      TITLE   = {Solving Roots of Polynomial Equations of Degree 2 and 3 with Real Coefficients},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/polyeq_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {347--350},
      NUMBER  = {{\bf 2}})

@ARTICLE(FUZZY_1.ABS,
      AUTHOR  = {Mitsuishi, Takashi and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {The Concept of Fuzzy Set and Membership Function and Basic Properties of Fuzzy Set Operation},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/fuzzy_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {351--356},
      NUMBER  = {{\bf 2}})

@ARTICLE(FUZZY_2.ABS,
      AUTHOR  = {Mitsuishi, Takashi and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Basic Properties of Fuzzy Set Operation and Membership Function},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/fuzzy_2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {357--362},
      NUMBER  = {{\bf 2}})

@ARTICLE(HAHNBAN1.ABS,
      AUTHOR  = {Milewska, Anna Justyna},
      TITLE   = {The {H}ahn {B}anach Theorem in the Vector Space over the Field of Complex Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/hahnban1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {363--371},
      NUMBER  = {{\bf 2}})

@ARTICLE(YELLOW17.ABS,
      AUTHOR  = {Skorulski, Bart{\l}omiej},
      TITLE   = {The {T}ichonov {T}heorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/yellow17.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {373--376},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL32.ABS,
      AUTHOR  = {Gr\k{a}dzka, Ewa},
      TITLE   = {On the Order-consistent Topology of Complete and Uncomplete Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/waybel32.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {377--382},
      NUMBER  = {{\bf 2}})

@ARTICLE(PENCIL_1.ABS,
      AUTHOR  = {Naumowicz, Adam},
      TITLE   = {On {S}egre's Product of Partial Line Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/pencil_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {383--390},
      NUMBER  = {{\bf 2}})

@ARTICLE(POLYNOM4.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {The Evaluation of Polynomials},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/polynom4.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {391--395},
      NUMBER  = {{\bf 2}})

@ARTICLE(SCMPDS_8.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {The Construction and Computation of While-Loop Programs for {SCMPDS}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/scmpds_8.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {397--405},
      NUMBER  = {{\bf 2}})

@ARTICLE(SCPISORT.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {Insert Sort on {SCMPDS}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/scpisort.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {407--412},
      NUMBER  = {{\bf 2}})

@ARTICLE(SCPQSORT.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {Quick Sort on {SCMPDS}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/scpqsort.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {413--418},
      NUMBER  = {{\bf 2}})

@ARTICLE(SCPINVAR.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {Justifying the Correctness of the {F}ibonacci Sequence and the {E}uclide Algorithm by Loop-Invariant},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/scpinvar.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {419--427},
      NUMBER  = {{\bf 2}})

@ARTICLE(ORDERS_4.ABS,
      AUTHOR  = {Pruszy\'nska, Marta and Dudzicz, Marek},
      TITLE   = {On the Isomorphism between Finite Chains},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/orders_4.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {429--430},
      NUMBER  = {{\bf 2}})

@ARTICLE(LATTICE8.ABS,
      AUTHOR  = {{\L}api\'nski, Mariusz},
      TITLE   = {The {J}\'onsson {T}heorem about the Representation of Modular Lattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-2/lattice8.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {431--438},
      NUMBER  = {{\bf 2}})

@ARTICLE(HILBERT3.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {The Canonical Formulae},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/hilbert3.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {441--447},
      NUMBER  = {{\bf 3}})

@ARTICLE(HEYTING3.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {The Incompleteness of the Lattice of Substitutions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/heyting3.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {449--454},
      NUMBER  = {{\bf 3}})

@ARTICLE(COMPTRIG.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Trigonometric Form of Complex Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/comptrig.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {455--460},
      NUMBER  = {{\bf 3}})

@ARTICLE(POLYNOM5.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Fundamental Theorem of Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/polynom5.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {461--470},
      NUMBER  = {{\bf 3}})

@ARTICLE(FINSEQ_7.ABS,
      AUTHOR  = {Yamazaki, Hiroshi and Fujisawa, Yoshinori and Nakamura, Yatsuka},
      TITLE   = {On Replace Function and Swap Function for Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/finseq_7.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {471--474},
      NUMBER  = {{\bf 3}})

@ARTICLE(GATE_5.ABS,
      AUTHOR  = {Yamazaki, Hiroshi and Wasaki, Katsumi},
      TITLE   = {The Correctness of the High Speed Array Multiplier Circuits},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/gate_5.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {475--479},
      NUMBER  = {{\bf 3}})

@ARTICLE(JCT_MISC.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Some Lemmas for the {J}ordan Curve Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/jct_misc.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {481--484},
      NUMBER  = {{\bf 3}})

@ARTICLE(CARD_LAR.ABS,
      AUTHOR  = {Urban, Josef},
      TITLE   = {Mahlo and Inaccessible Cardinals},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/card_lar.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {485--489},
      NUMBER  = {{\bf 3}})

@ARTICLE(EXTREAL1.ABS,
      AUTHOR  = {Endou, Noboru and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Basic Properties of Extended Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/extreal1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {491--494},
      NUMBER  = {{\bf 3}})

@ARTICLE(MESFUNC1.ABS,
      AUTHOR  = {Endou, Noboru and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Definitions and Basic Properties of Measurable Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/mesfunc1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {495--500},
      NUMBER  = {{\bf 3}})

@ARTICLE(JORDAN1A.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Milewski, Robert and Naumowicz, Adam and Trybulec, Andrzej},
      TITLE   = {Gauges and Cages. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/jordan1a.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {501--509},
      NUMBER  = {{\bf 3}})

@ARTICLE(EXTREAL2.ABS,
      AUTHOR  = {Endou, Noboru and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Some Properties of Extended Real Numbers Operations: abs, min and max},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/extreal2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {511--516},
      NUMBER  = {{\bf 3}})

@ARTICLE(FUZZY_3.ABS,
      AUTHOR  = {Mitsuishi, Takashi and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {The Concept of Fuzzy Relation and Basic Properties of its Operation},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/fuzzy_3.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {517--524},
      NUMBER  = {{\bf 3}})

@ARTICLE(MESFUNC2.ABS,
      AUTHOR  = {Endou, Noboru and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {The Measurability of Extended Real Valued Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/mesfunc2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {525--529},
      NUMBER  = {{\bf 3}})

@ARTICLE(JORDAN1B.ABS,
      AUTHOR  = {Milewski, Robert and Trybulec, Andrzej and Korni{\l}owicz, Artur and Naumowicz, Adam},
      TITLE   = {Some Properties of Cells and Arcs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/jordan1b.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {531--535},
      NUMBER  = {{\bf 3}})

@ARTICLE(FINTOPO2.ABS,
      AUTHOR  = {Liu, Gang and Fuwa, Yasushi and Eguchi, Masayoshi},
      TITLE   = {Formal Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/fintopo2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {537--543},
      NUMBER  = {{\bf 3}})

@ARTICLE(JORDAN1C.ABS,
      AUTHOR  = {Grabowski, Adam and Korni{\l}owicz, Artur and Trybulec, Andrzej},
      TITLE   = {Some Properties of Cells and Gauges},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/jordan1c.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {545--548},
      NUMBER  = {{\bf 3}})

@ARTICLE(SPRECT_5.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {Again on the Order on a Special Polygon},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/sprect_5.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {549--553},
      NUMBER  = {{\bf 3}})

@ARTICLE(JORDAN1D.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Milewski, Robert},
      TITLE   = {Gauges and Cages. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/jordan1d.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {555--558},
      NUMBER  = {{\bf 3}})

@ARTICLE(BINOM.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {The Binomial Theorem for Algebraic Structures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/binom.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {559--564},
      NUMBER  = {{\bf 3}})

@ARTICLE(IDEAL_1.ABS,
      AUTHOR  = {Backer, Jonathan and Rudnicki, Piotr and Schwarzweller, Christoph},
      TITLE   = {Ring Ideals},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/ideal_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {565--582},
      NUMBER  = {{\bf 3}})

@ARTICLE(HILBASIS.ABS,
      AUTHOR  = {Backer, Jonathan and Rudnicki, Piotr},
      TITLE   = {Hilbert Basis Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/hilbasis.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {583--589},
      NUMBER  = {{\bf 3}})

@ARTICLE(DYNKIN.ABS,
      AUTHOR  = {Merkl, Franz},
      TITLE   = {Dynkin's Lemma in Measure Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/dynkin.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {591--595},
      NUMBER  = {{\bf 3}})

@ARTICLE(TAXONOM1.ABS,
      AUTHOR  = {Giero, Mariusz and Matuszewski, Roman},
      TITLE   = {Lower Tolerance. {P}reliminaries to {W}roclaw Taxonomy},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/taxonom1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {597--603},
      NUMBER  = {{\bf 3}})

@ARTICLE(YELLOW18.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Concrete Categories},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/yellow18.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {605--621},
      NUMBER  = {{\bf 3}})

@ARTICLE(PARTIT_2.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Classes of Independent Partitions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/partit_2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {623--625},
      NUMBER  = {{\bf 3}})

@ARTICLE(URYSOHN2.ABS,
      AUTHOR  = {Bia{\l}as, J\'ozef and Nakamura, Yatsuka},
      TITLE   = {Some Properties of Dyadic Numbers and Intervals},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/urysohn2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {627--630},
      NUMBER  = {{\bf 3}})

@ARTICLE(URYSOHN3.ABS,
      AUTHOR  = {Bia{\l}as, J\'ozef and Nakamura, Yatsuka},
      TITLE   = {The {U}rysohn Lemma},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/urysohn3.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {631--636},
      NUMBER  = {{\bf 3}})

@ARTICLE(POLYALG1.ABS,
      AUTHOR  = {Gr\k{a}dzka, Ewa},
      TITLE   = {The Algebra of Polynomials},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/polyalg1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {637--643},
      NUMBER  = {{\bf 3}})

@ARTICLE(CIRCTRM1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Circuit Generated by Terms and Circuit Calculating Terms},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-3/circtrm1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {645--657},
      NUMBER  = {{\bf 3}})

@ARTICLE(AMI_6.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Instructions of {\SCM}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/ami_6.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {659--663},
      NUMBER  = {{\bf 4}})

@ARTICLE(AMI_7.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Input and Output of Instructions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/ami_7.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {665--671},
      NUMBER  = {{\bf 4}})

@ARTICLE(SCMFSA10.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Instructions of {\SCMFSA}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/scmfsa10.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {673--679},
      NUMBER  = {{\bf 4}})

@ARTICLE(ROBBINS1.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {Robbins Algebras vs. {B}oolean Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/robbins1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {681--690},
      NUMBER  = {{\bf 4}})

@ARTICLE(FUZZY_4.ABS,
      AUTHOR  = {Endou, Noboru and Mitsuishi, Takashi and Ohkubo, Keiji},
      TITLE   = {Properties of Fuzzy Relation},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/fuzzy_4.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {691--695},
      NUMBER  = {{\bf 4}})

@ARTICLE(JGRAPH_2.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {On {O}utside {F}ashoda {M}eet {T}heorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/jgraph_2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {697--704},
      NUMBER  = {{\bf 4}})

@ARTICLE(COMPUT_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Rudnicki, Piotr},
      TITLE   = {The Set of Primitive Recursive Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/comput_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {705--720},
      NUMBER  = {{\bf 4}})

@ARTICLE(TURING_1.ABS,
      AUTHOR  = {Chen, Jing-Chao and Nakamura, Yatsuka},
      TITLE   = {Introduction to {T}uring Machines},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/turing_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {721--732},
      NUMBER  = {{\bf 4}})

@ARTICLE(YELLOW19.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Endou, Noboru and Sakai, Yuji},
      TITLE   = {On the Characterizations of Compactness},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/yellow19.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {733--738},
      NUMBER  = {{\bf 4}})

@ARTICLE(WAYBEL33.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Endou, Noboru},
      TITLE   = {Compactness of Lim-inf Topology},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/waybel33.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {739--743},
      NUMBER  = {{\bf 4}})

@ARTICLE(YELLOW20.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Miscellaneous Facts about Functors},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/yellow20.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {745--754},
      NUMBER  = {{\bf 4}})

@ARTICLE(YELLOW21.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Categorial Background for Duality Theory},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/yellow21.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {755--765},
      NUMBER  = {{\bf 4}})

@ARTICLE(WAYBEL34.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Duality Based on the {G}alois Connection. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/waybel34.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {767--778},
      NUMBER  = {{\bf 4}})

@ARTICLE(MSAFREE3.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Korni{\l}owicz, Artur},
      TITLE   = {Yet Another Construction of Free Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/msafree3.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {779--785},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN1E.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Upper and Lower Sequence of a Cage},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/jordan1e.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {787--790},
      NUMBER  = {{\bf 4}})

@ARTICLE(POLYNOM6.ABS,
      AUTHOR  = {Dzienis, Barbara},
      TITLE   = {On Polynomials with Coefficients in a Ring of Polynomials},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/polynom6.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {791--794},
      NUMBER  = {{\bf 4}})

@ARTICLE(PENCIL_2.ABS,
      AUTHOR  = {Naumowicz, Adam},
      TITLE   = {On Cosets in {S}egre's Product of Partial Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/pencil_2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {795--800},
      NUMBER  = {{\bf 4}})

@ARTICLE(JGRAPH_3.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {On the Simple Closed Curve Property of the Circle and the {F}ashoda {M}eet {T}heorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/jgraph_3.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {801--808},
      NUMBER  = {{\bf 4}})

@ARTICLE(PYTHTRIP.ABS,
      AUTHOR  = {Wiedijk, Freek},
      TITLE   = {Pythagorean Triples},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/pythtrip.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {809--812},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN1F.ABS,
      AUTHOR  = {Naumowicz, Adam},
      TITLE   = {Some Remarks on Finite Sequences on Go-boards},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/jordan1f.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {813--816},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN1G.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Upper and Lower Sequence on the Cage. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/jordan1g.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {817--823},
      NUMBER  = {{\bf 4}})

@ARTICLE(AFINSQ_1.ABS,
      AUTHOR  = {Tsunetou, Tetsuya and Bancerek, Grzegorz and Nakamura, Yatsuka},
      TITLE   = {Zero-Based Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/afinsq_1.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {825--829},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN1H.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {More on the External Approximation of a~Continuum},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/jordan1h.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {831--841},
      NUMBER  = {{\bf 4}})

@ARTICLE(TOPREAL8.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {More on the Finite Sequences on the Plane},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/topreal8.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {843--847},
      NUMBER  = {{\bf 4}})

@ARTICLE(POLYNOM7.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {More on Multivariate Polynomials: Monomials and Constant Polynomials},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/polynom7.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {849--855},
      NUMBER  = {{\bf 4}})

@ARTICLE(FSM_2.ABS,
      AUTHOR  = {Kunimune, Hisayoshi and Bancerek, Grzegorz and Nakamura, Yatsuka},
      TITLE   = {On State Machines of Calculating Type},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/fsm_2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {857--864},
      NUMBER  = {{\bf 4}})

@ARTICLE(TAXONOM2.ABS,
      AUTHOR  = {Giero, Mariusz},
      TITLE   = {Hierarchies and Classifications of Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2001-9/pdf9-4/taxonom2.pdf},
      YEAR    = {2001},
      VOLUME  = 9,
      PAGES   = {865--869},
      NUMBER  = {{\bf 4}})

@ARTICLE(JGRAPH_4.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {Fan Homeomorphisms in the Plane},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-1/jgraph_4.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {1--19},
      NUMBER  = {{\bf 1}})

@ARTICLE(RCOMP_2.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {Half Open Intervals in Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-1/rcomp_2.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {21--22},
      NUMBER  = {{\bf 1}})

@ARTICLE(JORDAN1I.ABS,
      AUTHOR  = {Naumowicz, Adam and Milewski, Robert},
      TITLE   = {Some Remarks on Clockwise Oriented Sequences on Go-boards},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-1/jordan1i.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {23--27},
      NUMBER  = {{\bf 1}})

@ARTICLE(DICKSON.ABS,
      AUTHOR  = {Lee, Gilbert and Rudnicki, Piotr},
      TITLE   = {Dickson's Lemma},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-1/dickson.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {29--37},
      NUMBER  = {{\bf 1}})

@ARTICLE(BAGORDER.ABS,
      AUTHOR  = {Lee, Gilbert and Rudnicki, Piotr},
      TITLE   = {On Ordering of Bags},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-1/bagorder.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {39--46},
      NUMBER  = {{\bf 1}})

@ARTICLE(CIRCCMB2.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Yamaguchi, Shin'nosuke and Shidama, Yasunari},
      TITLE   = {Combining of Multi Cell Circuits},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-1/circcmb2.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {47--64},
      NUMBER  = {{\bf 1}})

@ARTICLE(FACIRC_2.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Yamaguchi, Shin'nosuke and Wasaki, Katsumi},
      TITLE   = {Full Adder Circuit. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-1/facirc_2.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {65--71},
      NUMBER  = {{\bf 1}})

@ARTICLE(JORDAN1J.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Upper and Lower Sequence on the Cage, Upper and Lower Arcs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-2/jordan1j.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {73--80},
      NUMBER  = {{\bf 2}})

@ARTICLE(FIB_NUM.ABS,
      AUTHOR  = {Solovay, Robert M.},
      TITLE   = {Fibonacci Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-2/fib_num.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {81--83},
      NUMBER  = {{\bf 2}})

@ARTICLE(JORDAN11.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Preparing the Internal Approximations of Simple Closed Curves},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-2/jordan11.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {85--87},
      NUMBER  = {{\bf 2}})

@ARTICLE(JORDAN12.ABS,
      AUTHOR  = {Giero, Mariusz},
      TITLE   = {On the General Position of Special Polygons},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-2/jordan12.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {89--95},
      NUMBER  = {{\bf 2}})

@ARTICLE(JORDAN13.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {Introducing Spans},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-2/jordan13.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {97--98},
      NUMBER  = {{\bf 2}})

@ARTICLE(JGRAPH_5.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {General {F}ashoda {M}eet {T}heorem for Unit Circle},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-2/jgraph_5.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {99--109},
      NUMBER  = {{\bf 2}})

@ARTICLE(JORDAN14.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Properties of the Internal Approximation of {J}ordan's Curve},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-2/jordan14.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {111--115},
      NUMBER  = {{\bf 2}})

@ARTICLE(CIRCCMB3.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Naumowicz, Adam},
      TITLE   = {Preliminaries to Automatic Generation  of {M}izar Documentation for Circuits},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/circcmb3.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {117--133},
      NUMBER  = {{\bf 3}})

@ARTICLE(JORDAN15.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Properties of the Upper and Lower Sequence on the Cage},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/jordan15.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {135--143},
      NUMBER  = {{\bf 3}})

@ARTICLE(BORSUK_4.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {On the Decompositions of Intervals  and Simple Closed Curves},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/borsuk_4.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {145--151},
      NUMBER  = {{\bf 3}})

@ARTICLE(JORDAN1K.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {On the Minimal Distance Between Sets  in {E}uclidean Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/jordan1k.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {153--158},
      NUMBER  = {{\bf 3}})

@ARTICLE(TOPMETR3.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {Sequences of Metric Spaces and  an Abstract Intermediate Value Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/topmetr3.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {159--161},
      NUMBER  = {{\bf 3}})

@ARTICLE(JORDAN16.ABS,
      AUTHOR  = {Trybulec, Andrzej and Nakamura, Yatsuka},
      TITLE   = {On the Decomposition of a Simple Closed Curve into Two Arcs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/jordan16.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {163--167},
      NUMBER  = {{\bf 3}})

@ARTICLE(JORDAN17.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Ordering of Points on a Curve.  {P}art {III}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/jordan17.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {169--171},
      NUMBER  = {{\bf 3}})

@ARTICLE(JORDAN18.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Ordering of Points on a Curve.  {P}art {IV}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/jordan18.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {173--177},
      NUMBER  = {{\bf 3}})

@ARTICLE(OSALG_1.ABS,
      AUTHOR  = {Urban, Josef},
      TITLE   = {Order Sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/osalg_1.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {179--188},
      NUMBER  = {{\bf 3}})

@ARTICLE(OSALG_2.ABS,
      AUTHOR  = {Urban, Josef},
      TITLE   = {Subalgebras of an Order Sorted Algebra. {L}attice of Subalgebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/osalg_2.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {189--196},
      NUMBER  = {{\bf 3}})

@ARTICLE(OSALG_3.ABS,
      AUTHOR  = {Urban, Josef},
      TITLE   = {Homomorphisms of Order Sorted Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/osalg_3.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {197--200},
      NUMBER  = {{\bf 3}})

@ARTICLE(OSALG_4.ABS,
      AUTHOR  = {Urban, Josef},
      TITLE   = {Order Sorted Quotient Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/osalg_4.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {201--210},
      NUMBER  = {{\bf 3}})

@ARTICLE(OSAFREE.ABS,
      AUTHOR  = {Urban, Josef},
      TITLE   = {Free Order Sorted Universal Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2002-10/pdf10-3/osafree.pdf},
      YEAR    = {2002},
      VOLUME  = 10,
      PAGES   = {211--225},
      NUMBER  = {{\bf 3}})

@ARTICLE(RUSUB_1.ABS,
      AUTHOR  = {Endou, Noboru and Mitsuishi, Takashi and Shidama, Yasunari},
      TITLE   = {Subspaces and Cosets of Subspace of Real Unitary Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/rusub_1.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {1--7},
      NUMBER  = {{\bf 1}})

@ARTICLE(RUSUB_2.ABS,
      AUTHOR  = {Endou, Noboru and Mitsuishi, Takashi and Shidama, Yasunari},
      TITLE   = {Operations on Subspaces in Real Unitary Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/rusub_2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {9--16},
      NUMBER  = {{\bf 1}})

@ARTICLE(RUSUB_3.ABS,
      AUTHOR  = {Endou, Noboru and Mitsuishi, Takashi and Shidama, Yasunari},
      TITLE   = {Linear Combinations in Real Unitary Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/rusub_3.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {17--22},
      NUMBER  = {{\bf 1}})

@ARTICLE(RUSUB_4.ABS,
      AUTHOR  = {Endou, Noboru and Mitsuishi, Takashi and Shidama, Yasunari},
      TITLE   = {Dimension of Real Unitary Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/rusub_4.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {23--28},
      NUMBER  = {{\bf 1}})

@ARTICLE(SIN_COS3.ABS,
      AUTHOR  = {Mitsuishi, Takashi and Endou, Noboru and Ohkubo, Keiji},
      TITLE   = {Trigonometric Functions on Complex Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/sin_cos3.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {29--32},
      NUMBER  = {{\bf 1}})

@ARTICLE(RUSUB_5.ABS,
      AUTHOR  = {Endou, Noboru and Mitsuishi, Takashi and Shidama, Yasunari},
      TITLE   = {Topology of Real Unitary Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/rusub_5.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {33--38},
      NUMBER  = {{\bf 1}})

@ARTICLE(ARMSTRNG.ABS,
      AUTHOR  = {Armstrong, William W. and Nakamura, Yatsuka and Rudnicki, Piotr},
      TITLE   = {Armstrong's Axioms},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/armstrng.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {39--51},
      NUMBER  = {{\bf 1}})

@ARTICLE(CONVEX1.ABS,
      AUTHOR  = {Endou, Noboru and Mitsuishi, Takashi and Shidama, Yasunari},
      TITLE   = {Convex Sets and Convex Combinations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/convex1.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {53--58},
      NUMBER  = {{\bf 1}})

@ARTICLE(VECTSP10.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Quotient Vector Spaces and Functionals},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/vectsp10.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {59--68},
      NUMBER  = {{\bf 1}})

@ARTICLE(BILINEAR.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Bilinear Functionals in Vector Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/bilinear.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {69--86},
      NUMBER  = {{\bf 1}})

@ARTICLE(HERMITAN.ABS,
      AUTHOR  = {Kotowicz, Jaros{\l}aw},
      TITLE   = {Hermitan Functionals. {C}anonical Construction of Scalar Product in Quotient Vector Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/hermitan.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {87--98},
      NUMBER  = {{\bf 1}})

@ARTICLE(NECKLACE.ABS,
      AUTHOR  = {Retel, Krzysztof},
      TITLE   = {The Class of Series -- Parallel Graphs. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/necklace.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {99--103},
      NUMBER  = {{\bf 1}})

@ARTICLE(TERMORD.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {Term Orders},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/termord.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {105--111},
      NUMBER  = {{\bf 1}})

@ARTICLE(POLYRED.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {Polynomial Reduction},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/polyred.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {113--123},
      NUMBER  = {{\bf 1}})

@ARTICLE(PNPROC_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Aoki, Mitsuru and Matsumoto, Akio and Shidama, Yasunari},
      TITLE   = {Processes in {P}etri nets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-1/pnproc_1.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {125--132},
      NUMBER  = {{\bf 1}})

@ARTICLE(RADIX_3.ABS,
      AUTHOR  = {Niimura, Masaaki and Fuwa, Yasushi},
      TITLE   = {Improvement of Radix-$2^k$ Signed-Digit Number for High Speed Circuit},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/radix_3.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {133--137},
      NUMBER  = {{\bf 2}})

@ARTICLE(RADIX_4.ABS,
      AUTHOR  = {Niimura, Masaaki and Fuwa, Yasushi},
      TITLE   = {High Speed Adder Algorithm with Radix-$2^k$ Sub Signed-Digit Number},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/radix_4.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {139--141},
      NUMBER  = {{\bf 2}})

@ARTICLE(GRAPH_5.ABS,
      AUTHOR  = {Chen, Jing-Chao and Nakamura, Yatsuka},
      TITLE   = {The Underlying Principle of {D}ijkstra's Shortest Path Algorithm},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/graph_5.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {143--152},
      NUMBER  = {{\bf 2}})

@ARTICLE(HAUSDORF.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {On the {H}ausdorff Distance Between Compact Subsets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/hausdorf.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {153--157},
      NUMBER  = {{\bf 2}})

@ARTICLE(CHAIN_1.ABS,
      AUTHOR  = {Wiedijk, Freek},
      TITLE   = {Chains on a Grating in {E}uclidean Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/chain_1.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {159--167},
      NUMBER  = {{\bf 2}})

@ARTICLE(BHSP_5.ABS,
      AUTHOR  = {Yamazaki, Hiroshi and Shidama, Yasunari and Nakamura, Yatsuka},
      TITLE   = {Bessel's Inequality},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/bhsp_5.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {169--173},
      NUMBER  = {{\bf 2}})

@ARTICLE(BINARI_4.ABS,
      AUTHOR  = {Kunimune, Hisayoshi and Nakamura, Yatsuka},
      TITLE   = {A Representation of Integers by Binary Arithmetics and Addition of Integers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/binari_4.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {175--178},
      NUMBER  = {{\bf 2}})

@ARTICLE(EUCLID_2.ABS,
      AUTHOR  = {Kanchun and Nakamura, Yatsuka},
      TITLE   = {The Inner Product of Finite Sequences  and of Points of $n$-dimensional Topological Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/euclid_2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {179--183},
      NUMBER  = {{\bf 2}})

@ARTICLE(POLYEQ_2.ABS,
      AUTHOR  = {Liang, Xiquan},
      TITLE   = {Solving Roots of Polynomial Equation  of Degree 4 with Real Coefficients},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/polyeq_2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {185--187},
      NUMBER  = {{\bf 2}})

@ARTICLE(WAYBEL35.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Morphisms Into Chains. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/waybel35.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {189--195},
      NUMBER  = {{\bf 2}})

@ARTICLE(BVFUNC25.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Propositional Calculus for {B}oolean Valued Functions. {P}art {VII}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/bvfunc25.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {197--199},
      NUMBER  = {{\bf 2}})

@ARTICLE(OPOSET_1.ABS,
      AUTHOR  = {Moschner, Markus},
      TITLE   = {Basic Notions and Properties of Orthoposets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-2/oposet_1.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {201--210},
      NUMBER  = {{\bf 2}})

@ARTICLE(JGRAPH_6.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {General {F}ashoda Meet Theorem for Unit Circle and Square},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/jgraph_6.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {213--224},
      NUMBER  = {{\bf 3}})

@ARTICLE(BHSP_6.ABS,
      AUTHOR  = {Yamazaki, Hiroshi and Suzuki, Yasumasa and Inou\'e, Takao and Shidama, Yasunari},
      TITLE   = {On Some Properties of Real {H}ilbert Space. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/bhsp_6.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {225--229},
      NUMBER  = {{\bf 3}})

@ARTICLE(FSCIRC_2.ABS,
      AUTHOR  = {Yamaguchi, Shin'nosuke and Bancerek, Grzegorz and Wasaki, Katsumi},
      TITLE   = {Full Subtracter Circuit. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/fscirc_2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {231--236},
      NUMBER  = {{\bf 3}})

@ARTICLE(GRAPHSP.ABS,
      AUTHOR  = {Chen, Jing-Chao},
      TITLE   = {Dijkstra's Shortest Path Algorithm},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/graphsp.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {237--247},
      NUMBER  = {{\bf 3}})

@ARTICLE(RSSPACE.ABS,
      AUTHOR  = {Endou, Noboru and Suzuki, Yasumasa and Shidama, Yasunari},
      TITLE   = {Real Linear Space of Real Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/rsspace.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {249--253},
      NUMBER  = {{\bf 3}})

@ARTICLE(RSSPACE2.ABS,
      AUTHOR  = {Endou, Noboru and Suzuki, Yasumasa and Shidama, Yasunari},
      TITLE   = {Hilbert Space of Real Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/rsspace2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {255--257},
      NUMBER  = {{\bf 3}})

@ARTICLE(INTPRO_1.ABS,
      AUTHOR  = {Inou\'e, Takao},
      TITLE   = {Intuitionistic Propositional Calculus in the Extended Framework with Modal Operator. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/intpro_1.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {259--266},
      NUMBER  = {{\bf 3}})

@ARTICLE(CONVEX2.ABS,
      AUTHOR  = {Endou, Noboru and Suzuki, Yasumasa and Shidama, Yasunari},
      TITLE   = {Some Properties for Convex Combinations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/convex2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {267--270},
      NUMBER  = {{\bf 3}})

@ARTICLE(BHSP_7.ABS,
      AUTHOR  = {Yamazaki, Hiroshi and Suzuki, Yasumasa and Inou\'{e}, Takao and Shidama, Yasunari},
      TITLE   = {On Some Properties of Real {H}ilbert Space. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/bhsp_7.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {271--273},
      NUMBER  = {{\bf 3}})

@ARTICLE(COMPLEX2.ABS,
      AUTHOR  = {Chang, Wenpai and Nakamura, Yatsuka and Rudnicki, Piotr},
      TITLE   = {Inner Products and Angles of Complex Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/complex2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {275--280},
      NUMBER  = {{\bf 3}})

@ARTICLE(EUCLID_3.ABS,
      AUTHOR  = {Kubo, Akihiro and Nakamura, Yatsuka},
      TITLE   = {Angle and Triangle in {E}uclidean Topological Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/euclid_3.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {281--287},
      NUMBER  = {{\bf 3}})

@ARTICLE(NECKLA_2.ABS,
      AUTHOR  = {Retel, Krzysztof},
      TITLE   = {The Class of Series-Parallel Graphs.  {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/neckla_2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {289--291},
      NUMBER  = {{\bf 3}})

@ARTICLE(GROEB_1.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {Characterization and Existence of {G}r\"{o}bner Bases},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/groeb_1.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {293--301},
      NUMBER  = {{\bf 3}})

@ARTICLE(GROEB_2.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {Construction of {G}r\"{o}bner bases.  {S}-Polynomials and Standard Representations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/groeb_2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {303--312},
      NUMBER  = {{\bf 3}})

@ARTICLE(BORSUK_5.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {On the Subcontinua of a Real Line},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/borsuk_5.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {313--322},
      NUMBER  = {{\bf 3}})

@ARTICLE(KURATO_1.ABS,
      AUTHOR  = {Bagi\'nska, Lilla Krystyna and Grabowski, Adam},
      TITLE   = {On the {K}uratowski Closure-Complement Problem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/kurato_1.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {323--329},
      NUMBER  = {{\bf 3}})

@ARTICLE(CONVEX3.ABS,
      AUTHOR  = {Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Convex Hull, Set of Convex Combinations and Convex Cone},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/convex3.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {331--333},
      NUMBER  = {{\bf 3}})

@ARTICLE(ROBBINS2.ABS,
      AUTHOR  = {Truszkowska, Wioletta and Grabowski, Adam},
      TITLE   = {On the Two Short Axiomatizations of Ortholattices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-3/robbins2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {335--340},
      NUMBER  = {{\bf 3}})

@ARTICLE(MEMBERED.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {On the Sets Inhabited by Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/membered.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {341--347},
      NUMBER  = {{\bf 4}})

@ARTICLE(CONVFUN1.ABS,
      AUTHOR  = {Ivanov, Grigory E.},
      TITLE   = {Definition of Convex Function  and {J}ensen's Inequality},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/convfun1.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {349--354},
      NUMBER  = {{\bf 4}})

@ARTICLE(ABCMIZ_0.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {On Semilattice Structure of {M}izar Types},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/abcmiz_0.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {355--369},
      NUMBER  = {{\bf 4}})

@ARTICLE(EUCLID_4.ABS,
      AUTHOR  = {Kubo, Akihiro},
      TITLE   = {Lines in $n$-Dimensional {E}uclidean Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/euclid_4.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {371--376},
      NUMBER  = {{\bf 4}})

@ARTICLE(RSSPACE3.ABS,
      AUTHOR  = {Suzuki, Yasumasa and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Banach Space of Absolute Summable  Real Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/rsspace3.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {377--380},
      NUMBER  = {{\bf 4}})

@ARTICLE(EUCLID_5.ABS,
      AUTHOR  = {Kanchun and Yamazaki, Hiroshi and Nakamura, Yatsuka},
      TITLE   = {Cross Products and Tripple Vector Products  in 3-dimensional {E}uclidean Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/euclid_5.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {381--383},
      NUMBER  = {{\bf 4}})

@ARTICLE(MATRIX_4.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Yamazaki, Hiroshi},
      TITLE   = {Calculation of Matrices of Field Elements. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/matrix_4.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {385--391},
      NUMBER  = {{\bf 4}})

@ARTICLE(LFUZZY_0.ABS,
      AUTHOR  = {Mitsuishi, Takashi and Bancerek, Grzegorz},
      TITLE   = {Lattice of Fuzzy Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/lfuzzy_0.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {393--398},
      NUMBER  = {{\bf 4}})

@ARTICLE(KURATO_2.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {On the {K}uratowski Limit Operators},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/kurato_2.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {399--409},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN_A.ABS,
      AUTHOR  = {Trybulec, Andrzej},
      TITLE   = {On the Segmentation of a Simple Closed Curve},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/jordan_a.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {411--416},
      NUMBER  = {{\bf 4}})

@ARTICLE(BINARI_5.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {On the Calculus of Binary Arithmetics},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/binari_5.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {417--419},
      NUMBER  = {{\bf 4}})

@ARTICLE(SCMPDS_9.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {{SCMPDS} Is Not Standard},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/scmpds_9.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {421--424},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN19.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {On the Upper and Lower Approximations of the Curve},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2003-11/pdf11-4/jordan19.pdf},
      YEAR    = {2003},
      VOLUME  = 11,
      PAGES   = {425--430},
      NUMBER  = {{\bf 4}})

@ARTICLE(RFINSEQ2.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {Sorting Operators for Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/rfinseq2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {1--4},
      NUMBER  = {{\bf 1}})

@ARTICLE(RADIX_5.ABS,
      AUTHOR  = {Niimura, Masaaki and Fuwa, Yasushi},
      TITLE   = {Magnitude Relation Properties of Radix-$2^k$ {SD} Number},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/radix_5.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {5--8},
      NUMBER  = {{\bf 1}})

@ARTICLE(RADIX_6.ABS,
      AUTHOR  = {Niimura, Masaaki and Fuwa, Yasushi},
      TITLE   = {High Speed Modulo Calculation Algorithm with Radix-$2^k$ {SD} Number},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/radix_6.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {9--13},
      NUMBER  = {{\bf 1}})

@ARTICLE(LFUZZY_1.ABS,
      AUTHOR  = {Mitsuishi, Takashi and Bancerek, Grzegorz},
      TITLE   = {Transitive Closure of Fuzzy Relations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/lfuzzy_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {15--20},
      NUMBER  = {{\bf 1}})

@ARTICLE(ROUGHS_1.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {Basic Properties of Rough Sets and Rough Membership Function},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/roughs_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {21--28},
      NUMBER  = {{\bf 1}})

@ARTICLE(PRGCOR_1.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {Correctness of Non Overwriting Programs. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/prgcor_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {29--32},
      NUMBER  = {{\bf 1}})

@ARTICLE(AMISTD_3.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {A Tree of Execution of a Macroinstruction},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/amistd_3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {33--37},
      NUMBER  = {{\bf 1}})

@ARTICLE(LOPBAN_1.ABS,
      AUTHOR  = {Shidama, Yasunari},
      TITLE   = {Banach Space of Bounded Linear Operators},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/lopban_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {39--48},
      NUMBER  = {{\bf 1}})

@ARTICLE(UPROOTS.ABS,
      AUTHOR  = {Rudnicki, Piotr},
      TITLE   = {Little {B}ezout Theorem (Factor Theorem)},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/uproots.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {49--58},
      NUMBER  = {{\bf 1}})

@ARTICLE(UNIROOTS.ABS,
      AUTHOR  = {Arneson, Broderick and Rudnicki, Piotr},
      TITLE   = {Primitive Roots of Unity and Cyclotomic Polynomials},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/uniroots.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {59--67},
      NUMBER  = {{\bf 1}})

@ARTICLE(WEDDWITT.ABS,
      AUTHOR  = {Arneson, Broderick and Baaz, Matthias and Rudnicki, Piotr},
      TITLE   = {Witt's Proof of the {W}edderburn Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-1/weddwitt.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {69--75},
      NUMBER  = {{\bf 1}})

@ARTICLE(RSSPACE4.ABS,
      AUTHOR  = {Suzuki, Yasumasa},
      TITLE   = {Banach Space of Bounded Real Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/rsspace4.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {77--83},
      NUMBER  = {{\bf 2}})

@ARTICLE(POLYEQ_3.ABS,
      AUTHOR  = {Ding, Yuzhong and Liang, Xiquan},
      TITLE   = {Solving Roots of Polynomial Equation  of Degree 2 and 3 with Complex Coefficients},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/polyeq_3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {85--92},
      NUMBER  = {{\bf 2}})

@ARTICLE(CLVECT_1.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Complex Linear Space and Complex Normed Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/clvect_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {93--102},
      NUMBER  = {{\bf 2}})

@ARTICLE(LOPBAN_2.ABS,
      AUTHOR  = {Shidama, Yasunari},
      TITLE   = {The {B}anach Algebra of Bounded Linear Operators},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/lopban_2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {103--108},
      NUMBER  = {{\bf 2}})

@ARTICLE(CSSPACE.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Complex Linear Space of Complex Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/csspace.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {109--117},
      NUMBER  = {{\bf 2}})

@ARTICLE(JORDAN20.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {Behaviour of an Arc Crossing a Line},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/jordan20.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {119--124},
      NUMBER  = {{\bf 2}})

@ARTICLE(FINTOPO3.ABS,
      AUTHOR  = {Tanaka, Masami and Nakamura, Yatsuka},
      TITLE   = {Some Set Series  in Finite Topological Spaces.  {F}undamental Concepts for Image Processing},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/fintopo3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {125--129},
      NUMBER  = {{\bf 2}})

@ARTICLE(LOPBAN_3.ABS,
      AUTHOR  = {Shidama, Yasunari},
      TITLE   = {The Series on {B}anach Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/lopban_3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {131--138},
      NUMBER  = {{\bf 2}})

@ARTICLE(SIN_COS4.ABS,
      AUTHOR  = {Pacharapokin, Chanapat and Kanchun and Yamazaki, Hiroshi},
      TITLE   = {Formulas and Identities of Trigonometric Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/sin_cos4.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {139--141},
      NUMBER  = {{\bf 2}})

@ARTICLE(NECKLA_3.ABS,
      AUTHOR  = {Retel, Krzysztof},
      TITLE   = {The Class of Series-Parallel Graphs.  {P}art {III}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/neckla_3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {143--149},
      NUMBER  = {{\bf 2}})

@ARTICLE(SCMRING4.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {Relocability for {SCM} over Ring},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/scmring4.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {151--157},
      NUMBER  = {{\bf 2}})

@ARTICLE(CLVECT_2.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Convergent Sequences in Complex Unitary Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/clvect_2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {159--165},
      NUMBER  = {{\bf 2}})

@ARTICLE(RECDEF_2.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Recursive Definitions. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/recdef_2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {167--172},
      NUMBER  = {{\bf 2}})

@ARTICLE(LOPBAN_4.ABS,
      AUTHOR  = {Shidama, Yasunari},
      TITLE   = {The Exponential Function on {B}anach Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/lopban_4.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {173--177},
      NUMBER  = {{\bf 2}})

@ARTICLE(NAT_3.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Rudnicki, Piotr},
      TITLE   = {Fundamental {T}heorem of {A}rithmetic},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/nat_3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {179--186},
      NUMBER  = {{\bf 2}})

@ARTICLE(CSSPACE2.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Hilbert Space of Complex Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/csspace2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {187--190},
      NUMBER  = {{\bf 2}})

@ARTICLE(CSSPACE3.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Banach Space of Absolute Summable Complex Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/csspace3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {191--194},
      NUMBER  = {{\bf 2}})

@ARTICLE(TAYLOR_1.ABS,
      AUTHOR  = {Shidama, Yasunari},
      TITLE   = {The {T}aylor Expansions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/taylor_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {195--200},
      NUMBER  = {{\bf 2}})

@ARTICLE(CLOPBAN1.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Complex {B}anach Space of Bounded Linear Operators},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/clopban1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {201--209},
      NUMBER  = {{\bf 2}})

@ARTICLE(CSSPACE4.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Complex {B}anach Space of Bounded Complex Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/csspace4.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {211--218},
      NUMBER  = {{\bf 2}})

@ARTICLE(FINSEQ_8.ABS,
      AUTHOR  = {Fukura, Hirofumi and Nakamura, Yatsuka},
      TITLE   = {Concatenation of Finite Sequences Reducing Overlapping Part and an Argument  of Separators of Sequential Files},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/finseq_8.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {219--224},
      NUMBER  = {{\bf 2}})

@ARTICLE(CLVECT_3.ABS,
      AUTHOR  = {Suzuki, Yasumasa and Endou, Noboru},
      TITLE   = {Cauchy Sequence of Complex Unitary Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-2/clvect_3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {225--229},
      NUMBER  = {{\bf 2}})

@ARTICLE(CFUNCDOM.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Complex Valued Functions Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/cfuncdom.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {231--235},
      NUMBER  = {{\bf 3}})

@ARTICLE(CLOPBAN2.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Banach Algebra of Bounded Complex Linear Operators},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/clopban2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {237--242},
      NUMBER  = {{\bf 3}})

@ARTICLE(SIN_COS5.ABS,
      AUTHOR  = {Ding, Yuzhong and Liang, Xiquan},
      TITLE   = {Formulas and Identities of Trigonometric Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/sin_cos5.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {243--246},
      NUMBER  = {{\bf 3}})

@ARTICLE(POLYEQ_4.ABS,
      AUTHOR  = {Ding, Yuzhong and Liang, Xiquan},
      TITLE   = {Solving Roots of the Special Polynomial Equation with Real Coefficients},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/polyeq_4.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {247--250},
      NUMBER  = {{\bf 3}})

@ARTICLE(BORSUK_6.ABS,
      AUTHOR  = {Grabowski, Adam and Korni{\l}owicz, Artur},
      TITLE   = {Algebraic Properties of Homotopies},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/borsuk_6.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {251--260},
      NUMBER  = {{\bf 3}})

@ARTICLE(TOPALG_1.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Shidama, Yasunari and Grabowski, Adam},
      TITLE   = {The Fundamental Group},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/topalg_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {261--268},
      NUMBER  = {{\bf 3}})

@ARTICLE(NFCONT_1.ABS,
      AUTHOR  = {Nishiyama, Takaya and Ohkubo, Keiji and Shidama, Yasunari},
      TITLE   = {The Continuous Functions on Normed Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/nfcont_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {269--275},
      NUMBER  = {{\bf 3}})

@ARTICLE(NFCONT_2.ABS,
      AUTHOR  = {Nishiyama, Takaya and Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {The Uniform Continuity of Functions on Normed Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/nfcont_2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {277--279},
      NUMBER  = {{\bf 3}})

@ARTICLE(CLOPBAN3.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Series on Complex {B}anach Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/clopban3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {281--288},
      NUMBER  = {{\bf 3}})

@ARTICLE(CLOPBAN4.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Exponential Function on Complex {B}anach Algebra},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/clopban4.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {289--293},
      NUMBER  = {{\bf 3}})

@ARTICLE(TOPALG_2.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Fundamental Group of Convex Subspaces of ${\calE}^n_{\rmT}$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/topalg_2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {295--299},
      NUMBER  = {{\bf 3}})

@ARTICLE(TOPREAL9.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {Intersections of Intervals and Balls in ${\calE}^n_{\rmT}$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/topreal9.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {301--306},
      NUMBER  = {{\bf 3}})

@ARTICLE(FIB_NUM2.ABS,
      AUTHOR  = {Jastrz\c{e}bska, Magdalena and Grabowski, Adam},
      TITLE   = {Some Properties of {F}ibonacci Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/fib_num2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {307--313},
      NUMBER  = {{\bf 3}})

@ARTICLE(HALLMAR1.ABS,
      AUTHOR  = {Romanowicz, Ewa and Grabowski, Adam},
      TITLE   = {The {H}all {M}arriage {T}heorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/hallmar1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {315--320},
      NUMBER  = {{\bf 3}})

@ARTICLE(NDIFF_1.ABS,
      AUTHOR  = {Imura, Hiroshi and Kimura, Morishige and Shidama, Yasunari},
      TITLE   = {The Differentiable Functions on Normed Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/ndiff_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {321--327},
      NUMBER  = {{\bf 3}})

@ARTICLE(FIB_NUM3.ABS,
      AUTHOR  = {Wojtecki, Piotr and Grabowski, Adam},
      TITLE   = {Lucas Numbers and Generalized {F}ibonacci Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/fib_num3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {329--333},
      NUMBER  = {{\bf 3}})

@ARTICLE(LATSUM_1.ABS,
      AUTHOR  = {Romanowicz, Katarzyna and Grabowski, Adam},
      TITLE   = {The Operation of Addition of Relational Structures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/latsum_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {335--339},
      NUMBER  = {{\bf 3}})

@ARTICLE(NAGATA_1.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {The {N}agata-{S}mirnov Theorem. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/nagata_1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {341--346},
      NUMBER  = {{\bf 3}})

@ARTICLE(GROUP_8.ABS,
      AUTHOR  = {Geleijnse, Gijs and Bancerek, Grzegorz},
      TITLE   = {Properties of Groups},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/group_8.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {347--350},
      NUMBER  = {{\bf 3}})

@ARTICLE(CATALAN1.ABS,
      AUTHOR  = {Cz\k{e}stochowska, Dorota and Grabowski, Adam},
      TITLE   = {Catalan Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/catalan1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {351--353},
      NUMBER  = {{\bf 3}})

@ARTICLE(SHEFFER1.ABS,
      AUTHOR  = {Kozarkiewicz, Violetta and Grabowski, Adam},
      TITLE   = {Axiomatization of {B}oolean Algebras Based on {S}heffer Stroke},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/sheffer1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {355--361},
      NUMBER  = {{\bf 3}})

@ARTICLE(SHEFFER2.ABS,
      AUTHOR  = {{\L}ukaszuk, Aneta and Grabowski, Adam},
      TITLE   = {Short {S}heffer Stroke-Based Single Axiom for {B}oolean Algebras},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/sheffer2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {363--370},
      NUMBER  = {{\bf 3}})

@ARTICLE(NDIFF_2.ABS,
      AUTHOR  = {Imura, Hiroshi and Sakai, Yuji and Shidama, Yasunari},
      TITLE   = {Differentiable Functions on Normed Linear Spaces. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/ndiff_2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {371--374},
      NUMBER  = {{\bf 3}})

@ARTICLE(PRGCOR_2.ABS,
      AUTHOR  = {Nishiyama, Takaya and Fukura, Hirofumi and Nakamura, Yatsuka},
      TITLE   = {Logical Correctness of Vector Calculation Programs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/prgcor_2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {375--380},
      NUMBER  = {{\bf 3}})

@ARTICLE(FINTOPO4.ABS,
      AUTHOR  = {Imura, Hiroshi and Tanaka, Masami and Nakamura, Yatsuka},
      TITLE   = {Continuous Mappings between Finite and One-Dimensional Finite Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/fintopo4.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {381--384},
      NUMBER  = {{\bf 3}})

@ARTICLE(NAGATA_2.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {The {N}agata-{S}mirnov Theorem. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/nagata_2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {385--389},
      NUMBER  = {{\bf 3}})

@ARTICLE(TOPALG_3.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Isomorphism of Fundamental Groups},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/topalg_3.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {391--396},
      NUMBER  = {{\bf 3}})

@ARTICLE(VFUNCT_2.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Algebra of Complex Vector Valued Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/vfunct_2.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {397--401},
      NUMBER  = {{\bf 3}})

@ARTICLE(NCFCONT1.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Continuous Functions on Real and Complex Normed Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/ncfcont1.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {403--419},
      NUMBER  = {{\bf 3}})

@ARTICLE(TOPALG_4.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Fundamental Groups of Products of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2004-12/pdf12-3/topalg_4.pdf},
      YEAR    = {2004},
      VOLUME  = 12,
      PAGES   = {421--425},
      NUMBER  = {{\bf 3}})

@ARTICLE(SERIES_2.ABS,
      AUTHOR  = {Liang, Ming and Ding, Yuzhong},
      TITLE   = {Partial Sum of Some Series},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/series_2.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {1--4},
      NUMBER  = {{\bf 1}})

@ARTICLE(SUBSTUT1.ABS,
      AUTHOR  = {Braselmann, Patrick and Koepke, Peter},
      TITLE   = {Substitution in First-Order Formulas: Elementary Properties},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/substut1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {5--15},
      NUMBER  = {{\bf 1}})

@ARTICLE(SUBLEMMA.ABS,
      AUTHOR  = {Braselmann, Patrick and Koepke, Peter},
      TITLE   = {Coincidence Lemma and Substitution Lemma},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/sublemma.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {17--26},
      NUMBER  = {{\bf 1}})

@ARTICLE(SUBSTUT2.ABS,
      AUTHOR  = {Braselmann, Patrick and Koepke, Peter},
      TITLE   = {Substitution in First-Order Formulas.  {P}art {II}.
                 {T}he Construction of First-Order Formulas},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/substut2.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {27--32},
      NUMBER  = {{\bf 1}})

@ARTICLE(CALCUL_1.ABS,
      AUTHOR  = {Braselmann, Patrick and Koepke, Peter},
      TITLE   = {A Sequent Calculus for First-Order Logic},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/calcul_1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {33--39},
      NUMBER  = {{\bf 1}})

@ARTICLE(CALCUL_2.ABS,
      AUTHOR  = {Braselmann, Patrick and Koepke, Peter},
      TITLE   = {Consequences of the Sequent Calculus},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/calcul_2.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {41--44},
      NUMBER  = {{\bf 1}})

@ARTICLE(HENMODEL.ABS,
      AUTHOR  = {Braselmann, Patrick and Koepke, Peter},
      TITLE   = {Equivalences of Inconsistency and {H}enkin Models},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/henmodel.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {45--48},
      NUMBER  = {{\bf 1}})

@ARTICLE(GOEDELCP.ABS,
      AUTHOR  = {Braselmann, Patrick and Koepke, Peter},
      TITLE   = {G{\"o}del's Completeness Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/goedelcp.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {49--53},
      NUMBER  = {{\bf 1}})

@ARTICLE(BVFUNC26.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {Propositional Calculus for Boolean Valued Functions.
                {P}art {VIII}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/bvfunc26.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {55--58},
      NUMBER  = {{\bf 1}})

@ARTICLE(HOLDER_1.ABS,
      AUTHOR  = {Suzuki, Yasumasa},
      TITLE   = {H\"older's Inequality and {M}inkowski's Inequality},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/holder_1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {59--61},
      NUMBER  = {{\bf 1}})

@ARTICLE(LP_SPACE.ABS,
      AUTHOR  = {Suzuki, Yasumasa},
      TITLE   = {The {B}anach Space $l^p$},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/lp_space.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {63--66},
      NUMBER  = {{\bf 1}})

@ARTICLE(MESFUNC3.ABS,
      AUTHOR  = {Shidama, Yasunari and Endou, Noboru},
      TITLE   = {Lebesgue Integral of Simple Valued Function},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/mesfunc3.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {67--71},
      NUMBER  = {{\bf 1}})

@ARTICLE(SIN_COS6.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {Inverse Trigonometric Functions Arcsin and Arccos},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/sin_cos6.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {73--79},
      NUMBER  = {{\bf 1}})

@ARTICLE(JORDAN21.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On Some Points of a Simple Closed Curve},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/jordan21.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {81--87},
      NUMBER  = {{\bf 1}})

@ARTICLE(JORDAN22.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Grabowski, Adam},
      TITLE   = {On Some Points of a Simple Closed Curve. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/jordan22.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {89--91},
      NUMBER  = {{\bf 1}})

@ARTICLE(NCFCONT2.ABS,
      AUTHOR  = {Endou, Noboru},
      TITLE   = {Uniform Continuity of Functions on Normed Complex Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/ncfcont2.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {93--98},
      NUMBER  = {{\bf 1}})

@ARTICLE(RLTOPSP1.ABS,
      AUTHOR  = {Byli\'nski, Czes{\l}aw},
      TITLE   = {Introduction to Real Linear Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/rltopsp1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {99--107},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPREALA.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {Some Properties of Rectangles on the Plane},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/topreala.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {109--115},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPREALB.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {Some Properties of Circles on the Plane},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/toprealb.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {117--124},
      NUMBER  = {{\bf 1}})

@ARTICLE(PENCIL_3.ABS,
      AUTHOR  = {Naumowicz, Adam},
      TITLE   = {On the Characterization of Collineations of the
                {S}egre Product of Strongly Connected Partial Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/pencil_3.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {125--131},
      NUMBER  = {{\bf 1}})

@ARTICLE(PENCIL_4.ABS,
      AUTHOR  = {Naumowicz, Adam},
      TITLE   = {Spaces of Pencils, {G}rassmann Spaces, and Generalized {V}eronese Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/pencil_4.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {133--138},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPGEN_1.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {On the Boundary and Derivative of a Set},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/topgen_1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {139--146},
      NUMBER  = {{\bf 1}})

@ARTICLE(GROEB_3.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {Construction of {G}r\"obner Bases:
                Avoiding S-Polynomials -- {B}uchberger's First Criterium},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/groeb_3.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {147--156},
      NUMBER  = {{\bf 1}})

@ARTICLE(MATRIX_5.ABS,
      AUTHOR  = {Chang, Wenpai and Yamazaki, Hiroshi and Nakamura, Yatsuka},
      TITLE   = {A Theory of Matrices of Complex Elements},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/matrix_5.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {157--162},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPGEN_2.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {On the Characteristic and Weight of a Topological Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/topgen_2.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {163--169},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPGEN_3.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {On Constructing Topological Spaces and {S}orgenfrey Line},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/topgen_3.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {171--179},
      NUMBER  = {{\bf 1}})

@ARTICLE(PARTFUN3.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {On the Real Valued Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/partfun3.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {181--187},
      NUMBER  = {{\bf 1}})

@ARTICLE(ROBBINS3.ABS,
      AUTHOR  = {Grabowski, Adam and Moschner, Markus},
      TITLE   = {Formalization of Ortholattices via~Orthoposets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-1/robbins3.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {189--197},
      NUMBER  = {{\bf 1}})

@ARTICLE(JGRAPH_7.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej},
      TITLE   = {The {F}ashoda Meet Theorem for Rectangles},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/jgraph_7.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {199--219},
      NUMBER  = {{\bf 2}})

@ARTICLE(MATHMORP.ABS,
      AUTHOR  = {Ding, Yuzhong and Liang, Xiquan},
      TITLE   = {Preliminaries to Mathematical Morphology and Its Properties},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/mathmorp.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {221--225},
      NUMBER  = {{\bf 2}})

@ARTICLE(JORDAN23.ABS,
      AUTHOR  = {Milewski, Robert},
      TITLE   = {Subsequences of Almost, Weakly and Poorly One-to-one Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/jordan23.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {227--233},
      NUMBER  = {{\bf 2}})

@ARTICLE(GLIB_000.ABS,
      AUTHOR  = {Lee, Gilbert and Rudnicki, Piotr},
      TITLE   = {Alternative Graph Structures},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/glib_000.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {235--252},
      NUMBER  = {{\bf 2}})

@ARTICLE(GLIB_001.ABS,
      AUTHOR  = {Lee, Gilbert},
      TITLE   = {Walks in Graphs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/glib_001.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {253--269},
      NUMBER  = {{\bf 2}})

@ARTICLE(GLIB_002.ABS,
      AUTHOR  = {Lee, Gilbert},
      TITLE   = {Trees and Graph Components},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/glib_002.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {271--277},
      NUMBER  = {{\bf 2}})

@ARTICLE(GLIB_003.ABS,
      AUTHOR  = {Lee, Gilbert},
      TITLE   = {Weighted and Labeled Graphs},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/glib_003.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {279--293},
      NUMBER  = {{\bf 2}})

@ARTICLE(GLIB_004.ABS,
      AUTHOR  = {Lee, Gilbert and Rudnicki, Piotr},
      TITLE   = {Correctness of {D}ijkstra's Shortest Path and {P}rim's Minimum Spanning Tree Algorithms},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/glib_004.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {295--304},
      NUMBER  = {{\bf 2}})

@ARTICLE(GLIB_005.ABS,
      AUTHOR  = {Lee, Gilbert},
      TITLE   = {Correctnesss of {F}ord-{F}ulkerson's Maximum Flow Algorithm},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/glib_005.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {305--314},
      NUMBER  = {{\bf 2}})

@ARTICLE(RCOMP_3.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Properties of Connected Subsets  of the Real Line},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/rcomp_3.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {315--323},
      NUMBER  = {{\bf 2}})

@ARTICLE(TOPALG_5.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {The Fundamental Group of the Circle},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/topalg_5.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {325--331},
      NUMBER  = {{\bf 2}})

@ARTICLE(BROUWER.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {Brouwer Fixed Point Theorem for Disks on the Plane},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/brouwer.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {333--336},
      NUMBER  = {{\bf 2}})

@ARTICLE(STIRL2_1.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Stirling Numbers of the Second Kind},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/stirl2_1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {337--345},
      NUMBER  = {{\bf 2}})

@ARTICLE(SETLIM_1.ABS,
      AUTHOR  = {Zhang, Bo and Yamazaki, Hiroshi and Nakamura, Yatsuka},
      TITLE   = {Limit of Sequence of Subsets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/setlim_1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {347--352},
      NUMBER  = {{\bf 2}})

@ARTICLE(ISOMICHI.ABS,
      AUTHOR  = {Jastrz\c{e}bska, Magdalena and Grabowski, Adam},
      TITLE   = {The Properties of Supercondensed Sets, Subcondensed Sets and Condensed Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-2/isomichi.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {353--359},
      NUMBER  = {{\bf 2}})

@ARTICLE(RELSET_2.ABS,
      AUTHOR  = {Retel, Krzysztof},
      TITLE   = {Properties of First and Second Order Cutting of Binary Relations},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/relset_2.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {361--365},
      NUMBER  = {{\bf 3}})

@ARTICLE(COMPLSP2.ABS,
      AUTHOR  = {Chang, Wenpai and Yamazaki, Hiroshi and Nakamura, Yatsuka},
      TITLE   = {The Inner Product and Conjugate of Finite Sequences of Complex Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/complsp2.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {367--373},
      NUMBER  = {{\bf 3}})

@ARTICLE(RINFSUP1.ABS,
      AUTHOR  = {Zhang, Bo and Yamazaki, Hiroshi and Nakamura, Yatsuka},
      TITLE   = {Inferior Limit and Superior Limit of Sequences of Real Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/rinfsup1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {375--381},
      NUMBER  = {{\bf 3}})

@ARTICLE(SIN_COS7.ABS,
      AUTHOR  = {Ge, Fuguo and Liang, Xiquan and Ding, Yuzhong},
      TITLE   = {Formulas and Identities of Inverse Hyperbolic Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/sin_cos7.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {383--387},
      NUMBER  = {{\bf 3}})

@ARTICLE(EUCLIDLP.ABS,
      AUTHOR  = {Kubo, Akihiro},
      TITLE   = {Lines on Planes in $n$-Dimensional {E}uclidean Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/euclidlp.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {389--397},
      NUMBER  = {{\bf 3}})

@ARTICLE(CARD_FIN.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Cardinal Numbers and Finite Sets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/card_fin.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {399--406},
      NUMBER  = {{\bf 3}})

@ARTICLE(SETLIM_2.ABS,
      AUTHOR  = {Zhang, Bo and Yamazaki, Hiroshi and Nakamura, Yatsuka},
      TITLE   = {Some Equations Related to the Limit  of Sequence of Subsets},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/setlim_2.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {407--412},
      NUMBER  = {{\bf 3}})

@ARTICLE(SERIES_3.ABS,
      AUTHOR  = {Ge, Fuguo and Liang, Xiquan},
      TITLE   = {On the Partial Product of Series and Related Basic Inequalities},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/series_3.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {413--416},
      NUMBER  = {{\bf 3}})

@ARTICLE(FINTOPO5.ABS,
      AUTHOR  = {Tanaka, Masami and Imura, Hiroshi and Nakamura, Yatsuka},
      TITLE   = {Homeomorphism between Finite Topological Spaces, Two-Dimensional Lattice Spaces and a Fixed Point Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/fintopo5.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {417--419},
      NUMBER  = {{\bf 3}})

@ARTICLE(TAYLOR_2.ABS,
      AUTHOR  = {Nishino, Akira and Shidama, Yasunari},
      TITLE   = {The {M}aclaurin Expansions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/taylor_2.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {421--425},
      NUMBER  = {{\bf 3}})

@ARTICLE(FDIFF_4.ABS,
      AUTHOR  = {Zhang, Yan and Liang, Xiquan},
      TITLE   = {Several Differentiable Formulas of Special Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-3/fdiff_4.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {427--434},
      NUMBER  = {{\bf 3}})

@ARTICLE(PROB_3.ABS,
      AUTHOR  = {Zhang, Bo and Yamazaki, Hiroshi and Nakamura, Yatsuka},
      TITLE   = {Set Sequences and Monotone Class},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/prob_3.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {435--441},
      NUMBER  = {{\bf 4}})

@ARTICLE(FILEREC1.ABS,
      AUTHOR  = {Fukura, Hirofumi and Nakamura, Yatsuka},
      TITLE   = {A Theory of Sequential Files},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/filerec1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {443--446},
      NUMBER  = {{\bf 4}})

@ARTICLE(CIRCLED1.ABS,
      AUTHOR  = {Zhai, Fahui and Cao, Jianbing and Liang, Xiquan},
      TITLE   = {Circled Sets, Circled Hull, and Circled Family},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/circled1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {447--451},
      NUMBER  = {{\bf 4}})

@ARTICLE(TOPGEN_4.ABS,
      AUTHOR  = {Grabowski, Adam},
      TITLE   = {On the {B}orel Families of Subsets of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/topgen_4.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {453--461},
      NUMBER  = {{\bf 4}})

@ARTICLE(MESFUNC4.ABS,
      AUTHOR  = {Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Linearity of {L}ebesgue Integral of Simple Valued Function},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/mesfunc4.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {463--465},
      NUMBER  = {{\bf 4}})

@ARTICLE(JGRAPH_8.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Trybulec, Andrzej and Korni{\l}owicz, Artur},
      TITLE   = {The {F}ashoda Meet Theorem for Continuous Mappings},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/jgraph_8.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {467--469},
      NUMBER  = {{\bf 4}})

@ARTICLE(TIETZE.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Bancerek, Grzegorz and Naumowicz, Adam},
      TITLE   = {Tietze {E}xtension {T}heorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/tietze.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {471--475},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN24.ABS,
      AUTHOR  = {Naumowicz, Adam and Bancerek, Grzegorz},
      TITLE   = {Homeomorphisms of {J}ordan Curves},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/jordan24.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {477--480},
      NUMBER  = {{\bf 4}})

@ARTICLE(JORDAN.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Jordan Curve Theorem},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/jordan.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {481--491},
      NUMBER  = {{\bf 4}})

@ARTICLE(MATRIXC1.ABS,
      AUTHOR  = {Chang, Wenpai and Yamazaki, Hiroshi and Nakamura, Yatsuka},
      TITLE   = {The Inner Product and Conjugate of Matrix of Complex Numbers},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/matrixc1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {493--499},
      NUMBER  = {{\bf 4}})

@ARTICLE(SERIES_4.ABS,
      AUTHOR  = {Cao, Jianbing and Zhai, Fahui and Liang, Xiquan},
      TITLE   = {Partial Sum and Partial Product of Some Series},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/series_4.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {501--503},
      NUMBER  = {{\bf 4}})

@ARTICLE(FDIFF_5.ABS,
      AUTHOR  = {Cao, Jianbing and Zhai, Fahui and Liang, Xiquan},
      TITLE   = {Some Differentiable Formulas of Special Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/fdiff_5.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {505--509},
      NUMBER  = {{\bf 4}})

@ARTICLE(SIN_COS8.ABS,
      AUTHOR  = {Pacharapokin, Chanapat and Yamazaki, Hiroshi},
      TITLE   = {Formulas and Identities of Hyperbolic Functions},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/sin_cos8.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {511--513},
      NUMBER  = {{\bf 4}})

@ARTICLE(TOPGEN_5.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Niemytzki Plane - an Example of {T}ychonoff Space Which Is Not {${T}_4$}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/topgen_5.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {515--524},
      NUMBER  = {{\bf 4}})

@ARTICLE(SERIES_5.ABS,
      AUTHOR  = {Ge, Fuguo and Liang, Xiquan},
      TITLE   = {On the Partial Product and Partial Sum of Series and Related Basic Inequalities},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/series_5.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {525--528},
      NUMBER  = {{\bf 4}})

@ARTICLE(FDIFF_6.ABS,
      AUTHOR  = {Zhang, Yan and Li, Bo and Liang, Xiquan},
      TITLE   = {Several Differentiable Formulas of Special Functions. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/fdiff_6.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {529--535},
      NUMBER  = {{\bf 4}})

@ARTICLE(BINARI_6.ABS,
      AUTHOR  = {Kobayashi, Shunichi},
      TITLE   = {On the Calculus of Binary Arithmetics. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/binari_6.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {537--540},
      NUMBER  = {{\bf 4}})

@ARTICLE(MATRIX_6.ABS,
      AUTHOR  = {Yue, Xiaopeng and Liang, Xiquan and Sun, Zhongpin},
      TITLE   = {Some Properties of Some Special Matrices},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/matrix_6.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {541--547},
      NUMBER  = {{\bf 4}})

@ARTICLE(GFACIRC1.ABS,
      AUTHOR  = {Yamaguchi, Shin'nosuke and Wasaki, Katsumi and Shimoi, Nobuhiro},
      TITLE   = {Generalized Full Adder Circuits ({GFA}s). {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/gfacirc1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {549--571},
      NUMBER  = {{\bf 4}})

@ARTICLE(RING_1.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Quotient Rings},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/ring_1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {573--576},
      NUMBER  = {{\bf 4}})

@ARTICLE(REAL_NS1.ABS,
      AUTHOR  = {Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Completeness of the Real {E}uclidean Space},
      JOURNAL = {Formalized Mathematics},
      URL     = {http://fm.mizar.org/2005-13/pdf13-4/real_ns1.pdf},
      YEAR    = {2005},
      VOLUME  = 13,
      PAGES   = {577--580},
      NUMBER  = {{\bf 4}})

@ARTICLE(MATRIX_7.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {Determinant of Some Matrices  of Field Elements},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0001-4},
      VOLUME  = 14,
      PAGES   = {1--5},
      NUMBER  = {{\bf 1}})

@ARTICLE(MATRIX_8.ABS,
      AUTHOR  = {Yue, Xiaopeng and Hu, Dahai and Liang, Xiquan},
      TITLE   = {Some Properties of Some Special Matrices.  {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0002-3},
      VOLUME  = 14,
      PAGES   = {7--12},
      NUMBER  = {{\bf 1}})

@ARTICLE(MATRIX_9.ABS,
      AUTHOR  = {Romanowicz, Ewa and Grabowski, Adam},
      TITLE   = {On the Permanent of a Matrix},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0003-2},
      VOLUME  = 14,
      PAGES   = {13--20},
      NUMBER  = {{\bf 1}})

@ARTICLE(MATRIXR1.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Tamura, Nobuyuki and Chang, Wenpai},
      TITLE   = {A Theory of Matrices of Real Elements},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0004-1},
      VOLUME  = 14,
      PAGES   = {21--28},
      NUMBER  = {{\bf 1}})

@ARTICLE(MOEBIUS1.ABS,
      AUTHOR  = {Jastrz\c{e}bska, Magdalena and Grabowski, Adam},
      TITLE   = {On the Properties of the {M}\"obius Function},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0005-0},
      VOLUME  = 14,
      PAGES   = {29--36},
      NUMBER  = {{\bf 1}})

@ARTICLE(FDIFF_7.ABS,
      AUTHOR  = {Li, Bo and Zhang, Yan and Liang, Xiquan},
      TITLE   = {Several Differentiation Formulas  of Special Functions. {P}art {III}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0006-z},
      VOLUME  = 14,
      PAGES   = {37--45},
      NUMBER  = {{\bf 1}})

@ARTICLE(NAT_4.ABS,
      AUTHOR  = {Riccardi, Marco},
      TITLE   = {Pocklington's Theorem and {B}ertrand's Postulate},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0007-y},
      VOLUME  = 14,
      PAGES   = {47--52},
      NUMBER  = {{\bf 2}})

@ARTICLE(MESFUNC5.ABS,
      AUTHOR  = {Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Integral of Measurable Function},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0008-x},
      VOLUME  = 14,
      PAGES   = {53--70},
      NUMBER  = {{\bf 2}})

@ARTICLE(REAL_3.ABS,
      AUTHOR  = {Li, Bo and Zhang, Yan and Korni{\l}owicz, Artur},
      TITLE   = {Simple Continued Fractions  and Their Convergents},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0009-9},
      VOLUME  = 14,
      PAGES   = {71--78},
      NUMBER  = {{\bf 3}})

@ARTICLE(CHORD.ABS,
      AUTHOR  = {Arneson, Broderick and Rudnicki, Piotr},
      TITLE   = {Chordal Graphs},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0010-3},
      VOLUME  = 14,
      PAGES   = {79--92},
      NUMBER  = {{\bf 3}})

@ARTICLE(FINTOPO6.ABS,
      AUTHOR  = {Nakamura, Yatsuka},
      TITLE   = {Connectedness and Continuous Sequences  in Finite Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0011-2},
      VOLUME  = 14,
      PAGES   = {93--100},
      NUMBER  = {{\bf 3}})

@ARTICLE(MATRPROB.ABS,
      AUTHOR  = {Zhang, Bo and Nakamura, Yatsuka},
      TITLE   = {The Definition of Finite Sequences  and Matrices of Probability, and Addition of Matrices of Real Elements},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0012-1},
      VOLUME  = 14,
      PAGES   = {101--108},
      NUMBER  = {{\bf 3}})

@ARTICLE(FDIFF_8.ABS,
      AUTHOR  = {Li, Bo and Wang, Peng},
      TITLE   = {Several Differentiation Formulas  of Special Functions. {P}art {IV}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0013-0},
      VOLUME  = 14,
      PAGES   = {109--114},
      NUMBER  = {{\bf 3}})

@ARTICLE(DIFF_1.ABS,
      AUTHOR  = {Li, Bo and Zhang, Yan and Liang, Xiquan},
      TITLE   = {Difference and Difference Quotient},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0014-z},
      VOLUME  = 14,
      PAGES   = {115--119},
      NUMBER  = {{\bf 3}})

@ARTICLE(POLYNOM8.ABS,
      AUTHOR  = {Treyderowski, Krzysztof and Schwarzweller, Christoph},
      TITLE   = {Multiplication of Polynomials  using {D}iscrete {F}ourier {T}ransformation},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0015-y},
      VOLUME  = 14,
      PAGES   = {121--128},
      NUMBER  = {{\bf 4}})

@ARTICLE(MATRIX10.ABS,
      AUTHOR  = {Liang, Xiquan and Ge, Fuguo and Yue, Xiaopeng},
      TITLE   = {Some Special Matrices of Real Elements  and Their Properties},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0016-x},
      VOLUME  = 14,
      PAGES   = {129--134},
      NUMBER  = {{\bf 4}})

@ARTICLE(HURWITZ.ABS,
      AUTHOR  = {Schwarzweller, Christoph and Rowi\'{n}ska-Schwarzweller, Agnieszka},
      TITLE   = {Schur's Theorem on the Stability of Networks},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0017-9},
      VOLUME  = 14,
      PAGES   = {135--142},
      NUMBER  = {{\bf 4}})

@ARTICLE(MESFUNC6.ABS,
      AUTHOR  = {Shidama, Yasunari and Endou, Noboru},
      TITLE   = {Integral of Real-Valued  Measurable Function},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0018-8},
      VOLUME  = 14,
      PAGES   = {143--152},
      NUMBER  = {{\bf 4}})

@ARTICLE(CATALAN2.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {The {C}atalan Numbers. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0019-7},
      VOLUME  = 14,
      PAGES   = {153--159},
      NUMBER  = {{\bf 4}})

@ARTICLE(QUATERNI.ABS,
      AUTHOR  = {Liang, Xiquan and Ge, Fuguo},
      TITLE   = {The Quaternion Numbers},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0020-1},
      VOLUME  = 14,
      PAGES   = {161--169},
      NUMBER  = {{\bf 4}})

@ARTICLE(MODELC_1.ABS,
      AUTHOR  = {Ishida, Kazuhisa},
      TITLE   = {Model Checking. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0021-0},
      VOLUME  = 14,
      PAGES   = {171--186},
      NUMBER  = {{\bf 4}})

@ARTICLE(LEXBFS.ABS,
      AUTHOR  = {Arneson, Broderick and Rudnicki, Piotr},
      TITLE   = {Recognizing Chordal Graphs: Lex {BFS} and {MCS}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0022-z},
      VOLUME  = 14,
      PAGES   = {187--205},
      NUMBER  = {{\bf 4}})

@ARTICLE(INTEGRA6.ABS,
      AUTHOR  = {Endou, Noboru and Shidama, Yasunari and Yamazaki, Masahiko},
      TITLE   = {Integrability and the Integral  of Partial Functions from {$\mathbb{R}$} into {$\mathbb{R}$}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0023-y},
      VOLUME  = 14,
      PAGES   = {207--212},
      NUMBER  = {{\bf 4}})

@ARTICLE(NORMSP_2.ABS,
      AUTHOR  = {Endou, Noboru and Shidama, Yasunari and Okamura, Katsumasa},
      TITLE   = {Baire's Category Theorem and Some Spaces  Generated from Real Normed Space},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0024-x},
      VOLUME  = 14,
      PAGES   = {213--219},
      NUMBER  = {{\bf 4}})

@ARTICLE(NUMERAL1.ABS,
      AUTHOR  = {Naumowicz, Adam},
      TITLE   = {On the Representation of Natural Numbers  in Positional Numeral Systems},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0025-9},
      VOLUME  = 14,
      PAGES   = {221--223},
      NUMBER  = {{\bf 4}})

@ARTICLE(PROB_4.ABS,
      AUTHOR  = {Zhang, Bo and Yamazaki, Hiroshi and Nakamura, Yatsuka},
      TITLE   = {The Relevance of Measure and Probability, and Definition of Completeness of Probability},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2006},
      DOI     = {10.2478/v10037-006-0026-8},
      VOLUME  = 14,
      PAGES   = {225--229},
      NUMBER  = {{\bf 4}})

@ARTICLE(BCIALG_1.ABS,
      AUTHOR  = {Ding, Yuzhong},
      TITLE   = {Several Classes of {BCI}-algebras  and their Properties},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0001-z},
      VOLUME  = 15,
      PAGES   = {1--9},
      NUMBER  = {{\bf 1}})

@ARTICLE(FLANG_1.ABS,
      AUTHOR  = {Trybulec, Micha{\l}},
      TITLE   = {Formal Languages -- Concatenation and Closure},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0002-y},
      VOLUME  = 15,
      PAGES   = {11--15},
      NUMBER  = {{\bf 1}})

@ARTICLE(MATRIX11.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Basic Properties of Determinants  of Square Matrices over a Field},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0003-x},
      VOLUME  = 15,
      PAGES   = {17--25},
      NUMBER  = {{\bf 1}})

@ARTICLE(COMBGRAS.ABS,
      AUTHOR  = {Owsiejczuk, Andrzej},
      TITLE   = {Combinatorial {G}rassmannians},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0004-9},
      VOLUME  = 15,
      PAGES   = {27--33},
      NUMBER  = {{\bf 2}})

@ARTICLE(GROUP_9.ABS,
      AUTHOR  = {Riccardi, Marco},
      TITLE   = {The {J}ordan-{H}\"older Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0005-8},
      VOLUME  = 15,
      PAGES   = {35--51},
      NUMBER  = {{\bf 2}})

@ARTICLE(FLANG_2.ABS,
      AUTHOR  = {Trybulec, Micha{\l}},
      TITLE   = {Regular Expression Quantifiers -- $m$ to $n$ Occurrences},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0006-7},
      VOLUME  = 15,
      PAGES   = {53--58},
      NUMBER  = {{\bf 2}})

@ARTICLE(INTEGRA7.ABS,
      AUTHOR  = {Shidama, Yasunari and Endou, Noboru and Wasaki, Katsumi},
      TITLE   = {Riemann Indefinite Integral of Functions  of Real Variable},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0007-6},
      VOLUME  = 15,
      PAGES   = {59--63},
      NUMBER  = {{\bf 2}})

@ARTICLE(PDIFF_1.ABS,
      AUTHOR  = {Endou, Noboru and Shidama, Yasunari and Miyajima, Keiichi},
      TITLE   = {Partial Differentiation on Normed Linear Spaces ${\calR}^n$},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0008-5},
      VOLUME  = 15,
      PAGES   = {65--72},
      NUMBER  = {{\bf 2}})

@ARTICLE(FDIFF_9.ABS,
      AUTHOR  = {Wang, Peng and Li, Bo},
      TITLE   = {Several Differentiation Formulas of Special Functions. {P}art {V}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0009-4},
      VOLUME  = 15,
      PAGES   = {73--79},
      NUMBER  = {{\bf 3}})

@ARTICLE(PRVECT_2.ABS,
      AUTHOR  = {Endou, Noboru and Shidama, Yasunari and Miyajima, Keiichi},
      TITLE   = {The Product Space of Real Normed Spaces and its Properties},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0010-y},
      VOLUME  = 15,
      PAGES   = {81--85},
      NUMBER  = {{\bf 3}})

@ARTICLE(AOFA_000.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Mizar Analysis of Algorithms: {P}reliminaries},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0011-x},
      VOLUME  = 15,
      PAGES   = {87--110},
      NUMBER  = {{\bf 3}})

@ARTICLE(ENTROPY1.ABS,
      AUTHOR  = {Zhang, Bo and Nakamura, Yatsuka},
      TITLE   = {Definition and some Properties of Information Entropy},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0012-9},
      VOLUME  = 15,
      PAGES   = {111--119},
      NUMBER  = {{\bf 3}})

@ARTICLE(REWRITE2.ABS,
      AUTHOR  = {Trybulec, Micha{\l}},
      TITLE   = {String Rewriting Systems},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0013-8},
      VOLUME  = 15,
      PAGES   = {121--126},
      NUMBER  = {{\bf 3}})

@ARTICLE(MATRIXR2.ABS,
      AUTHOR  = {Tamura, Nobuyuki and Nakamura, Yatsuka},
      TITLE   = {Determinant and Inverse of Matrices  of Real Elements},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0014-7},
      VOLUME  = 15,
      PAGES   = {127--136},
      NUMBER  = {{\bf 3}})

@ARTICLE(RANKNULL.ABS,
      AUTHOR  = {Alama, Jesse},
      TITLE   = {The Rank+Nullity Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0015-6},
      VOLUME  = 15,
      PAGES   = {137--142},
      NUMBER  = {{\bf 3}})

@ARTICLE(LAPLACE.ABS,
      AUTHOR  = {P\k{a}k, Karol and Trybulec, Andrzej},
      TITLE   = {Laplace Expansion},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0016-5},
      VOLUME  = 15,
      PAGES   = {143--150},
      NUMBER  = {{\bf 3}})

@ARTICLE(MATRIX12.ABS,
      AUTHOR  = {Liang, Xiquan and Sun, Tao and Hu, Dahai},
      TITLE   = {Some Properties of Line and Column Operations on Matrices},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0017-4},
      VOLUME  = 15,
      PAGES   = {151--157},
      NUMBER  = {{\bf 3}})

@ARTICLE(GROUP_10.ABS,
      AUTHOR  = {Riccardi, Marco},
      TITLE   = {The {S}ylow Theorems},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0018-3},
      VOLUME  = 15,
      PAGES   = {159--165},
      NUMBER  = {{\bf 3}})

@ARTICLE(COMPACT1.ABS,
      AUTHOR  = {Byli{\'n}ski, Czes{\l}aw},
      TITLE   = {Alexandroff One Point Compactification},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0019-2},
      VOLUME  = 15,
      PAGES   = {167--170},
      NUMBER  = {{\bf 4}})

@ARTICLE(ARROW.ABS,
      AUTHOR  = {Wiedijk, Freek},
      TITLE   = {Arrow's Impossibility Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0020-9},
      VOLUME  = 15,
      PAGES   = {171--174},
      NUMBER  = {{\bf 4}})

@ARTICLE(BCIALG_2.ABS,
      AUTHOR  = {Ding, Yuzhong and Pang, Zhiyong},
      TITLE   = {Congruences and Quotient Algebras  of {BCI}-algebras},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0021-8},
      VOLUME  = 15,
      PAGES   = {175--180},
      NUMBER  = {{\bf 4}})

@ARTICLE(INT_4.ABS,
      AUTHOR  = {Liang, Xiquan and Yan, Li and Zhao, Junjie},
      TITLE   = {Linear Congruence Relation  and Complete Residue Systems},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0022-7},
      VOLUME  = 15,
      PAGES   = {181--187},
      NUMBER  = {{\bf 4}})

@ARTICLE(INTEGRA8.ABS,
      AUTHOR  = {Peng, Cuiying and Ge, Fuguo and Liang, Xiquan},
      TITLE   = {Several Integrability Formulas  of Special Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0023-6},
      VOLUME  = 15,
      PAGES   = {189--198},
      NUMBER  = {{\bf 4}})

@ARTICLE(MATRIX13.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Basic Properties of the Rank  of Matrices over a Field},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0024-5},
      VOLUME  = 15,
      PAGES   = {199--211},
      NUMBER  = {{\bf 4}})

@ARTICLE(PCS_0.ABS,
      AUTHOR  = {Grue, Klaus E. and Korni{\l}owicz, Artur},
      TITLE   = {Basic Operations on Preordered  Coherent Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0025-4},
      VOLUME  = 15,
      PAGES   = {213--230},
      NUMBER  = {{\bf 4}})

@ARTICLE(RINFSUP2.ABS,
      AUTHOR  = {Yamazaki, Hiroshi and Endou, Noboru and Shidama, Yasunari and Okazaki, Hiroyuki},
      TITLE   = {Inferior Limit, Superior Limit and Convergence of Sequences of Extended Real Numbers},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0026-3},
      VOLUME  = 15,
      PAGES   = {231--236},
      NUMBER  = {{\bf 4}})

@ARTICLE(BCIALG_3.ABS,
      AUTHOR  = {Sun, Tao and Hu, Dahai and Liang, Xiquan},
      TITLE   = {Several Classes of {BCK}-algebras  and their Properties},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0027-2},
      VOLUME  = 15,
      PAGES   = {237--242},
      NUMBER  = {{\bf 4}})

@ARTICLE(FDIFF_10.ABS,
      AUTHOR  = {Li, Bo and Wang, Pan},
      TITLE   = {Several Differentiation Formulas  of Special Functions. {P}art {VI}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2007},
      DOI     = {10.2478/v10037-007-0028-1},
      VOLUME  = 15,
      PAGES   = {243--250},
      NUMBER  = {{\bf 4}})

@ARTICLE(BSPACE.ABS,
      AUTHOR  = {Alama, Jesse},
      TITLE   = {The Vector Space of Subsets of a Set Based on Symmetric Difference},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0001-7},
      VOLUME  = 16,
      PAGES   = {1--5},
      NUMBER  = {{\bf 1}})

@ARTICLE(POLYFORM.ABS,
      AUTHOR  = {Alama, Jesse},
      TITLE   = {Euler's Polyhedron Formula},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0002-6},
      VOLUME  = 16,
      PAGES   = {7--17},
      NUMBER  = {{\bf 1}})

@ARTICLE(LOPBAN_5.ABS,
      AUTHOR  = {Sakurai, Hideki and Kunimune, Hisayoshi and Shidama, Yasunari},
      TITLE   = {Uniform Boundedness Principle},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0003-5},
      VOLUME  = 16,
      PAGES   = {19--21},
      NUMBER  = {{\bf 1}})

@ARTICLE(INT_5.ABS,
      AUTHOR  = {Yan, Li and Liang, Xiquan and Zhao, Junjie},
      TITLE   = {Gauss Lemma and Law of Quadratic Reciprocity},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0004-4},
      VOLUME  = 16,
      PAGES   = {23--28},
      NUMBER  = {{\bf 1}})

@ARTICLE(FLANG_3.ABS,
      AUTHOR  = {Trybulec, Micha{\l}},
      TITLE   = {Regular Expression Quantifiers -- at least $m$ Occurrences},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0005-3},
      VOLUME  = 16,
      PAGES   = {29--33},
      NUMBER  = {{\bf 1}})

@ARTICLE(COMPL_SP.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Complete Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0006-2},
      VOLUME  = 16,
      PAGES   = {35--43},
      NUMBER  = {{\bf 1}})

@ARTICLE(DIFF_2.ABS,
      AUTHOR  = {Li, Bo and Zhuang, Yanping and Liang, Xiquan},
      TITLE   = {Difference and Difference Quotient. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0007-1},
      VOLUME  = 16,
      PAGES   = {45--49},
      NUMBER  = {{\bf 1}})

@ARTICLE(MESFUNC7.ABS,
      AUTHOR  = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {The First Mean Value Theorem for Integrals},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0008-0},
      VOLUME  = 16,
      PAGES   = {51--55},
      NUMBER  = {{\bf 1}})

@ARTICLE(MESFUNC8.ABS,
      AUTHOR  = {Endou, Noboru and Shidama, Yasunari and Narita, Keiko},
      TITLE   = {Egoroff's Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0009-z},
      VOLUME  = 16,
      PAGES   = {57--63},
      NUMBER  = {{\bf 1}})

@ARTICLE(BCIALG_4.ABS,
      AUTHOR  = {Sun, Tao and Zhao, Junjie and Liang, Xiquan},
      TITLE   = {BCI-algebras with Condition ({S})  and their Properties},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0010-6},
      VOLUME  = 16,
      PAGES   = {65--71},
      NUMBER  = {{\bf 1}})

@ARTICLE(GFACIRC2.ABS,
      AUTHOR  = {Wasaki, Katsumi},
      TITLE   = {Stability of $n$-Bit Generalized Full Adder Circuits ({GFA}s). {P}art~{II}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0011-5},
      VOLUME  = 16,
      PAGES   = {73--80},
      NUMBER  = {{\bf 1}})

@ARTICLE(MATRIX15.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Solutions of Linear Equations},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0012-4},
      VOLUME  = 16,
      PAGES   = {81--90},
      NUMBER  = {{\bf 1}})

@ARTICLE(HELLY.ABS,
      AUTHOR  = {Enright, Jessica and Rudnicki, Piotr},
      TITLE   = {Helly Property for Subtrees},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0013-3},
      VOLUME  = 16,
      PAGES   = {91--96},
      NUMBER  = {{\bf 2}})

@ARTICLE(EUCLID_6.ABS,
      AUTHOR  = {Riccardi, Marco},
      TITLE   = {Heron's Formula and {P}tolemy's Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0014-2},
      VOLUME  = 16,
      PAGES   = {97--101},
      NUMBER  = {{\bf 2}})

@ARTICLE(INT_7.ABS,
      AUTHOR  = {Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {Uniqueness of Factoring an Integer and Multiplicative Group ${{\mathbb Z}}/p{{\mathbb Z}^\ast}$},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0015-1},
      VOLUME  = 16,
      PAGES   = {103--107},
      NUMBER  = {{\bf 2}})

@ARTICLE(BCIIDEAL.ABS,
      AUTHOR  = {Wu, Chenglong and Ding, Yuzhong},
      TITLE   = {Ideals of {BCI}-algebras and their Properties},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0016-0},
      VOLUME  = 16,
      PAGES   = {109--114},
      NUMBER  = {{\bf 2}})

@ARTICLE(C0SP1.ABS,
      AUTHOR  = {Shidama, Yasunari and Suzuki, Hikofumi and Endou, Noboru},
      TITLE   = {Banach Algebra of Bounded Functionals},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0017-z},
      VOLUME  = 16,
      PAGES   = {115--122},
      NUMBER  = {{\bf 2}})

@ARTICLE(CONVEX4.ABS,
      AUTHOR  = {Matsuzaki, Hidenori and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Convex Sets and Convex Combinations  on Complex Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0018-y},
      VOLUME  = 16,
      PAGES   = {123--133},
      NUMBER  = {{\bf 2}})

@ARTICLE(QUATERN2.ABS,
      AUTHOR  = {Ge, Fuguo},
      TITLE   = {Inner Products, Group, Ring  of Quaternion Numbers},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0019-x},
      VOLUME  = 16,
      PAGES   = {135--139},
      NUMBER  = {{\bf 2}})

@ARTICLE(HFDIFF_1.ABS,
      AUTHOR  = {Zhao, Junjie and Liang, Xiquan and Yan, Li},
      TITLE   = {Several Higher Differentiation Formulas  of Special Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0020-4},
      VOLUME  = 16,
      PAGES   = {141--145},
      NUMBER  = {{\bf 2}})

@ARTICLE(SIN_COS9.ABS,
      AUTHOR  = {Liang, Xiquan and Xie, Bing},
      TITLE   = {Inverse Trigonometric Functions  Arctan and Arccot},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0021-3},
      VOLUME  = 16,
      PAGES   = {147--158},
      NUMBER  = {{\bf 2}})

@ARTICLE(SINCOS10.ABS,
      AUTHOR  = {Xie, Bing and Liang, Xiquan and Ge, Fuguo},
      TITLE   = {Inverse Trigonometric Functions  Arcsec and Arccosec},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0022-2},
      VOLUME  = 16,
      PAGES   = {159--165},
      NUMBER  = {{\bf 2}})

@ARTICLE(MESFUNC9.ABS,
      AUTHOR  = {Endou, Noboru and Narita, Keiko and Shidama, Yasunari},
      TITLE   = {The {L}ebesgue Monotone Convergence Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0023-1},
      VOLUME  = 16,
      PAGES   = {167--175},
      NUMBER  = {{\bf 2}})

@ARTICLE(AOFA_I00.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Mizar Analysis of Algorithms: Algorithms over Integers},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0024-0},
      VOLUME  = 16,
      PAGES   = {177--194},
      NUMBER  = {{\bf 2}})

@ARTICLE(MATRIX14.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Oniumi, Kunio and Chang, Wenpai},
      TITLE   = {Invertibility of Matrices of Field Elements},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0025-z},
      VOLUME  = 16,
      PAGES   = {195--202},
      NUMBER  = {{\bf 2}})

@ARTICLE(RAMSEY_1.ABS,
      AUTHOR  = {Riccardi, Marco},
      TITLE   = {Ramsey's Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0026-y},
      VOLUME  = 16,
      PAGES   = {203--205},
      NUMBER  = {{\bf 2}})

@ARTICLE(ABCMIZ_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Towards the Construction of a Model of {M}izar Concepts},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0027-x},
      VOLUME  = 16,
      PAGES   = {207--230},
      NUMBER  = {{\bf 2}})

@ARTICLE(MODELC_2.ABS,
      AUTHOR  = {Ishida, Kazuhisa},
      TITLE   = {Model Checking. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0028-9},
      VOLUME  = 16,
      PAGES   = {231--245},
      NUMBER  = {{\bf 3}})

@ARTICLE(INT_6.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {Modular Integer Arithmetic},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0029-8},
      VOLUME  = 16,
      PAGES   = {247--252},
      NUMBER  = {{\bf 3}})

@ARTICLE(BCIALG_5.ABS,
      AUTHOR  = {Sun, Tao and Pan, Weibo and Wu, Chenglong and Liang, Xiquan},
      TITLE   = {General Theory of Quasi-Commutative {BCI}-algebras},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0030-2},
      VOLUME  = 16,
      PAGES   = {253--258},
      NUMBER  = {{\bf 3}})

@ARTICLE(MATRIXJ1.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Block Diagonal Matrices},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0031-1},
      VOLUME  = 16,
      PAGES   = {259--267},
      NUMBER  = {{\bf 3}})

@ARTICLE(MATRLIN2.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Linear Map of Matrices},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0032-0},
      VOLUME  = 16,
      PAGES   = {269--275},
      NUMBER  = {{\bf 3}})

@ARTICLE(ROBBINS4.ABS,
      AUTHOR  = {M\k{a}dra, El\.zbieta and Grabowski, Adam},
      TITLE   = {Orthomodular Lattices},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0033-z},
      VOLUME  = 16,
      PAGES   = {277--282},
      NUMBER  = {{\bf 3}})

@ARTICLE(AFINSQ_2.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Ito, Hisashi},
      TITLE   = {Basic Properties and Concept  of Selected Subsequence  of Zero Based Finite Sequences},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0034-y},
      VOLUME  = 16,
      PAGES   = {283--288},
      NUMBER  = {{\bf 3}})

@ARTICLE(VECTSP11.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Eigenvalues of a Linear Transformation},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0035-x},
      VOLUME  = 16,
      PAGES   = {289--295},
      NUMBER  = {{\bf 4}})

@ARTICLE(MATRIXJ2.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Jordan Matrix Decomposition},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0036-9},
      VOLUME  = 16,
      PAGES   = {297--303},
      NUMBER  = {{\bf 4}})

@ARTICLE(MESFUN10.ABS,
      AUTHOR  = {Endou, Noboru and Narita, Keiko and Shidama, Yasunari},
      TITLE   = {Fatou's Lemma and the {L}ebesgue's Convergence Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0037-8},
      VOLUME  = 16,
      PAGES   = {305--309},
      NUMBER  = {{\bf 4}})

@ARTICLE(INTEGR10.ABS,
      AUTHOR  = {Yamazaki, Masahiko and Yamazaki, Hiroshi and Shidama, Yasunari},
      TITLE   = {Extended {R}iemann Integral of Functions  of Real Variable  and One-sided {L}aplace Transform},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0038-7},
      VOLUME  = 16,
      PAGES   = {311--317},
      NUMBER  = {{\bf 4}})

@ARTICLE(MESFUN6C.ABS,
      AUTHOR  = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Integral of Complex-Valued Measurable Function},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0039-6},
      VOLUME  = 16,
      PAGES   = {319--324},
      NUMBER  = {{\bf 4}})

@ARTICLE(MATROID0.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Shidama, Yasunari},
      TITLE   = {Introduction to Matroids},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0040-0},
      VOLUME  = 16,
      PAGES   = {325--332},
      NUMBER  = {{\bf 4}})

@ARTICLE(PDIFF_2.ABS,
      AUTHOR  = {Xie, Bing and Liang, Xiquan and Li, Hongwei},
      TITLE   = {Partial Differentiation of Real Binary Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0041-z},
      VOLUME  = 16,
      PAGES   = {333--338},
      NUMBER  = {{\bf 4}})

@ARTICLE(MODELC_3.ABS,
      AUTHOR  = {Ishida, Kazuhisa and Shidama, Yasunari},
      TITLE   = {Model Checking. {P}art {III}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0042-y},
      VOLUME  = 16,
      PAGES   = {339--353},
      NUMBER  = {{\bf 4}})

@ARTICLE(MATRIX16.ABS,
      AUTHOR  = {Yue, Xiaopeng and Liang, Xiquan},
      TITLE   = {Basic Properties of Circulant Matrices  and Anti-Circular Matrices},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0043-x},
      VOLUME  = 16,
      PAGES   = {355--360},
      NUMBER  = {{\bf 4}})

@ARTICLE(LPSPACE1.ABS,
      AUTHOR  = {Watase, Yasushige and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {On ${L}^1$ Space Formed by Real-Valued Partial Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0044-9},
      VOLUME  = 16,
      PAGES   = {361--369},
      NUMBER  = {{\bf 4}})

@ARTICLE(BCIALG_6.ABS,
      AUTHOR  = {Ding, Yuzhong and Ge, Fuguo and Wu, Chenglong},
      TITLE   = {{BCI}-homomorphisms},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0045-8},
      VOLUME  = 16,
      PAGES   = {371--376},
      NUMBER  = {{\bf 4}})

@ARTICLE(FTACELL1.ABS,
      AUTHOR  = {Wasaki, Katsumi},
      TITLE   = {Stability of the 4-2 Binary Addition Circuit Cells. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0046-7},
      VOLUME  = 16,
      PAGES   = {377--387},
      NUMBER  = {{\bf 4}})

@ARTICLE(FDIFF_11.ABS,
      AUTHOR  = {Ge, Fuguo and Xie, Bing},
      TITLE   = {Several Differentiation Formulas of Special Functions. {P}art {VII}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0047-6},
      VOLUME  = 16,
      PAGES   = {389--399},
      NUMBER  = {{\bf 4}})

@ARTICLE(LOPBAN_6.ABS,
      AUTHOR  = {Sakurai, Hideki and Kunimune, Hisayoshi and Shidama, Yasunari},
      TITLE   = {Open Mapping Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2008},
      DOI     = {10.2478/v10037-008-0048-5},
      VOLUME  = 16,
      PAGES   = {401--403},
      NUMBER  = {{\bf 4}})

@ARTICLE(EUCLID_7.ABS,
      AUTHOR  = {Nakamura, Yatsuka and Korni{\l}owicz, Artur and Oya, Nagato and Shidama, Yasunari},
      TITLE   = {The Real Vector Spaces of Finite Sequences are Finite Dimensional},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0001-2},
      VOLUME  = 17,
      PAGES   = {1--9},
      NUMBER  = {{\bf 1}})

@ARTICLE(INTEGRA9.ABS,
      AUTHOR  = {Li, Bo and Zhuang, Yanping and Xie, Bing and Wang, Pan},
      TITLE   = {Several Integrability Formulas  of Some Functions,  Orthogonal Polynomials  and Norm Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0002-1},
      VOLUME  = 17,
      PAGES   = {11--21},
      NUMBER  = {{\bf 1}})

@ARTICLE(INTEGR11.ABS,
      AUTHOR  = {Li, Bo and Zhuang, Yanping and Men, Yanhong and Liang, Xiquan},
      TITLE   = {Several Integrability Formulas of Special Functions. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0003-0},
      VOLUME  = 17,
      PAGES   = {23--35},
      NUMBER  = {{\bf 1}})

@ARTICLE(PETRI_2.ABS,
      AUTHOR  = {Jitsukawa, Mitsuru and Kawamoto, Pauline N. and Shidama, Yasunari and Nakamura, Yatsuka},
      TITLE   = {Cell {P}etri Net Concepts},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0004-z},
      VOLUME  = 17,
      PAGES   = {37--42},
      NUMBER  = {{\bf 1}})

@ARTICLE(VALUED_2.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Arithmetic Operations on Functions from Sets into Functional Sets},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0005-y},
      VOLUME  = 17,
      PAGES   = {43--60},
      NUMBER  = {{\bf 1}})

@ARTICLE(QUATERN3.ABS,
      AUTHOR  = {Li, Bo and Wang, Pan and Liang, Xiquan and Zhuang, Yanping},
      TITLE   = {Some Operations on Quaternion Numbers},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0006-x},
      VOLUME  = 17,
      PAGES   = {61--65},
      NUMBER  = {{\bf 2}})

@ARTICLE(CFDIFF_1.ABS,
      AUTHOR  = {Pacharapokin, Chanapat and Yamazaki, Hiroshi and Shidama, Yasunari and Nakamura, Yatsuka},
      TITLE   = {Complex Function Differentiability},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0007-9},
      VOLUME  = 17,
      PAGES   = {67--72},
      NUMBER  = {{\bf 2}})

@ARTICLE(KOLMOG01.ABS,
      AUTHOR  = {Doll, Agnes},
      TITLE   = {Kolmogorov's Zero-One Law},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0008-8},
      VOLUME  = 17,
      PAGES   = {73--77},
      NUMBER  = {{\bf 2}})

@ARTICLE(PDIFF_3.ABS,
      AUTHOR  = {Xie, Bing and Liang, Xiquan and Shen, Xiuzhuan},
      TITLE   = {Second-Order Partial Differentiation  of Real Binary Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0009-7},
      VOLUME  = 17,
      PAGES   = {79--87},
      NUMBER  = {{\bf 2}})

@ARTICLE(MESFUN7C.ABS,
      AUTHOR  = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {The Measurability of Complex-Valued Functional Sequences},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0010-1},
      VOLUME  = 17,
      PAGES   = {89--97},
      NUMBER  = {{\bf 2}})

@ARTICLE(MEMBER_1.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Collective Operations on Number-Membered Sets},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0011-0},
      VOLUME  = 17,
      PAGES   = {99--115},
      NUMBER  = {{\bf 2}})

@ARTICLE(POLYEQ_5.ABS,
      AUTHOR  = {Riccardi, Marco},
      TITLE   = {Solution of Cubic and Quartic Equations},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0012-z},
      VOLUME  = 17,
      PAGES   = {117--122},
      NUMBER  = {{\bf 2}})

@ARTICLE(NAT_5.ABS,
      AUTHOR  = {Riccardi, Marco},
      TITLE   = {The Perfect Number Theorem  and {W}ilson's Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0013-y},
      VOLUME  = 17,
      PAGES   = {123--128},
      NUMBER  = {{\bf 2}})

@ARTICLE(RANDOM_1.ABS,
      AUTHOR  = {Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {Probability on Finite Set and Real-Valued Random Variables},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0014-x},
      VOLUME  = 17,
      PAGES   = {129--136},
      NUMBER  = {{\bf 2}})

@ARTICLE(MESFUN9C.ABS,
      AUTHOR  = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Lebesgue's Convergence Theorem  of Complex-Valued Function},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0015-9},
      VOLUME  = 17,
      PAGES   = {137--145},
      NUMBER  = {{\bf 2}})

@ARTICLE(CFDIFF_2.ABS,
      AUTHOR  = {Yamazaki, Hiroshi and Shidama, Yasunari and Pacharapokin, Chanapat and Nakamura, Yatsuka},
      TITLE   = {The {C}auchy-{R}iemann Differential Equations of Complex Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0016-8},
      VOLUME  = 17,
      PAGES   = {147--149},
      NUMBER  = {{\bf 2}})

@ARTICLE(GR_CY_3.ABS,
      AUTHOR  = {Arai, Kenichi and Okazaki, Hiroyuki},
      TITLE   = {Properties of Primes and Multiplicative Group of a Field},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0017-7},
      VOLUME  = 17,
      PAGES   = {151--155},
      NUMBER  = {{\bf 2}})

@ARTICLE(MEASURE8.ABS,
      AUTHOR  = {Endou, Noboru and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {{H}opf Extension Theorem of Measure},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0018-6},
      VOLUME  = 17,
      PAGES   = {157--162},
      NUMBER  = {{\bf 2}})

@ARTICLE(REWRITE3.ABS,
      AUTHOR  = {Trybulec, Micha{\l}},
      TITLE   = {Labelled State Transition Systems},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0019-5},
      VOLUME  = 17,
      PAGES   = {163--171},
      NUMBER  = {{\bf 2}})

@ARTICLE(DIST_1.ABS,
      AUTHOR  = {Okazaki, Hiroyuki},
      TITLE   = {Probability on Finite and Discrete Set and Uniform Distribution},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0020-z},
      VOLUME  = 17,
      PAGES   = {173--178},
      NUMBER  = {{\bf 2}})

@ARTICLE(INTEGR15.ABS,
      AUTHOR  = {Miyajima, Keiichi and Shidama, Yasunari},
      TITLE   = {Riemann Integral of Functions  from $\mathbb{R}$ into $\calR^n$},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0021-y},
      VOLUME  = 17,
      PAGES   = {179--185},
      NUMBER  = {{\bf 2}})

@ARTICLE(FUNCT_8.ABS,
      AUTHOR  = {Li, Bo and Men, Yanhong},
      TITLE   = {Basic Properties of Even and Odd Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0022-x},
      VOLUME  = 17,
      PAGES   = {187--192},
      NUMBER  = {{\bf 2}})

@ARTICLE(FSM_3.ABS,
      AUTHOR  = {Trybulec, Micha{\l}},
      TITLE   = {Equivalence of Deterministic and Nondeterministic Epsilon Automata},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0023-9},
      VOLUME  = 17,
      PAGES   = {193--199},
      NUMBER  = {{\bf 2}})

@ARTICLE(METRIZTS.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Basic Properties of Metrizable Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0024-8},
      VOLUME  = 17,
      PAGES   = {201--205},
      NUMBER  = {{\bf 3}})

@ARTICLE(TOPDIM_1.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Small Inductive Dimension of Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0025-7},
      VOLUME  = 17,
      PAGES   = {207--212},
      NUMBER  = {{\bf 3}})

@ARTICLE(GROUP_11.ABS,
      AUTHOR  = {Liang, Xiquan and Li, Dailu},
      TITLE   = {On Rough Subgroup of a Group},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0026-6},
      VOLUME  = 17,
      PAGES   = {213--217},
      NUMBER  = {{\bf 3}})

@ARTICLE(TOPDIM_2.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Small Inductive Dimension of Topological Spaces. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0027-5},
      VOLUME  = 17,
      PAGES   = {219--222},
      NUMBER  = {{\bf 3}})

@ARTICLE(DILWORTH.ABS,
      AUTHOR  = {Rudnicki, Piotr},
      TITLE   = {Dilworth's Decomposition Theorem for Posets},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0028-4},
      VOLUME  = 17,
      PAGES   = {223--232},
      NUMBER  = {{\bf 4}})

@ARTICLE(INTEGR1C.ABS,
      AUTHOR  = {Yamazaki, Masahiko and Yamazaki, Hiroshi and Wasaki, Katsumi and Shidama, Yasunari},
      TITLE   = {Complex Integral},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0029-3},
      VOLUME  = 17,
      PAGES   = {233--236},
      NUMBER  = {{\bf 4}})

@ARTICLE(INTERVA1.ABS,
      AUTHOR  = {Grabowski, Adam and Jastrz\k{e}bska, Magdalena},
      TITLE   = {On the Lattice of Intervals and Rough Sets},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0030-x},
      VOLUME  = 17,
      PAGES   = {237--244},
      NUMBER  = {{\bf 4}})

@ARTICLE(FUNCT_9.ABS,
      AUTHOR  = {Li, Bo and Men, Yanhong and Li, Dailu and Liang, Xiquan},
      TITLE   = {Basic Properties of Periodic Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0031-9},
      VOLUME  = 17,
      PAGES   = {245--248},
      NUMBER  = {{\bf 4}})

@ARTICLE(ORDINAL5.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Epsilon Numbers and {C}antor Normal Form},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2009},
      DOI     = {10.2478/v10037-009-0032-8},
      VOLUME  = 17,
      PAGES   = {249--256},
      NUMBER  = {{\bf 4}})
      
@ARTICLE(EUCLID_8.ABS,
      AUTHOR  = {Liang, Xiquan and Zhao, Piqing and Bai, Ou},
      TITLE   = {{V}ector Functions and their Differentiation Formulas in 3-dimensional {E}uclidean Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0001-2},
      VOLUME  = 18,
      PAGES   = {1--10},
      NUMBER  = {{\bf 1}})

@ARTICLE(C0SP2.ABS,
      AUTHOR  = {Kanazashi, Katuhiko and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {{B}anach Algebra of Continuous Functionals and the Space of Real-Valued Continuous Functionals with Bounded Support},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0002-1},
      VOLUME  = 18,
      PAGES   = {11--16},
      NUMBER  = {{\bf 1}})

@ARTICLE(ALGSTR_4.ABS,
      AUTHOR  = {Riccardi, Marco},
      TITLE   = {{F}ree Magmas},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0003-0},
      VOLUME  = 18,
      PAGES   = {17--26},
      NUMBER  = {{\bf 1}})

@ARTICLE(INTEGR12.ABS,
      AUTHOR  = {Li, Bo and Ma, Na},
      TITLE   = {{I}ntegrability Formulas. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0004-z},
      VOLUME  = 18,
      PAGES   = {27--37},
      NUMBER  = {{\bf 1}})

@ARTICLE(PDIFF_4.ABS,
      AUTHOR  = {Inou\'e, Takao and Xie, Bing and Liang, Xiquan},
      TITLE   = {{P}artial Differentiation of Real Ternary Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0005-y},
      VOLUME  = 18,
      PAGES   = {39--46},
      NUMBER  = {{\bf 1}})

@ARTICLE(POSET_1.ABS,
      AUTHOR  = {Ishida, Kazuhisa and Shidama, Yasunari},
      TITLE   = {{F}ixpoint Theorem for Continuous Functions on Chain-Complete Posets},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0006-x},
      VOLUME  = 18,
      PAGES   = {47--51},
      NUMBER  = {{\bf 1}})

@ARTICLE(GRNILP_1.ABS,
      AUTHOR  = {Li, Dailu and Liang, Xiquan and Men, Yanhong},
      TITLE   = {{N}ilpotent Groups},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0007-9},
      VOLUME  = 18,
      PAGES   = {53--56},
      NUMBER  = {{\bf 1}})

@ARTICLE(DIFF_3.ABS,
      AUTHOR  = {Liang, Xiquan and Tang, Ling},
      TITLE   = {{D}ifference and Difference Quotient. {P}art {III}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0008-8},
      VOLUME  = 18,
      PAGES   = {57--64},
      NUMBER  = {{\bf 1}})

@ARTICLE(ABCMIZ_A.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {{A} Model of {M}izar Concepts -- Unification},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0009-7},
      VOLUME  = 18,
      PAGES   = {65--75},
      NUMBER  = {{\bf 1}})

@ARTICLE(FIB_NUM4.ABS,
      AUTHOR  = {Jastrz\k{e}bska, Magdalena},
      TITLE   = {{R}epresentation of the {F}ibonacci and {L}ucas Numbers in Terms of Floor and Ceiling},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0010-1},
      VOLUME  = 18,
      PAGES   = {77--80},
      NUMBER  = {{\bf 1}})

@ARTICLE(EUCLID_9.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {{T}he Correspondence Between $n$-dimensional {E}uclidean Space and the Product of $n$ Real Lines},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0011-0},
      VOLUME  = 18,
      PAGES   = {81--85},
      NUMBER  = {{\bf 1}})

@ARTICLE(RLAFFIN1.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {{A}ffine Independence in Vector Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0012-z},
      VOLUME  = 18,
      PAGES   = {87--93},
      NUMBER  = {{\bf 1}})

@ARTICLE(SIMPLEX0.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {{A}bstract Simplicial Complexes},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0013-y},
      VOLUME  = 18,
      PAGES   = {95--106},
      NUMBER  = {{\bf 1}})

@ARTICLE(RVSUM_2.ABS,
      AUTHOR  = {Miyajima, Keiichi and Kato, Takahiro},
      TITLE   = {{T}he Sum and Product of Finite Sequences of Complex Numbers},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0014-x},
      VOLUME  = 18,
      PAGES   = {107--111},
      NUMBER  = {{\bf 2}})

@ARTICLE(PDIFF_5.ABS,
      AUTHOR  = {Inou\'e, Takao},
      TITLE   = {{S}econd-Order Partial Differentiation of Real Ternary Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0015-9},
      VOLUME  = 18,
      PAGES   = {113--127},
      NUMBER  = {{\bf 2}})

@ARTICLE(INTEGR13.ABS,
      AUTHOR  = {Li, Bo and Ma, Na and Liang, Xiquan},
      TITLE   = {{I}ntegrability Formulas. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0016-8},
      VOLUME  = 18,
      PAGES   = {129--141},
      NUMBER  = {{\bf 2}})

@ARTICLE(INTEGR14.ABS,
      AUTHOR  = {Li, Bo and Ma, Na},
      TITLE   = {{I}ntegrability Formulas. {P}art {III}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0017-7},
      VOLUME  = 18,
      PAGES   = {143--157},
      NUMBER  = {{\bf 2}})

@ARTICLE(LPSPACE2.ABS,
      AUTHOR  = {Watase, Yasushige and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {{O}n ${L}^p$ Space Formed by Real-Valued Partial Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0018-6},
      VOLUME  = 18,
      PAGES   = {159--169},
      NUMBER  = {{\bf 3}})

@ARTICLE(TOPS_4.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {{M}iscellaneous Facts about Open Functions and Continuous Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0019-5},
      VOLUME  = 18,
      PAGES   = {171--174},
      NUMBER  = {{\bf 3}})

@ARTICLE(TOPREALC.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {{O}n the Continuity of Some Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0020-z},
      VOLUME  = 18,
      PAGES   = {175--183},
      NUMBER  = {{\bf 3}})

@ARTICLE(RLAFFIN2.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {{T}he Geometric Interior in Real Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0021-y},
      VOLUME  = 18,
      PAGES   = {185--188},
      NUMBER  = {{\bf 3}})

@ARTICLE(SIMPLEX1.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {{S}perner's Lemma},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0022-x},
      VOLUME  = 18,
      PAGES   = {189--196},
      NUMBER  = {{\bf 4}})

@ARTICLE(CARDFIN2.ABS,
      AUTHOR  = {Kaliszyk, Cezary},
      TITLE   = {{C}ounting Derangements, Non Bijective Functions and the Birthday Problem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0023-9},
      VOLUME  = 18,
      PAGES   = {197--200},
      NUMBER  = {{\bf 4}})

@ARTICLE(INTEGR16.ABS,
      AUTHOR  = {Miyajima, Keiichi and Kato, Takahiro and Shidama, Yasunari},
      TITLE   = {{R}iemann Integral of Functions $\mathbb{R}$ into $\mathbb{C}$},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0024-8},
      VOLUME  = 18,
      PAGES   = {201--206},
      NUMBER  = {{\bf 4}})

@ARTICLE(PDIFF_6.ABS,
      AUTHOR  = {Inou\'e, Takao and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {{D}ifferentiation of Vector-Valued Functions on $n$-Dimensional Real Normed Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0025-7},
      VOLUME  = 18,
      PAGES   = {207--212},
      NUMBER  = {{\bf 4}})

@ARTICLE(RANDOM_2.ABS,
      AUTHOR  = {Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {{P}robability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2010},
      DOI     = {10.2478/v10037-010-0026-6},
      VOLUME  = 18,
      PAGES   = {213--217},
      NUMBER  = {{\bf 4}})

@ARTICLE(PDIFF_7.ABS,
      AUTHOR  = {Inou\'e, Takao and Naumowicz, Adam and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Partial Differentiation of Vector-Valued Functions on $n$-Dimensional Real Normed Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0001-x},
      VOLUME  = 19,
      PAGES   = {1--9},
      NUMBER  = {{\bf 1}})

@ARTICLE(GROUPP_1.ABS,
      AUTHOR  = {Liang, Xiquan and Li, Dailu},
      TITLE   = {Some Properties of $p$-Groups and Commutative $p$-Groups},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0002-9},
      VOLUME  = 19,
      PAGES   = {11--15},
      NUMBER  = {{\bf 1}})

@ARTICLE(INTEGR18.ABS,
      AUTHOR  = {Miyajima, Keiichi and Kato, Takahiro and Shidama, Yasunari},
      TITLE   = {Riemann Integral of Functions from $\mathbb{R}$ into Real Normed Space},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0003-8},
      VOLUME  = 19,
      PAGES   = {17--22},
      NUMBER  = {{\bf 1}})

@ARTICLE(GROUP_12.ABS,
      AUTHOR  = {Okazaki, Hiroyuki and Arai, Kenichi and Shidama, Yasunari},
      TITLE   = {Normal Subgroup of Product of Groups},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0004-7},
      VOLUME  = 19,
      PAGES   = {23--26},
      NUMBER  = {{\bf 1}})

@ARTICLE(MYCIELSK.ABS,
      AUTHOR  = {Rudnicki, Piotr and Stewart, Lorna},
      TITLE   = {{T}he {M}ycielskian of a Graph},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0005-6},
      VOLUME  = 19,
      PAGES   = {27--34},
      NUMBER  = {{\bf 1}})

@ARTICLE(DIFF_4.ABS,
      AUTHOR  = {Liang, Xiquan and Tang, Ling and Jiang, Xichun},
      TITLE   = {{D}ifference and Difference Quotient. {P}art {IV}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0006-5},
      VOLUME  = 19,
      PAGES   = {35--39},
      NUMBER  = {{\bf 1}})

@ARTICLE(MFOLD_1.ABS,
      AUTHOR  = {Riccardi, Marco},
      TITLE   = {The Definition of Topological Manifolds},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0007-4},
      VOLUME  = 19,
      PAGES   = {41--44},
      NUMBER  = {{\bf 1}})

@ARTICLE(NFCONT_3.ABS,
      AUTHOR  = {Okazaki, Hiroyuki and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {More on Continuous Functions on Normed Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0008-3},
      VOLUME  = 19,
      PAGES   = {45--49},
      NUMBER  = {{\bf 1}})

@ARTICLE(PRVECT_3.ABS,
      AUTHOR  = {Okazaki, Hiroyuki and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Cartesian Products of Family of Real Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0009-2},
      VOLUME  = 19,
      PAGES   = {51--59},
      NUMBER  = {{\bf 1}})

@ARTICLE(RLVECT_X.ABS,
      AUTHOR  = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {Formalization of Integral Linear Space},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0010-9},
      VOLUME  = 19,
      PAGES   = {61--64},
      NUMBER  = {{\bf 1}})

@ARTICLE(PDIFF_8.ABS,
      AUTHOR  = {Inou\'e, Takao and Naumowicz, Adam and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {Partial Differentiation, Differentiation and Continuity on $n$-Dimensional Real Normed Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0011-8},
      VOLUME  = 19,
      PAGES   = {65--68},
      NUMBER  = {{\bf 2}})

@ARTICLE(NDIFF_3.ABS,
      AUTHOR  = {Okazaki, Hiroyuki and Endou, Noboru and Narita, Keiko and Shidama, Yasunari},
      TITLE   = {Differentiable Functions into Real Normed Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0012-7},
      VOLUME  = 19,
      PAGES   = {69--72},
      NUMBER  = {{\bf 2}})

@ARTICLE(CGAMES_1.ABS,
      AUTHOR  = {Nittka, Robin},
      TITLE   = {Conway's Games and Some of their Basic Properties},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0013-6},
      VOLUME  = 19,
      PAGES   = {73--81},
      NUMBER  = {{\bf 2}})

@ARTICLE(ORDINAL6.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Veblen Hierarchy},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0014-5},
      VOLUME  = 19,
      PAGES   = {83--92},
      NUMBER  = {{\bf 2}})

@ARTICLE(EXCHSORT.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Sorting by Exchanging},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0015-4},
      VOLUME  = 19,
      PAGES   = {93--102},
      NUMBER  = {{\bf 2}})

@ARTICLE(MATRTOP1.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {{L}inear Transformations of {E}uclidean Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0016-3},
      VOLUME  = 19,
      PAGES   = {103--108},
      NUMBER  = {{\bf 2}})

@ARTICLE(MATRTOP2.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {{L}inear Transformations of {E}uclidean Topological Spaces. {P}art {II}},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0017-2},
      VOLUME  = 19,
      PAGES   = {109--112},
      NUMBER  = {{\bf 2}})

@ARTICLE(LTLAXIO1.ABS,
      AUTHOR  = {Giero, Mariusz},
      TITLE   = {The Axiomatization of Propositional Linear Time Temporal Logic},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0018-1},
      VOLUME  = 19,
      PAGES   = {113--119},
      NUMBER  = {{\bf 2}})

@ARTICLE(CC0SP1.ABS,
      AUTHOR  = {Kanazashi, Katuhiko and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {Banach Algebra of Bounded Complex-Valued Functionals},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0019-0},
      VOLUME  = 19,
      PAGES   = {121--126},
      NUMBER  = {{\bf 2}})

@ARTICLE(MAZURULM.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {{M}azur-{U}lam Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0020-7},
      VOLUME  = 19,
      PAGES   = {127--130},
      NUMBER  = {{\bf 3}})

@ARTICLE(EC_PF_1.ABS,
      AUTHOR  = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {Set of Points on Elliptic Curve in Projective Coordinates},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0021-6},
      VOLUME  = 19,
      PAGES   = {131--138},
      NUMBER  = {{\bf 3}})

@ARTICLE(RLAFFIN3.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Continuity of Barycentric Coordinates in {E}uclidean Topological Spaces},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0022-5},
      VOLUME  = 19,
      PAGES   = {139--144},
      NUMBER  = {{\bf 3}})

@ARTICLE(SIMPLEX2.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {{B}rouwer Fixed Point Theorem for Simplexes},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0023-4},
      VOLUME  = 19,
      PAGES   = {145--150},
      NUMBER  = {{\bf 3}})

@ARTICLE(BROUWER2.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {Brouwer Fixed Point Theorem in the General Case},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0024-3},
      VOLUME  = 19,
      PAGES   = {151--153},
      NUMBER  = {{\bf 3}})

@ARTICLE(FOMODEL0.ABS,
      AUTHOR  = {Caminati, Marco B.},
      TITLE   = {Preliminaries to Classical First Order Model Theory},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0025-2},
      VOLUME  = 19,
      PAGES   = {155--167},
      NUMBER  = {{\bf 3}})

@ARTICLE(FOMODEL1.ABS,
      AUTHOR  = {Caminati, Marco B.},
      TITLE   = {Definition of First Order Language with Arbitrary Alphabet. {S}yntax of Terms, Atomic Formulas and their Subterms},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0026-1},
      VOLUME  = 19,
      PAGES   = {169--178},
      NUMBER  = {{\bf 3}})

@ARTICLE(FOMODEL2.ABS,
      AUTHOR  = {Caminati, Marco B.},
      TITLE   = {First Order Languages: Further Syntax and Semantics},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0027-0},
      VOLUME  = 19,
      PAGES   = {179--192},
      NUMBER  = {{\bf 3}})

@ARTICLE(FOMODEL3.ABS,
      AUTHOR  = {Caminati, Marco B.},
      TITLE   = {Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0028-z},
      VOLUME  = 19,
      PAGES   = {193--203},
      NUMBER  = {{\bf 3}})

@ARTICLE(FOMODEL4.ABS,
      AUTHOR  = {Caminati, Marco B.},
      TITLE   = {Sequent Calculus, Derivability, Provability. {G}\"odel's Completeness Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0029-y},
      VOLUME  = 19,
      PAGES   = {205--222},
      NUMBER  = {{\bf 3}})

@ARTICLE(CAYLEY.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {Cayley's Theorem},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0030-5},
      VOLUME  = 19,
      PAGES   = {223--225},
      NUMBER  = {{\bf 4}})

@ARTICLE(BOR_CANT.ABS,
      AUTHOR  = {Jaeger, Peter},
      TITLE   = {{B}orel-{C}antelli Lemma},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0031-4},
      VOLUME  = 19,
      PAGES   = {227--232},
      NUMBER  = {{\bf 4}})

@ARTICLE(NFCONT_4.ABS,
      AUTHOR  = {Narita, Keiko and Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {More on the Continuity of Real Functions},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0032-3},
      VOLUME  = 19,
      PAGES   = {233--239},
      NUMBER  = {{\bf 4}})

@ARTICLE(STACKS_1.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {Representation Theorem for Stacks},
      JOURNAL = {Formalized Mathematics},
      YEAR    = {2011},
      DOI     = {10.2478/v10037-011-0033-2},
      VOLUME  = 19,
      PAGES   = {241--250},
      NUMBER  = {{\bf 4}})

@ARTICLE(FINANCE1.ABS,
      AUTHOR  = {Jaeger, Peter},
      TITLE   = {{E}lementary Introduction to Stochastic Finance in Discrete Time},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0001-5},
      VOLUME  = 20,
      PAGES   = {1--5},
      NUMBER  = {{\bf 1}})

@ARTICLE(FVALUAT1.ABS,
      AUTHOR  = {Bancerek, Grzegorz and Kobayashi, Hidetsune and Korni{\l}owicz, Artur},
      TITLE   = {{V}aluation Theory. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0002-4},
      VOLUME  = 20,
      PAGES   = {7--14},
      NUMBER  = {{\bf 1}})

@ARTICLE(CC0SP2.ABS,
      AUTHOR  = {Kanazashi, Katuhiko and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {{F}unctional Space ${\bmC}(\Omega)$, ${\bmCz(\Omega)}$},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0003-3},
      VOLUME  = 20,
      PAGES   = {15--22},
      NUMBER  = {{\bf 1}})

@ARTICLE(MATRTOP3.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {{T}he Rotation Group},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0004-2},
      VOLUME  = 20,
      PAGES   = {23--29},
      NUMBER  = {{\bf 1}})

@ARTICLE(NDIFF_5.ABS,
      AUTHOR  = {Shidama, Yasunari},
      TITLE   = {{D}ifferentiable Functions on Normed Linear Spaces},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0005-1},
      VOLUME  = 20,
      PAGES   = {31--40},
      NUMBER  = {{\bf 1}})

@ARTICLE(MFOLD_2.ABS,
      AUTHOR  = {Riccardi, Marco},
      TITLE   = {{P}lanes and Spheres as Topological Manifolds. {S}tereographic Projection},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0006-0},
      VOLUME  = 20,
      PAGES   = {41--45},
      NUMBER  = {{\bf 1}})

@ARTICLE(ZMODUL01.ABS,
      AUTHOR  = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = { {$\mathbb Z$}-modules},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0007-z},
      VOLUME  = 20,
      PAGES   = {47--59},
      NUMBER  = {{\bf 1}})

@ARTICLE(MORPH_01.ABS,
      AUTHOR  = {Yamazaki, Hiroshi and Byli\'nski, Czes\l aw and Wasaki, Katsumi},
      TITLE   = {{M}orphology for Image Processing. {P}art {I}},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0008-y},
      VOLUME  = 20,
      PAGES   = {61--63},
      NUMBER  = {{\bf 1}})

@ARTICLE(NDIFF_4.ABS,
      AUTHOR  = {Narita, Keiko and Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {{T}he Differentiable Functions  from $\mathbb{R}$ into ${\calR}^n$},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0009-x},
      VOLUME  = 20,
      PAGES   = {65--71},
      NUMBER  = {{\bf 1}})

@ARTICLE(MATRIX17.ABS,
      AUTHOR  = {Liang, Xiquan and Wang, Tao},
      TITLE   = {{S}ome Basic Properties of Some Special Matrices. {P}art {III}},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0010-4},
      VOLUME  = 20,
      PAGES   = {73--77},
      NUMBER  = {{\bf 1}})

@ARTICLE(INTEGR19.ABS,
      AUTHOR  = {Miyajima, Keiichi and Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {{R}iemann Integral of Functions from $\mathbb{R}$ into $n$-dimensional Real Normed Space},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0011-3},
      VOLUME  = 20,
      PAGES   = {79--86},
      NUMBER  = {{\bf 1}})

@ARTICLE(EC_PF_2.ABS,
      AUTHOR  = {Futa, Yuichi and Okazaki, Hiroyuki and Mizushima, Daichi and Shidama, Yasunari},
      TITLE   = {{O}perations of Points on Elliptic Curve in Projective Coordinates},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0012-2},
      VOLUME  = 20,
      PAGES   = {87--95},
      NUMBER  = {{\bf 1}})

@ARTICLE(TOPALG_6.ABS,
      AUTHOR  = {Riccardi, Marco and Korni{\l}owicz, Artur},
      TITLE   = {{F}undamental Group of $n$-sphere for $n \geq 2$},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0013-1},
      VOLUME  = 20,
      PAGES   = {97--104},
      NUMBER  = {{\bf 2}})

@ARTICLE(BORSUK_7.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur and Riccardi, Marco},
      TITLE   = {{T}he {B}orsuk-{U}lam Theorem},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0014-0},
      VOLUME  = 20,
      PAGES   = {105--112},
      NUMBER  = {{\bf 2}})

@ARTICLE(PDIFF_9.ABS,
      AUTHOR  = {Endou, Noboru and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {{H}igher-Order Partial Differentiation},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0015-z},
      VOLUME  = 20,
      PAGES   = {113--124},
      NUMBER  = {{\bf 2}})

@ARTICLE(DESCIP_1.ABS,
      AUTHOR  = {Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {{F}ormalization of the Data Encryption Standard},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0016-y},
      VOLUME  = 20,
      PAGES   = {125--146},
      NUMBER  = {{\bf 2}})

@ARTICLE(MMLQUERY.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {{S}emantics of {MML} Query},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0017-x},
      VOLUME  = 20,
      PAGES   = {147--155},
      NUMBER  = {{\bf 2}})

@ARTICLE(MENELAUS.ABS,
      AUTHOR  = {Shminke, Boris A.},
      TITLE   = {{R}outh's, {M}enelaus' and Generalized {C}eva's Theorems},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0018-9},
      VOLUME  = 20,
      PAGES   = {157--159},
      NUMBER  = {{\bf 2}})

@ARTICLE(SCMYCIEL.ABS,
      AUTHOR  = {Rudnicki, Piotr and Stewart, Lorna},
      TITLE   = {{S}imple Graphs as Simplicial Complexes: the {M}ycielskian of a Graph},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0019-8},
      VOLUME  = 20,
      PAGES   = {161--174},
      NUMBER  = {{\bf 2}})

@ARTICLE(NTALGO_1.ABS,
      AUTHOR  = {Okazaki, Hiroyuki and Aoki, Yosiki and Shidama, Yasunari},
      TITLE   = {{E}xtended {E}uclidean Algorithm and {CRT} Algorithm},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0020-2},
      VOLUME  = 20,
      PAGES   = {175--179},
      NUMBER  = {{\bf 2}})

@ARTICLE(RATFUNC1.ABS,
      AUTHOR  = {Schwarzweller, Christoph},
      TITLE   = {{I}ntroduction to Rational Functions},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0021-1},
      VOLUME  = 20,
      PAGES   = {181--191},
      NUMBER  = {{\bf 2}})

@ARTICLE(QC_TRANS.ABS,
      AUTHOR  = {Schl\"oder, Julian J. and Koepke, Peter},
      TITLE   = {{T}ransition of Consistency and Satisfiability under Language Extensions},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0022-0},
      VOLUME  = 20,
      PAGES   = {193--197},
      NUMBER  = {{\bf 3}})

@ARTICLE(GOEDCPUC.ABS,
      AUTHOR  = {Schl\"oder, Julian J. and Koepke, Peter},
      TITLE   = {{T}he {G}\"odel Completeness Theorem for Uncountable Languages},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0023-z},
      VOLUME  = 20,
      PAGES   = {199--203},
      NUMBER  = {{\bf 3}})

@ARTICLE(ZMODUL02.ABS,
      AUTHOR  = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {{Q}uotient Module of {$\mathbb Z$}-module},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0024-y},
      VOLUME  = 20,
      PAGES   = {205--214},
      NUMBER  = {{\bf 3}})

@ARTICLE(LTLAXIO2.ABS,
      AUTHOR  = {Giero, Mariusz},
      TITLE   = {{T}he Derivations of Temporal Logic Formulas},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0025-x},
      VOLUME  = 20,
      PAGES   = {215--219},
      NUMBER  = {{\bf 3}})

@ARTICLE(LTLAXIO3.ABS,
      AUTHOR  = {Giero, Mariusz},
      TITLE   = {{T}he Properties of Sets of Temporal Logic Subformulas},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0026-9},
      VOLUME  = 20,
      PAGES   = {221--226},
      NUMBER  = {{\bf 3}})

@ARTICLE(LTLAXIO4.ABS,
      AUTHOR  = {Giero, Mariusz},
      TITLE   = {{W}eak Completeness Theorem for Propositional Linear Time Temporal Logic},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0027-8},
      VOLUME  = 20,
      PAGES   = {227--234},
      NUMBER  = {{\bf 3}})

@ARTICLE(FRIENDS1.ABS,
      AUTHOR  = {P\k{a}k, Karol},
      TITLE   = {{T}he Friendship Theorem},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0028-7},
      VOLUME  = 20,
      PAGES   = {235--237},
      NUMBER  = {{\bf 3}})

@ARTICLE(MSAFREE4.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {{F}ree Term Algebras},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0029-6},
      VOLUME  = 20,
      PAGES   = {239--256},
      NUMBER  = {{\bf 3}})

@ARTICLE(DIST_2.ABS,
      AUTHOR  = {Okazaki, Hiroyuki},
      TITLE   = {{P}osterior Probability on Finite Set},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0030-0},
      VOLUME  = 20,
      PAGES   = {257--263},
      NUMBER  = {{\bf 4}})

@ARTICLE(INT_8.ABS,
      AUTHOR  = {Ma, Na and Liang, Xiquan},
      TITLE   = {{B}asic Properties of Primitive Root and Order Function},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0031-z},
      VOLUME  = 20,
      PAGES   = {265--269},
      NUMBER  = {{\bf 4}})

@ARTICLE(LOPBAN_7.ABS,
      AUTHOR  = {Sakurai, Hideki and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {{B}anach's Continuous Inverse Theorem and Closed Graph Theorem},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0032-y},
      VOLUME  = 20,
      PAGES   = {271--274},
      NUMBER  = {{\bf 4}})

@ARTICLE(ZMODUL03.ABS,
      AUTHOR  = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {{F}ree {$\mathbb Z$}-module},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0033-x},
      VOLUME  = 20,
      PAGES   = {275--280},
      NUMBER  = {{\bf 4}})

@ARTICLE(CAYLDICK.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {{C}ayley-{D}ickson Construction},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0034-9},
      VOLUME  = 20,
      PAGES   = {281--290},
      NUMBER  = {{\bf 4}})

@ARTICLE(ORDEQ_01.ABS,
      AUTHOR  = {Miyajima, Keiichi and Korni{\l}owicz, Artur and Shidama, Yasunari},
      TITLE   = {{C}ontracting Mapping on Normed Linear Space},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0035-8},
      VOLUME  = 20,
      PAGES   = {291--301},
      NUMBER  = {{\bf 4}})

@ARTICLE(ALTCAT_5.ABS,
      AUTHOR  = {Korni{\l}owicz, Artur},
      TITLE   = {{P}roducts in Categories without Uniqueness of {\bf cod} and {\bf dom}},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0036-7},
      VOLUME  = 20,
      PAGES   = {303--307},
      NUMBER  = {{\bf 4}})

@ARTICLE(AOFA_A00.ABS,
      AUTHOR  = {Bancerek, Grzegorz},
      TITLE   = {{P}rogram Algebra over an Algebra},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0037-6},
      VOLUME  = 20,
      PAGES   = {309--341},
      NUMBER  = {{\bf 4}})

@ARTICLE(GROUP_14.ABS,
      AUTHOR  = {Arai, Kenichi and Okazaki, Hiroyuki and Shidama, Yasunari},
      TITLE   = {{I}somorphisms of Direct Products of Finite Cyclic Groups},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0038-5},
      VOLUME  = 20,
      PAGES   = {343--347},
      NUMBER  = {{\bf 4}})

@ARTICLE(LPSPACC1.ABS,
      AUTHOR  = {Watase, Yasushige and Endou, Noboru and Shidama, Yasunari},
      TITLE   = {{O}n $L^1$ Space Formed by Complex-Valued Partial Functions},
      JOURNAL = {Formalized Mathematics},
      ISSN = {1426-2630},
      EISSN = {1898-9934},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0039-4},
      VOLUME  = 20,
      PAGES   = {349--357},
      NUMBER  = {{\bf 4}})
    
@ARTICLE(AOFA_A01.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 1}},
    PAGES = {1--23},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0001},
    VERSION = {8.0.01 5.5.1167},    
    TITLE = {{A}nalysis of Algorithms: An Example of a Sort Algorithm},
    AUTHOR = {Bancerek, Grzegorz},
    ADDRESS1 = {Association of Mizar Users\\Bia\l ystok, Poland},
    SUMMARY = {We analyse three algorithms: exponentiation by squaring, calculation of maximum, and sorting by exchanging in terms of program algebra over an algebra.   },
    MSC2010 = {03B35},
    SECTION1 = {Exponentiation by Squaring Revisited},
    SECTION2 = {Calculation of Maximum},
    SECTION3 = {Sorting by Exchanging},
    INTERNALREFS = {AFINSQ_1;AOFA_000;AOFA_A00;BINOP_1;CARD_1;EXCHSORT;FINSEQ_1;FINSEQ_2;FINSEQ_4;FINSET_1;FUNCOP_1;FUNCT_1;FUNCT_2;FUNCT_7;INSTALG1;INT_1;
LATTICE3;LFUZZY_0;MARGREL1;MSAFREE;MSAFREE1;MSAFREE3;MSAFREE4;MSATERM;MSUALG_1;MSUALG_2;MSUALG_3;NEWTON;ORDERS_2;ORDINAL1;PARTFUN1;PBOOLE;
RELAT_1;SUBSET_1;TREES_2;TREES_4;UNIALG_1;WAYBEL_0;WAYBEL_1;},
    KEYWORDS = {sort algorithm;},
    SUBMITTED = {November 9, 2012})

@ARTICLE(CKSPACE1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 1}},
    PAGES = {25--31},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0002},
    VERSION = {8.0.01 5.5.1167},
    TITLE = {{T}he {$C^k$} Space},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285.},
    AUTHOR = {Kanazashi, Katuhiko and Okazaki, Hiroyuki and Shidama, Yasunari},
    ADDRESS1 = {Shizuoka City, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we formalize continuous differentiability of real-valued functions on $n$-dimensional real normed linear spaces. Next, we give a definition of the $C^k$ space according to \cite{Kosaku:1996}.   },
    MSC2010 = {03B35},
    SECTION1 = {Definition of Continuously Differentiable Functions and Some Properties},
    SECTION2 = {Definition of the $C^k$ Space},
    EXTERNALREFS = {Kosaku:1996;},
    INTERNALREFS = {CARD_1;COMPLEX1;EUCLID;FINSEQ_1;FINSEQ_2;FINSET_1;FUNCOP_1;FUNCT_1;FUNCT_2;INT_1;MEMBERED;ORDINAL1;PARTFUN1;PDIFF_7;PDIFF_9;
RAT_1;REAL_1;RELAT_1;RELSET_1;SEQFUNC;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {real normed linear space;},
    SUBMITTED = {November 9, 2012})
    
@ARTICLE(RANDOM_3.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 1}},
    PAGES = {33--39},
    YEAR = {2013},
    VERSION = {8.1.01 5.7.1169},
    DOI = {10.2478/forma-2013-0003},
    TITLE = {{R}andom Variables and Product of Probability Spaces},
    ANNOTE = {The 1st author was supported by JSPS KAKENHI 21240001, and the 2nd author was supported by JSPS KAKENHI 22300285.},
    AUTHOR = {Okazaki, Hiroyuki and Shidama, Yasunari},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {We have been working on the formalization of the probability and the randomness. In \cite{RANDOM_1.ABS} and \cite{RANDOM_2.ABS}, we formalized some theorems concerning the real-valued random variables and the product of two probability spaces. In this article, we present the generalized formalization of \cite{RANDOM_1.ABS} and \cite{RANDOM_2.ABS}. First, we formalize the random variables of arbitrary set and prove the equivalence between random variable on $\Sigma$, Borel sets and a real-valued random variable on $\Sigma$. Next, we formalize the product of countably infinite probability spaces.   },
    MSC2010 = {03B35},    
    SECTION1 = {Random Variables},
    SECTION2 = {Product of Probability Spaces},
    INTERNALREFS = {CARD_1;DIST_1;FINANCE1;FINSEQ_1;FINSET_1;FINSUB_1;FUNCT_1;FUNCT_2;MEASURE1;NAT_1;ORDINAL1;PARTFUN1;PROB_1;PROB_4;
RANDOM_1;RANDOM_2;RELAT_1;RELSET_1;SETFAM_1;SUBSET_1;SUPINF_2;ZFMISC_1;},
    KEYWORDS = {random variables;Borel sets;probability space;},
    SUBMITTED = {December 1, 2012})
    
@ARTICLE(MMLQUER2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 1}},
    PAGES = {41--46},
    YEAR = {2013},
    VERSION = {8.1.01 5.7.1169},
    DOI = {10.2478/forma-2013-0004},
    TITLE = {{S}emantics of {MML} {Q}uery - Ordering},
    AUTHOR = {Bancerek, Grzegorz},
    ADDRESS1 = {Association of Mizar Users\\Bia{\l}ystok, Poland},
    SUMMARY = {Semantics of order directives of MML Query is presented. The formalization is done according to \cite{Bancerek2006}.   },
    MSC2010 = {03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Composition of Orders},
    SECTION3 = {{\tt number of} Ordering},
    SECTION4 = {Ordering by Resources},
    SECTION5 = {Ordering by Number of Iteration},
    SECTION6 = {{\tt value of} Ordering},
    EXTERNALREFS = {Bancerek2006;},
    INTERNALREFS = {CARD_1;FINSEQ_1;FINSET_1;FUNCT_1;FUNCT_2;FUNCT_7;MMLQUERY;ORDINAL1;ORDINAL4;PARTFUN1;RELAT_1;RELAT_2;RELSET_1;
REWRITE1;SETFAM_1;SUBSET_1;TOLER_1;ZFMISC_1;},
    KEYWORDS = {MML Query;},
    SUBMITTED = {December 1, 2012})
    
@ARTICLE(HURWITZ2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 1}},
    PAGES = {47--53},
    YEAR = {2013},
    VERSION = {8.1.01 5.8.1171},    
    DOI = {10.2478/forma-2013-0005},
    TITLE = {{A} Test for the Stability of Networks},
    AUTHOR = {Rowi\'nska-Schwarzweller, Agnieszka and Schwarzweller, Christoph},
    ADDRESS1 = {Chair of Display Technology\\University of Stuttgart\\Allmandring 3b, 70596 Stuttgart, Germany},
    ADDRESS2 = {Institute of Computer Science\\University of Gdansk\\Wita Stwosza 57, 80-952 Gdansk, Poland},
    SUMMARY = {A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks.\\ In this article we prove that a polynomial $p$ can be shown to be Hurwitz by checking whether the rational function $e(p)/o(p)$ can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here $e(p)$ and $o(p)$ denote the even and the odd part of $p$ \cite{Unb93}.   },
    MSC2010 = {03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Even and Odd Part of Polynomials},
    SECTION3 = {Even and Odd Polynomials and Rational Functions},
    SECTION4 = {Real Positive Polynomials and Rational Functions},
    SECTION5 = {The Routh-Schur Stability Criterion},
    EXTERNALREFS = {Unb93;},
    INTERNALREFS = {ABIAN;ALGSTR_1;BINOP_1;COMPLEX1;COMPLFLD;FUNCT_1;FUNCT_2;GROUP_1;HURWITZ;INT_1;NORMSP_1;ORDINAL1;PARTFUN1;POLYNOM1;
POLYNOM3;POLYNOM4;POLYNOM5;RATFUNC1;RAT_1;RELAT_1;RLVECT_1;SQUARE_1;SUBSET_1;VECTSP_1;VFUNCT_1;ZFMISC_1;},
    KEYWORDS = {Hurwitz polynomial;electrical impedance;Routh-Schur stability criterion;},
    SUBMITTED = {January 17, 2013}) 
   
@ARTICLE(ROUGHS_2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 1}},
    PAGES = {55--64},
    YEAR = {2013},
    VERSION = {8.1.01 5.8.1171},
    DOI = {10.2478/forma-2013-0006},
    TITLE = {{R}elational Formal Characterization of Rough Sets},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok, Poland},
    SUMMARY = {The notion of a rough set, developed by Pawlak \cite{Pawlak1982}, is an important tool to describe situation of incomplete or partially unknown information. In this article, which is essentially the continuation of \cite{ROUGHS_1.ABS}, we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library). Here we drop the classical equivalence- and tolerance-based models of rough sets \cite{SkowronS96} trying to formalize some parts of \cite{Zhu:2007} following also \cite{Yao96} in some sense (Propositions 1--8, Corr. 1 and 2; the complete description is available in the Mizar script). Our main problem was that informally, there is a direct correspondence between relations and underlying properties, in our approach however \cite{GrabowskiJ10}, which uses relational structures rather than relations, we had to switch between classical (based on pure set theory) and abstract (using the notion of a structure) parts of the Mizar Mathematical Library. Our next step will be translation of these properties into the pure language of Mizar attributes.   },
    MSC2010 = {03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Missing Ordinary Properties of Binary Relations},
    SECTION3 = {Approximations Revisited},
    SECTION4 = {General Properties of Approximations},
    SECTION5 = {Auxiliary Operations on Approximation Operators},
    SECTION6 = {Towards Topological Models of Rough Sets},
    SECTION7 = {Formalization of Zhu's Paper \cite{Zhu:2007}},
    EXTERNALREFS = {Pawlak1982;SkowronS96;Zhu:2007;Yao96;GrabowskiJ10;},
    INTERNALREFS = {DOMAIN_1;EQREL_1;FINSET_1;FUNCT_1;FUNCT_2;ORDERS_2;PARTFUN1;PRE_TOPC;RELAT_1;ROUGHS_1;SUBSET_1;TOPS_1;YELLOW_3;ZFMISC_1;},
    KEYWORDS = {Rough sets},
    SUBMITTED = {January 17, 2013})  
    
@ARTICLE(GROUP_17.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 1}},
    PAGES = {65--74},
    YEAR = {2013},
    VERSION = {8.1.01 5.9.1172},    
    DOI = {10.2478/forma-2013-0007},
    TITLE = {{I}somorphisms of Direct Products of Finite Commutative Groups},
    ANNOTE = {The 1st author was supported by JSPS KAKENHI 21240001, and the 3rd author was supported by JSPS KAKENHI 22300285.},
    AUTHOR = {Okazaki, Hiroyuki and Yamazaki, Hiroshi and Shidama, Yasunari},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {We have been working on the formalization of groups. In \cite{GROUP_14.ABS}, we encoded some theorems concerning the product of cyclic groups. In this article, we present the generalized formalization of \cite{GROUP_14.ABS}. First, we show that every finite commutative group which order is composite number is isomorphic to a direct product of finite commutative groups which orders are relatively prime. Next, we describe finite direct products of finite commutative groups.   },
    MSC2010 = {03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Direct Product of Finite Commutative Groups},
    SECTION3 = {Finite Direct Products of Finite Commutative Groups},
    INTERNALREFS = {CARD_1;CARD_3;DOMAIN_1;FINSEQ_1;FINSET_1;FUNCOP_1;FUNCT_1;FUNCT_2;FUNCT_4;GROUP_1;GROUP_2;GROUP_3;GROUP_4;GROUP_6;GROUP_7;GROUP_14;
INT_1;INT_2;MONOID_0;NAT_1;NAT_3;NEWTON;ORDINAL1;PARTFUN1;PBOOLE;PRALG_1;RELAT_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {finite commutative groups},
    SUBMITTED = {January 31, 2013})    

@ARTICLE(NBVECTSP.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 2}},
    PAGES = {75--81},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0008},
    VERSION = {8.1.01 5.13.1174},
    TITLE = {{$N$}-Dimensional Binary Vector Spaces},
    AUTHOR = {Arai, Kenichi and Okazaki, Hiroyuki},
    ADDRESS1 = {Tokyo University of Science\\Chiba, Japan},
    NOTE1 = {This research was presented during the 2013 International Conference on Foundations of Computer Science FCS'13 in Las Vegas, USA.},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {The binary set $\{0,1\}$ together with modulo-2 addition and multiplication is called a binary field, which is denoted by $\mathbb{F}_2$. The binary field $\mathbb{F}_2$ is defined in \cite{BSPACE.ABS}. A vector space over $\mathbb{F}_2$ is called a binary vector space. The set of all binary vectors of length $n$ forms an $n$-dimensional vector space $V_n$ over $\mathbb{F}_2$. Binary fields and $n$-dimensional binary vector spaces play an important role in practical computer science, for example, coding theory \cite{ECC:2006} and cryptology. In cryptology, binary fields and $n$-dimensional binary vector spaces are very important in proving the security of cryptographic systems \cite{Lai1994}. In this article we define the $n$-dimensional binary vector space $V_n$. Moreover, we formalize some facts about the $n$-dimensional binary vector space $V_n$.   },
    MSC2010 = {15A03 03B35},
    EXTERNALREFS = {ECC:2006;Lai1994;},
    INTERNALREFS = {BINOP_1;BSPACE;CARD_1;DESCIP_1;FINSEQ_1;FINSEQ_2;FINSET_1;FUNCOP_1;FUNCT_1;FUNCT_2;GROUP_1;MARGREL1;MATRLIN;
ORDINAL1;RELAT_1;RELSET_1;RLVECT_1;SUBSET_1;VECTSP_1;VECTSP_4;VECTSP_6;VECTSP_7;VECTSP_9;ZFMISC_1;},
    KEYWORDS = {formalization of binary vector space; },
    SUBMITTED = {April 17, 2013})    
    
@ARTICLE(TOPGEN_6.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 2}},
    PAGES = {83--85},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0009},
    VERSION = {8.1.01 5.13.1174},
    TITLE = {{S}ome Properties of the {S}orgenfrey Line and the {S}orgenfrey Plane},
    AUTHOR = {St. Arnaud, Adam and Rudnicki, Piotr},
    ADDRESS1 = {University of Alberta\\Edmonton, Canada},
    ADDRESS2 = {University of Alberta\\Edmonton, Canada},
    ACKNOWLEDGEMENT = {I would like to thank Piotr Rudnicki for taking me on as his summer student and being a mentor to me. Piotr was an incredibly caring, intelligent, funny, passionate human being. I am proud to know I was his last student, in a long line of students he has mentored and cared about throughout his life. Thank you Piotr, for the opportunity you gave me, and for the faith, confidence and trust you showed in me. I will miss you.},
    SUMMARY = {We first provide a modified version of the proof in \cite{TOPGEN_5.ABS} that the Sorgenfrey line is $T_1$. Here, we prove that it is in fact $T_2$, a stronger result. Next, we prove that all subspaces of ${\mathbb R}^{\bf 1}$ (that is the real line with the usual topology) are Lindel\"of. We utilize this result in the proof that the Sorgenfrey line is Lindel\"of, which is based on the proof found in \cite{Engelking:1968}. Next, we construct the Sorgenfrey plane, as the product topology of the Sorgenfrey line and itself. We prove that the Sorgenfrey plane is not Lindel\"of, and therefore the product space of two Lindel\"of spaces need not be Lindel\"of. Further, we note that the Sorgenfrey line is regular, following from \cite{TOPGEN_5.ABS}:59. Next, we observe that the Sorgenfrey line is normal since it is both regular and Lindel\"of. Finally, we prove that the Sorgenfrey plane is not normal, and hence the product of two normal spaces need not be normal. The proof that the Sorgenfrey plane is not normal and many of the lemmas leading up to this result are modelled after the proof in \cite{TOPGEN_5.ABS}, that the Niemytzki plane is not normal. Information was also gathered from \cite{SteenSeebach:1978}.   },
    MSC2010 = {54D15 54B10 03B35},
    EXTERNALREFS = {Engelking:1968;SteenSeebach:1978;},
    INTERNALREFS = {BORSUK_1;CARD_1;METRIZTS;PRE_TOPC;RAT_1;RCOMP_1;RELAT_1;SUBSET_1;TOPGEN_1;TOPGEN_3;TOPGEN_4;TOPGEN_5;TOPMETR;TOPS_1;ZFMISC_1;},
    KEYWORDS = {topological spaces; products of normal spaces; Sorgenfrey line; Sorgenfrey plane; Lindel\"of spaces; },
    SUBMITTED = {April 17, 2013})    

@ARTICLE(NUMERAL2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 2}},
    PAGES = {87--94},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0010},
    VERSION = {8.1.02 5.17.1179},
    TITLE = {{M}ore on Divisibility Criteria for Selected Primes},
    AUTHOR = {Naumowicz, Adam and Piliszek, Rados{\l}aw},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
    ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
      SUMMARY = {This paper is a continuation of \cite{NUMERAL1.ABS}, where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in \cite{Briggs2000}.   },
    MSC2010 = {11A63 03B35},
    SECTION1 = {Preliminaries on Finite Sequences},
    SECTION2 = {Lemmas on Some Divisibility Properties},
    SECTION3 = {Divisibility Criteria for Primes up to 101},
    EXTERNALREFS = {Briggs2000;},
    INTERNALREFS = {ABIAN;AFINSQ_1;AFINSQ_2;CARD_1;FIB_NUM2;FINSET_1;FUNCT_1;FUNCT_2;INT_1;INT_2;MEMBERED;NAT_1;NEWTON;NUMERAL1;
ORDINAL1;ORDINAL2;ORDINAL4;ORDINAL6;PARTFUN1;PARTFUN3;RECDEF_1;RELAT_1;RELSET_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {divisibility; divisibility rules; decimal digits; },
    SUBMITTED = {May 19, 2013})    
    
@ARTICLE(NDIFF_6.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 2}},
    PAGES = {95--102},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0011},
    VERSION = {8.1.02 5.17.1179},
    TITLE = {{D}ifferentiation in Normed Spaces},
    AUTHOR = {Endou, Noboru and Shidama, Yasunari},
    ADDRESS1 = {Gifu National College of Technology\\Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article we formalized the Fr\'echet differentiation. It is defined as a generalization of the differentiation of a real-valued function of a single real variable to more general functions whose domain and range are subsets of normed spaces \cite{Schwartz:1981}.   },
    MSC2010 = {58C20 46G05 03B35},
    EXTERNALREFS = {Schwartz:1981;},
    INTERNALREFS = {BINOP_1;CARD_3;FINSEQ_1;FINSET_1;FUNCT_1;FUNCT_2;INT_1;LOPBAN_1;NDIFF_1;NFCONT_1;NORMSP_1;
ORDINAL1;PARTFUN1;RELAT_1;RELSET_1;RLVECT_1;SUBSET_1;VFUNCT_1;ZFMISC_1;},
    KEYWORDS = {formalization of Fr\'echet derivative; Fr\'echet differentiability; },
    SUBMITTED = {May 19, 2013})    
    
@ARTICLE(NUMPOLY1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 2}},
    PAGES = {103--113},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0012},
    VERSION = {8.1.02 5.17.1179},
    TITLE = {{P}olygonal Numbers},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok\\Poland},
    SUMMARY = {In the article the formal characterization of triangular numbers (famous from \cite{Gauss:1986} and words ``EYPHKA! num = $\Delta + \Delta + \Delta$") \cite{Heath:1921} is given. Our primary aim was to formalize one of the items (\#42) from Wiedijk's Top 100 Mathematical Theorems list \cite{Freek-100-theorems}, namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar language evolved and attributes with arguments were implemented, we decided to extend these lines and we characterized polygonal numbers.\par We formalized centered polygonal numbers, the connection between triangular and square numbers, and also some equalities involving Mersenne primes and perfect numbers. We gave also explicit formula to obtain from the polygonal number its ordinal index. Also selected congruences modulo 10 were enumerated. Our work basically covers the Wikipedia item for triangular numbers and the Online Encyclopedia of Integer Sequences (\url{http://oeis.org/A000217}). \par An interesting related result \cite{Guy:1994} could be the proof of Lagrange's four-square theorem or Fermat's polygonal number theorem \cite{Weil:1983}.   },
    MSC2010 = {11E25 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Triangular Numbers},
    SECTION3 = {Polygonal Numbers},
    SECTION4 = {Centered Polygonal Numbers},
    SECTION5 = {On the Connection between Triangular and Other Polygonal Numbers},
    SECTION6 = {Sets of Polygonal Numbers},
    SECTION7 = {Some Well-known Properties},
    SECTION8 = {Reciprocals of Triangular Numbers},
    EXTERNALREFS = {Gauss:1986;Heath:1921;Freek-100-theorems;Guy:1994;Weil:1983;},
    INTERNALREFS = {ABIAN;COMSEQ_2;EC_PF_2;FINSEQ_1;FINSEQ_2;FINSET_1;FUNCT_1;FUNCT_2;GR_CY_3;INT_1;INT_2;MEMBERED;NAT_1;NAT_5;NEWTON;
ORDINAL1;PARTFUN1;PEPIN;PYTHTRIP;RAT_1;RELAT_1;RVSUM_1;SEQ_1;SEQ_2;SERIES_1;SQUARE_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {triangular number; polygonal number; reciprocals of triangular numbers; },
    SUBMITTED = {May 19, 2013})    
    
@ARTICLE(GAUSSINT.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 2}},
    PAGES = {115--125},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0013},
    VERSION = {8.1.02 5.17.1179},
    TITLE = {{G}aussian Integers},
    AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Mizushima, Daichi and Shidama, Yasunari},
    ADDRESS1 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    NOTE3 = {This research was presented during the 2012 International Symposium on Information Theory and its Applications (ISITA2012) in Honolulu, USA.},
    ADDRESS4 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers \cite{Weil:1979}. We also formalize ring (called Gaussian integer ring), $\mathbb Z$-module and $\mathbb Z$-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.   },
    MSC2010 = {11R04 03B35},
    SECTION1 = {Gaussian Integer Ring},
    SECTION2 = {$\mathbb Z$-Algebra},
    SECTION3 = {Quotient Field of Gaussian Integer Ring},
    SECTION4 = {Rational Field},
    SECTION5 = {Gaussian Rational Number Field},
    SECTION6 = {Gaussian Integer Ring is Euclidean},
    EXTERNALREFS = {Weil:1979;},
    INTERNALREFS = {BINOP_1;CARD_1;CARD_3;COMPLEX1;EC_PF_1;FINSET_1;FUNCT_1;FUNCT_2;GCD_1;GROUP_1;INT_1;INT_2;INT_3;MEMBERED;ORDINAL1;PARTFUN1;
QUOFIELD;RAT_1;REALSET1;RELAT_1;RLVECT_1;SQUARE_1;SUBSET_1;VECTSP_1;VECTSP_2;ZFMISC_1;ZMODUL01;},
    KEYWORDS = {formalization of Gaussian integers; algebraic integers; },
    SUBMITTED = {May 19, 2013})
    
@ARTICLE(TOPALG_7.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 2}},
    PAGES = {127--131},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0014},
    VERSION = {8.1.02 5.17.1179},
    TITLE = {{C}ommutativeness of Fundamental Groups of Topological Groups},
    AUTHOR = {Korni{\l}owicz, Artur},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
    SUMMARY = {In this article we prove that fundamental groups based at the unit point of topological groups are commutative \cite{Hatcher}.   },
    MSC2010 = {55Q52 03B35},
    EXTERNALREFS = {Hatcher;},
    INTERNALREFS = {BINOP_1;BORSUK_1;BORSUK_2;BORSUK_6;CONNSP_2;FUNCOP_1;FUNCT_1;FUNCT_2;GROUP_1;GROUP_2;MEMBERED;MONOID_0;PARTFUN1;PRE_TOPC;
RCOMP_1;REALSET1;RELAT_1;RELSET_1;SUBSET_1;TOPALG_1;TOPALG_2;TOPGRP_1;TOPMETR;TOPREALB;ZFMISC_1;},    
    KEYWORDS = {fundamental group; topological group; },
    SUBMITTED = {May 19, 2013})    
    
@ARTICLE(HUFFMAN1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 2}},
    PAGES = {133--143},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0015},
    VERSION = {8.1.02 5.17.1181},
    TITLE = {{C}onstructing Binary {H}uffman Tree},
    AUTHOR = {Okazaki, Hiroyuki and Futa, Yuichi and Shidama, Yasunari},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
    NOTE1 = {This work was supported by JSPS KAKENHI 21240001.},
    ADDRESS2 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    NOTE3 = {This work was supported by JSPS KAKENHI 22300285.},
    SUMMARY = {Huffman coding is one of a most famous entropy encoding methods for lossless data compression \cite{HUFFMAN:1952}. JPEG and ZIP formats employ variants of Huffman encoding as lossless compression algorithms. Huffman coding is a bijective map from source letters into leaves of the Huffman tree constructed by the algorithm. In this article we formalize an algorithm constructing a binary code tree, Huffman tree.   },
    MSC2010 = {14G50 68P30 03B35},
    SECTION1 = {Constructing Binary Decoded Trees},
    SECTION2 = {Binary Huffman Tree},
    EXTERNALREFS = {HUFFMAN:1952;},
    INTERNALREFS = {BINTREE1;CARD_1;DOMAIN_1;FINSEQ_1;FINSET_1;FUNCT_1;FUNCT_2;INT_1;MEMBERED;NAT_1;ORDINAL1;PROB_1;RANDOM_1;
RAT_1;RELAT_1;RELSET_1;SUBSET_1;TREES_1;TREES_2;TREES_3;TREES_4;ZFMISC_1;},
    KEYWORDS = {formalization of Huffman coding tree; source coding; },
    SUBMITTED = {June 18, 2013})
    
@ARTICLE(INTEGR20.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 2}},
    PAGES = {145--152},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0016},
    VERSION = {8.1.02 5.17.1181},
    TITLE = {{R}iemann Integral of Functions from {$\mathbb R$} into Real {B}anach Space},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285 and 23500029.},
    AUTHOR = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
    ADDRESS1 = {Hirosaki-city\\Aomori, Japan},
    ADDRESS2 = {Gifu National College of Technology\\Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article we deal with the Riemann integral of functions from $\mathbb{R}$ into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from $\mathbb{R}$ into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements of a real normed space and finite sequences of real numbers.\par In addition we proved some theorems about the convergence of sequences. We applied definitions introduced in the previous article \cite{INTEGR18.ABS} to the proof of integrability.   },
    MSC2010 = {26A42 03B35},
    SECTION1 = {Some Properties of Continuous Functions},
    SECTION2 = {Some Properties of Sequences},
    INTERNALREFS = {BINOP_1;CARD_1;COMPLEX1;COMSEQ_2;FINSEQ_1;FINSEQ_2;FINSET_1;FUNCT_1;FUNCT_2;INTEGR18;INTEGRA1;INTEGRA2;INTEGRA3;INTEGRA5;INT_1;
LOPBAN_1;MEASURE5;MEMBERED;NAT_1;NFCONT_3;NORMSP_1;ORDINAL1;PARTFUN1;RAT_1;RCOMP_1;RELAT_1;RELSET_1;RLVECT_1;RVSUM_1;SEQ_2;
SUBSET_1;ZFMISC_1;},
    KEYWORDS = {formalization of Riemann integral; },
    SUBMITTED = {June 18, 2013})
    
@ARTICLE(MOEBIUS2.ABS,
    JOURNAL = {Formalized Mathematics},
     ISSN = {1426-2630},
     EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 2}},
    PAGES = {153--162},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0017},
    VERSION = {8.1.02 5.18.1182},
    TITLE = {{O}n Square-Free Numbers},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok\\Poland},
    SUMMARY = {In the article the formal characterization of square-free numbers is shown; in this manner the paper is the continuation of \cite{MOEBIUS1.ABS}. Essentially, we prepared some lemmas for convenient work with numbers (including the proof that the sequence of prime reciprocals diverges \cite{PFTB}) according to \cite{HardyWright} which were absent in the Mizar Mathematical Library. Some of them were expressed in terms of clusters' registrations, enabling automatization machinery available in the Mizar system. Our main result of the article is in the final section; we proved that the lattice of positive divisors of a positive integer $n$ is Boolean if and only if $n$ is square-free.   },
    MSC2010 = {11A51 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Prime Numbers},
    SECTION3 = {Prime Reciprocals},
    SECTION4 = {Square Factors},
    SECTION5 = {Extracting Square-containing and Square-free Part of a Number},
    SECTION6 = {Binary Operations},
    SECTION7 = {On the Natural Divisors},
    SECTION8 = {The Lattice of Natural Divisors},
    EXTERNALREFS = {PFTB;HardyWright;},
    INTERNALREFS = {BINOP_1;CARD_1;CARD_3;COMSEQ_2;ENUMSET1;FINSEQ_1;FINSEQ_2;FINSET_1;FUNCT_1;FUNCT_2;INT_1;INT_2;INT_7;LATTICES;MEMBERED;
MOEBIUS1;NAT_1;NAT_3;NAT_LAT;NEWTON;ORDINAL1;PARTFUN1;PARTFUN3;PBOOLE;PEPIN;RAT_1;REALSET1;RELAT_1;RVSUM_1;SEQ_2;SERIES_1;
SETFAM_1;SQUARE_1;SUBSET_1;WSIERP_1;ZFMISC_1;},
    KEYWORDS = {square-free numbers; prime reciprocals; lattice of natural divisors; },
    SUBMITTED = {July 12, 2013}) 
 
 @ARTICLE(DBLSEQ_1.ABS,
     JOURNAL = {Formalized Mathematics},
     ISSN = {1426-2630},
     EISSN = {1898-9934},
     VOLUME = {21},
     NUMBER = {{\bf 3}},
     PAGES = {163--170},
     YEAR = {2013},
     DOI = {10.2478/forma-2013-0018},
     VERSION = {8.1.02 5.19.1189},
     TITLE = {{D}ouble Sequences and Limits},
     ANNOTE = {This work was supported by JSPS KAKENHI 23500029.},
     AUTHOR = {Endou, Noboru and Okazaki, Hiroyuki and Shidama, Yasunari},
     ADDRESS1 = {Gifu National College of Technology\\Japan},
     ADDRESS2 = {Shinshu University\\Nagano, Japan},
     ADDRESS3 = {Shinshu University\\Nagano, Japan},
     SUMMARY = {Double sequences are important extension of the ordinary notion of a sequence. In this article we formalized three types of limits of double sequences and the theory of these limits.   },
     MSC2010 = {54A20 03B35},
     INTERNALREFS = {BINOP_1;COMPLEX1;COMSEQ_2;FINSEQOP;FUNCOP_1;FUNCT_1;FUNCT_2;INT_1;MESFUNC9;NAT_1;ORDINAL1;
 PARTFUN1;RAT_1;RELAT_1;SEQ_2;SUBSET_1;ZFMISC_1;},
     KEYWORDS = {formalization of basic metric space; limits of double sequences; },
     SUBMITTED = {August 31, 2013})
     
 @ARTICLE(AESCIP_1.ABS,
     JOURNAL = {Formalized Mathematics},
     ISSN = {1426-2630},
     EISSN = {1898-9934},
     VOLUME = {21},
     NUMBER = {{\bf 3}},
     PAGES = {171--184},
     YEAR = {2013},
     DOI = {10.2478/forma-2013-0019},
     VERSION = {8.1.02 5.19.1189},
     TITLE = {{F}ormalization of the Advanced Encryption Standard. {P}art {I}},
     ANNOTE = {This work was supported by JSPS KAKENHI 21240001 and 22300285.},
     AUTHOR = {Arai, Kenichi and Okazaki, Hiroyuki},
     ADDRESS1 = {Tokyo University of Science\\Chiba, Japan},
     NOTE1 = {This research was presented during the 2012 International Conference on Foundations of Computer Science FCS'12 in Las Vegas, USA.},
     ADDRESS2 = {Shinshu University\\Nagano, Japan},
     SUMMARY = {In this article, we formalize the Advanced Encryption Standard (AES). AES, which is the most widely used symmetric cryptosystem in the world, is a block cipher that was selected by the National Institute of Standards and Technology (NIST) as an official Federal Information Processing Standard for the United States in 2001 \cite{FIPS:197}. AES is the successor to DES \cite{DESCIP_1.ABS}, which was formerly the most widely used symmetric cryptosystem in the world. We formalize the AES algorithm according to \cite{FIPS:197}. We then verify the correctness of the formalized algorithm that the ciphertext encoded by the AES algorithm can be decoded uniquely by the same key. Please note the following points about this formalization: the AES round process is composed of the {\tt SubBytes}, {\tt ShiftRows}, {\tt MixColumns}, and {\tt AddRoundKey} transformations (see \cite{FIPS:197}). In this formalization, the {\tt SubBytes} and {\tt MixColumns} transformations are given as permutations, because it is necessary to treat the finite field GF($2^8$) for those transformations. The formalization of AES that considers the finite field GF($2^8$) is formalized by the future article.   },
     MSC2010 = {68P25 94A60 03B35},
     SECTION1 = {Preliminaries},
     SECTION2 = {State Array},
     SECTION3 = {\tt SubBytes},
     SECTION4 = {\tt ShiftRows},
     SECTION5 = {\tt AddRoundKey},
     SECTION6 = {Key Expansion},
     SECTION7 = {Encryption and Decryption},
     EXTERNALREFS = {FIPS:197;},
     INTERNALREFS = {BINOP_1;CARD_1;DESCIP_1;FINSEQ_1;FINSEQ_2;FINSEQ_4;FINSEQ_6;FINSET_1;FUNCT_1;FUNCT_2;
 INT_1;MARGREL1;NAT_1;ORDINAL1;PARTFUN1;RELAT_1;SUBSET_1;ZFMISC_1;},
     KEYWORDS = {Mizar formalization; Advanced Encryption Standard (AES) algorithm; cryptology; },
     SUBMITTED = {October 7, 2013})    
     
 @ARTICLE(INTEGR21.ABS,
     JOURNAL = {Formalized Mathematics},
     ISSN = {1426-2630},
     EISSN = {1898-9934},
     VOLUME = {21},
     NUMBER = {{\bf 3}},
     PAGES = {185--191},
     YEAR = {2013},
     DOI = {10.2478/forma-2013-0020},
     VERSION = {8.1.02 5.19.1189},
     TITLE = {{T}he Linearity of {R}iemann Integral on Functions from {$\mathbb R$} into Real {B}anach Space},
     ANNOTE = {This work was supported by JSPS KAKENHI 22300285, 23500029.},
     AUTHOR = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
     ADDRESS1 = {Hirosaki-city\\Aomori, Japan},
     ADDRESS2 = {Gifu National College of Technology\\Japan},
     ADDRESS3 = {Shinshu University\\Nagano, Japan},
     SUMMARY = {In this article, we described basic properties of Riemann integral on functions from {$\mathbb R$} into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article \cite{INTEGR19.ABS} and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article \cite{INTEGR18.ABS} and the article \cite{INTEGR20.ABS} to the proof. Using the definition of the article \cite{INTEGR19.ABS}, we also proved some theorems on bounded functions.   },
     MSC2010 = {26A42 03B35},
     SECTION1 = {Some Properties of Bounded Functions},
     SECTION2 = {Some Properties of Integral of Continuous Functions},
     INTERNALREFS = {COMPLEX1;COMSEQ_2;FUNCT_1;FUNCT_2;INTEGR18;INTEGR19;INTEGR20;INTEGRA1;INTEGRA5;LOPBAN_1;MEASURE5;MEMBERED;
 NFCONT_3;NORMSP_1;PARTFUN1;RCOMP_1;RELAT_1;RELSET_1;RLVECT_1;SUBSET_1;VFUNCT_1;ZFMISC_1;},
     KEYWORDS = {formalization of Riemann integral; },
     SUBMITTED = {October 7, 2013})
     
 @ARTICLE(CAT_6.ABS,
     JOURNAL = {Formalized Mathematics},
     ISSN = {1426-2630},
     EISSN = {1898-9934},
     VOLUME = {21},
     NUMBER = {{\bf 3}},
     PAGES = {193--205},
     YEAR = {2013},
     DOI = {10.2478/forma-2013-0021},
     VERSION = {8.1.02 5.19.1189},
     TITLE = {{O}bject-Free Definition of Categories},
     AUTHOR = {Riccardi, Marco},
     ADDRESS1 = {Via del Pero 102\\54038 Montignoso\\Italy},
     SUMMARY = {Category theory was formalized in Mizar with two different approaches \cite{CAT_1.ABS}, \cite{ALTCAT_1.ABS} that correspond to those most commonly used \cite{MacLane:1}, \cite{Borceaux}. Since there is a one-to-one correspondence between objects and identity morphisms, some authors have used an approach that does not refer to objects as elements of the theory, and are usually indicated as object-free category \cite{Adamek:2009} or as arrows-only category \cite{MacLane:1}. In this article is proposed a new definition of an object-free category, introducing the two properties: left composable and right composable, and a simplification of the notation through a symbol, a binary relation between morphisms, that indicates whether the composition is defined. In the final part we define two functions that allow to switch from the two definitions, with and without objects, and it is shown that their composition produces isomorphic categories.   },
     MSC2010 = {18A05 03B35},
     SECTION1 = {Yet Another Definition of Category},
     SECTION2 = {Transform a Category in the Other},
     EXTERNALREFS = {Adamek:2009;MacLane:1;Borceaux;},
     INTERNALREFS = {ALTCAT_1;BINOP_1;CARD_1;CAT_1;CAT_2;FINSEQ_1;FINSET_1;FUNCT_1;FUNCT_2;FUNCT_4;GRAPH_1;
 ISOCAT_1;ORDINAL1;PARTFUN1;RELAT_1;RELSET_1;SETFAM_1;SUBSET_1;ZFMISC_1;},
     KEYWORDS = {object-free category; correspondence between different approaches to category; },
     SUBMITTED = {October 7, 2013})    
     
 @ARTICLE(GROUP_18.ABS,
     JOURNAL = {Formalized Mathematics},
     ISSN = {1426-2630},
     EISSN = {1898-9934},
     VOLUME = {21},
     NUMBER = {{\bf 3}},
     PAGES = {207--211},
     YEAR = {2013},
     DOI = {10.2478/forma-2013-0022},
     VERSION = {8.1.02 5.19.1189},
     TITLE = {{I}somorphisms of Direct Products of Cyclic Groups of Prime Power Order},
     AUTHOR = {Yamazaki, Hiroshi and Okazaki, Hiroyuki and Nakasho, Kazuhisa and Shidama, Yasunari},
     ADDRESS1 = {Shinshu University\\Nagano, Japan},
     ADDRESS2 = {Shinshu University\\Nagano, Japan},
     ADDRESS3 = {Shinshu University\\Nagano, Japan},
     ADDRESS4 = {Shinshu University\\Nagano, Japan},
     NOTE4 = {This work was supported by JSPS KAKENHI 22300285.},
     SUMMARY = {In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups \cite{GROUP_14.ABS}, \cite{GROUP_17.ABS}.   },
     MSC2010 = {13D99 06A75 03B35},
     SECTION1 = {Basic Properties of Cyclic Groups of Prime Power Order},
     SECTION2 = {Isomorphism of Cyclic Groups of Prime Power Order},
     INTERNALREFS = {CARD_1;DOMAIN_1;FINSEQ_1;FINSET_1;FUNCT_1;FUNCT_2;GROUP_1;GROUP_14;GROUP_17;GROUP_2;GROUP_3;GROUP_4;GROUP_6;GROUP_7;
 INT_1;INT_2;MEMBERED;MONOID_0;NAT_1;NEWTON;ORDINAL1;PARTFUN1;PBOOLE;PRALG_1;RAT_1;RELAT_1;RELSET_1;SEQ_4;SUBSET_1;ZFMISC_1;   },
     KEYWORDS = {formalization of the commutative cyclic group; prime power set; },
     SUBMITTED = {October 7, 2013})    
     
 @ARTICLE(LATTICEA.ABS,
     JOURNAL = {Formalized Mathematics},
     ISSN = {1426-2630},
     EISSN = {1898-9934},
     VOLUME = {21},
     NUMBER = {{\bf 3}},
     PAGES = {213--221},
     YEAR = {2013},
     DOI = {10.2478/forma-2013-0023},
     VERSION = {8.1.02 5.19.1189},
     TITLE = {{P}rime Filters and Ideals in Distributive Lattices},
     AUTHOR = {Grabowski, Adam},
     ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok\\Poland},
     SUMMARY = {The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the Mizar Mathematical Library, there are some attempts to formalize prime ideals and filters; one series of articles written as decoding \cite{CCL} proven some results; we tried however to follow \cite{LOPCLSET.ABS}, \cite{OPENLATT.ABS}, and \cite{LATTICE4.ABS}. All three were devoted to the Stone representation theorem \cite{StoneRepr} for Boolean or Heyting lattices. The main aim of the present article was to bridge this gap between general distributive lattices and Boolean algebras, having in mind that the more general approach will eventually replace the common proof of aforementioned articles.\footnote{As one of the anonymous reviewers pointed out, it would be interesting to show counterexamples showing that the assumptions of the distributivity and boundedness are necessary, and this will be our plan for future work as basic examples of non-distributive lattices are available as of now only as relational structures.} \par Because in Boolean algebras the notions of ultrafilters, prime filters and maximal filters coincide, we decided to construct some concrete examples of ultrafilters in nontrivial Boolean lattice. We proved also the Prime Ideal Theorem not as BPI (Boolean Prime Ideal), but in the more general setting.\par In the final section we present Nachbin theorems \cite{Nachbin}, \cite{Balbes} expressed both in terms of maximal and prime filters and as the unordered spectra of a lattice \cite{Gratzer2011}, \cite{Gratzer}. This shows that if the notion of maximal and prime filters coincide in the lattice, it is Boolean.   },
     MSC2010 = {06D05 03B35},
     SECTION1 = {Preliminaries},
     SECTION2 = {Examples of Filters in Nontrivial Boolean Lattices},
     SECTION3 = {On Prime and Maximal Filters and Ideals},
     SECTION4 = {Prime Ideal Theorem for Distributive Lattices},
     SECTION5 = {The Stone Representation},
     SECTION6 = {Pseudo Complements in Lattices},
     SECTION7 = {Nachbin's Theorem for Bounded Distributive Lattices},
     EXTERNALREFS = {CCL;StoneRepr;Nachbin;Balbes;Gratzer2011;Gratzer;},
     INTERNALREFS = {FILTER_0;FILTER_2;FUNCT_1;FUNCT_2;LATTICE3;LATTICE4;LATTICES;LOPCLSET;OPENLATT;PARTFUN1;PRE_TOPC;RELAT_1;
 RELSET_1;SETFAM_1;SUBSET_1;ZFMISC_1;},
     KEYWORDS = {prime filters; prime ideals; distributive lattices; },
     SUBMITTED = {October 7, 2013})    
     
 @ARTICLE(PREFER_1.ABS,
     JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
     VOLUME = {21},
     NUMBER = {{\bf 3}},
     PAGES = {223--233},
     YEAR = {2013},
     DOI = {10.2478/forma-2013-0024},
     VERSION = {8.1.02 5.19.1189},
     TITLE = {{I}ntroduction to Formal Preference Spaces},
     AUTHOR = {Niewiadomska, Eliza and Grabowski, Adam},
     ADDRESS1 = {Institute of Mathematics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok\\Poland},
     ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok\\Poland},
     SUMMARY = {In the article the formal characterization of preference spaces \cite{Arrow} is given. As the preference relation is one of the very basic notions of mathematical economics \cite{Hallden}, it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done -- see \cite{ARROW.ABS}). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished \cite{Kuzyka}. \par There are many approaches to preferences in literature. We modelled them in a rather illustrative way (similar structures were considered in \cite{PCS_0.ABS}): either the consumer (strictly) prefers an alternative, or they are of equal interest; he/she could also have no opinion of the choice. Then our structures are based on three relations on the (arbitrary, not necessarily finite) set of alternatives. The completeness property can however also be modelled, although we rather follow \cite{Aumann} which is more general \cite{Schumm}. Additionally we assume all three relations are disjoint and their set-theoretic union gives a whole universe of alternatives. \par We constructed some positive and negative examples of preference structures; the main aim of the article however is to give the characterization of consumer preference structures in terms of a binary relation, called characteristic relation \cite{Panek}, and to show the way the corresponding structure can be obtained only using this relation. Finally, we show the connection between tournament and total spaces and usual properties of the ordering relations.   },
     MSC2010 = {91B08 03B35},
     SECTION1 = {Preliminaries},
     SECTION2 = {Properties of Binary Relations},
     SECTION3 = {Preference Structures},
     SECTION4 = {Constructing Examples},
     SECTION5 = {Characteristic Relation of a Preference Space},
     SECTION6 = {Generating Preference Space from Arbitrary (Characteristic) Relation},
     SECTION7 = {Flat Preference Spaces},
     SECTION8 = {Tournament Preference Spaces},
     SECTION9 = {Total Preference Spaces},
     EXTERNALREFS = {Arrow;Hallden;Kuzyka;Aumann;Schumm;Panek;},
     INTERNALREFS = {ARROW;CARD_1;DOMAIN_1;ENUMSET1;EQREL_1;FINSET_1;ORDERS_1;ORDINAL1;PARTFUN1;PCS_0;
 RELAT_1;RELAT_2;RELSET_1;SUBSET_1;ZFMISC_1;},
     KEYWORDS = {preferences; preference spaces; social choice; },
     SUBMITTED = {October 7, 2013})

@ARTICLE(ALTCAT_6.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 4}},
    PAGES = {235--239},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0025},
    VERSION = {8.1.02 5.22.1191},
    TITLE = {{C}oproducts in Categories without Uniqueness of {\bf cod} and {\bf dom}},
    AUTHOR = {Goli\'nski, Maciej and Korni{\l}owicz, Artur},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
    ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
    SUMMARY = {The paper introduces coproducts in categories without uniqueness of {\bf cod} and {\bf dom}. It is proven that set-theoretical disjoint union is the coproduct in the category Ens \cite{SEMAD}.   },
    MSC2010 = {18A30 03B35},
    EXTERNALREFS = {SEMAD;},
    INTERNALREFS = {ALTCAT_1;ALTCAT_3;ALTCAT_5;CARD_3;FUNCOP_1;FUNCT_1;FUNCT_2;MSAFREE;PARTFUN1;PBOOLE;RELAT_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {coproducts; disjoined union; },
    SUBMITTED = {December 8, 2013})      
      
@ARTICLE(PETRI_3.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 4}},
    PAGES = {241--247},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0026},
    VERSION = {8.1.02 5.22.1191},
    TITLE = {{F}ormulation of Cell {P}etri Nets},
    AUTHOR = {Jitsukawa, Mitsuru and Kawamoto, Pauline N. and Shidama, Yasunari},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    ACKNOWLEDGEMENT = {We are thankful to Dr. Yatsuka Nakamura. He is the former professor of the Shinshu University. The completion of this article would not have be possible without the deep insight into the automatic proof verification system of Dr. Nakamura. Thank you.},
    SUMMARY = {Based on the Petri net definitions and theorems already formalized in the Mizar article \cite{PETRI.ABS}, in this article we were able to formalize the definition of cell Petri nets. It is based on \cite{Kawamoto-Nakamura:1996}. Colored Petri net has already been defined in \cite{PETRI_2.ABS}. In addition, the conditions of the firing rule and the colored set to this definition, that defines the cell Petri nets are further extended to CPNT.i further. The synthesis of two Petri nets was introduced in \cite{PETRI_2.ABS} and in this work the definition is extended to produce the synthesis of a family of colored Petri nets. Specifically, the extension to a CPNT family is performed by specifying how to link the outbound transitions of each colored Petri net to the place elements of other nets to form a neighborhood relationship. Finally, the activation of colored Petri nets was formalized.   },
    MSC2010 = {68-04 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Synthesis of CPNT and I},
    SECTION3 = {Extension to a Family of Colored Petri Nets},
    SECTION4 = {Definition of Cell Petri Nets},
    SECTION5 = {Activation of Petri Nets},
    EXTERNALREFS = {Kawamoto-Nakamura:1996;},
    INTERNALREFS = {CARD_3;DOMAIN_1;FINSET_1;FUNCT_1;FUNCT_2;FUNCT_4;INT_1;MSAFREE4;NAT_1;ORDINAL1;
PARTFUN1;PBOOLE;PETRI;PETRI_2;RELAT_1;RELSET_2;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {Petri net; system modelling; },
    SUBMITTED = {December 8, 2013})      
      
@ARTICLE(NDIFF_7.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 4}},
    PAGES = {249--260},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0027},
    VERSION = {8.1.02 5.22.1194},
    TITLE = {{I}sometric Differentiable Functions on Real Normed Space},
    ANNOTE = {This work was supported by JSPS KAKENHI 23500029 and 22300285.},
    AUTHOR = {Futa, Yuichi and Endou, Noboru and Shidama, Yasunari},
    ADDRESS1 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
    ADDRESS2 = {Gifu National College of Technology\\Gifu, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we formalize isometric differentiable functions on real normed space \cite{Schwartz:1981}, and their properties.   },
    MSC2010 = {58C20 46G05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Isometries},
    SECTION3 = {Isometric Differentiable Functions on Real Normed Space},
    EXTERNALREFS = {Schwartz:1981;},
    INTERNALREFS = {BINOP_1;FINSEQ_1;FINSET_1;FUNCT_1;FUNCT_2;LOPBAN_1;NDIFF_1;NDIFF_2;NDIFF_5;NFCONT_1;NORMSP_1;
ORDINAL1;PARTFUN1;PRVECT_2;PRVECT_3;RELAT_1;RELSET_1;RLVECT_1;SUBSET_1;VECTSP_1;VFUNCT_1;ZFMISC_1;},
    KEYWORDS = {isometric differentiable function; },
    SUBMITTED = {December 31, 2013})      
      
@ARTICLE(ORDEQ_02.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 4}},
    PAGES = {261--272},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0028},
    VERSION = {8.1.02 5.22.1194},
    TITLE = {{D}ifferential Equations on Functions from {$\mathbb R$} into Real {B}anach Space},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285 and 23500029.},
    AUTHOR = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
    ADDRESS1 = {Hirosaki-city\\Aomori, Japan},
    ADDRESS2 = {Gifu National College of Technology\\Gifu, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we describe the differential equations on functions from $\mathbb R$ into real Banach space. The descriptions are based on the article \cite{ORDEQ_01.ABS}. As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article \cite{NDIFF_4.ABS} and the article \cite{NDIFF_5.ABS}. And applying the theorems of Riemann integral introduced in the article \cite{INTEGR21.ABS}, we proved the ordinary differential equations on real Banach space. We referred to the methods of proof in \cite{Schwartz:1967}.   },
    MSC2010 = {46B99 34A99 03B35},
    SECTION1 = {Some Properties of Differentiable Functions on Real Normed Space},
    SECTION2 = {Differential Equations},
    EXTERNALREFS = {Schwartz:1967;},
    INTERNALREFS = {ABIAN;COMPLEX1;EUCLID;FINSEQ_1;FUNCT_1;FUNCT_2;INTEGR18;INTEGR21;INTEGRA5;INT_1;LOPBAN_1;MEMBERED;NAT_1;NDIFF_1;NDIFF_3;NDIFF_4;NDIFF_5;
NEWTON;NFCONT_1;NFCONT_2;NFCONT_3;NORMSP_1;ORDEQ_01;ORDINAL1;PARTFUN1;PDIFF_1;RAT_1;RCOMP_1;REAL_NS1;RELAT_1;RELSET_1;
RLTOPSP1;RLVECT_1;SUBSET_1;VECTSP_1;ZFMISC_1;},
    KEYWORDS = {formalization of differential equations; },
    SUBMITTED = {December 31, 2013})      
      
@ARTICLE(ZMODUL04.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {21},
    NUMBER = {{\bf 4}},
    PAGES = {273--282},
    YEAR = {2013},
    DOI = {10.2478/forma-2013-0029},
    VERSION = {8.1.02 5.22.1194},
    TITLE = {{S}ubmodule of free {$\mathbb Z$}-module},
    ANNOTE = {This work was supported by JSPS KAKENHI 21240001 and 22300285.},
    AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
    ADDRESS1 = {Japan Advanced Institute of Science and Technology\\Ishikawa, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we formalize a free $\mathbb Z$-module and its property. In particular, we formalize the vector space of rational field corresponding to a free $\mathbb Z$-module and prove formally that submodules of a free $\mathbb Z$-module are free. $\mathbb Z$-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm and cryptographic systems with lattice \cite{LATTICE2002}. Some theorems in this article are described by translating theorems in \cite{RLVECT_5.ABS} into theorems of $\mathbb Z$-module, however their proofs are different.   },
    MSC2010 = {13C10 15A03 03B35},
    SECTION1 = {Vector Space of Rational Field Generated by a Free $\mathbb Z$-module},
    SECTION2 = {Submodule of Free $\mathbb Z$-module},
    EXTERNALREFS = {LATTICE2002},
    INTERNALREFS = {BINOP_1;CARD_1;DOMAIN_1;EQREL_1;FINSEQ_1;FINSET_1;FUNCT_1;FUNCT_2;GAUSSINT;INT_1;INT_2;INT_3;MATRLIN;NAT_1;
ORDINAL1;PARTFUN1;RAT_1;RELAT_1;RELAT_2;RELSET_1;RLVECT_1;RLVECT_5;SUBSET_1;VECTSP_1;VECTSP_6;VECTSP_7;VECTSP_9;
ZFMISC_1;ZMODUL01;ZMODUL02;ZMODUL03;},
    KEYWORDS = {free Z-module; submodule of free Z-module; },
    SUBMITTED = {December 31, 2013}) 

@ARTICLE(POSET_2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 1}},
    PAGES = {1--10},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0001},
    VERSION = {8.1.02 5.22.1199},
    TITLE = {{D}efinition of Flat Poset and Existence Theorems for Recursive Call},
    AUTHOR = {Ishida, Kazuhisa and Shidama, Yasunari and Grabowski, Adam},
    ADDRESS1 = {Neyagawa-shi\\Osaka, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    NOTE2 = {My work was supported by JSPS KAKENHI 22300285.},
    ADDRESS3 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia\l ystok\\Poland},
    SUMMARY = {This text includes the definition and basic notions of product of posets, chain-complete and flat posets, flattening operation, and the existence theorems of recursive call using the flattening operator. First part of the article, devoted to product and flat posets has a purely mathematical quality. Definition 3 allows to construct a flat poset from arbitrary non-empty set \cite{Davey:2002} in order to provide formal apparatus which eanbles to work with recursive calls within the Mizar langauge. To achieve this we extensively use technical Mizar functors like \verb!BaseFunc! or \verb!RecFunc!. The remaining part builds the background for information engineering approach for lists, namely recursive call for posets \cite{Winskel:1993}. We formalized some facts from Chapter 8 of this book as an introduction to the next two sections where we concentrate on binary product of posets rather than on a more general case.   },
    MSC2010 = {06A11 68N30 03B35},
    SECTION1 = {Preliminaries from Poset Theory},
    SECTION2 = {On the Product of Posets},
    SECTION3 = {Definition of Flat Poset and Poset Flattening},
    SECTION4 = {Primaries for Existence Theorems of Recursive Call Using Flattening},
    SECTION5 = {Existence Theorem of Recursive Call for Single-equation},
    SECTION6 = {Existence Theorem of Recursive Calls for 2-equations},
    EXTERNALREFS = {Davey:2002;Winskel:1993;},
    INTERNALREFS = {CARD_1;DOMAIN_1;FINSET_1;FUNCT_1;FUNCT_2;FUNCT_3;LATTICE3;LATTICE7;ORDERS_2;ORDERS_3;
ORDINAL1;PARTFUN1;POSET_1;RELAT_1;RELSET_1;SUBSET_1;YELLOW_0;YELLOW_3;ZFMISC_1;},
    KEYWORDS = {flat posets; recursive calls for posets; flattening operator; },
    SUBMITTED = {February 11, 2014})
    
@ARTICLE(TIETZE_2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 1}},
    PAGES = {11--19},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0002},
    VERSION = {8.1.02 5.22.1199},
    TITLE = {{T}ietze Extension Theorem for $n$-dimensional Spaces},
    ANNOTE = {The paper has been financed by the resources of the Polish National Science Centre granted by decision no DEC-2012/07/N/ST6/02147.},
    AUTHOR = {P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
    SUMMARY = {In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of ${\cal E}^n$ with a non-empty interior. This theorem states that, if $T$ is a normal topological space, $X$ is a closed subset of $T$, and $A$ is a convex compact subset of ${\cal E}^n$ with a non-empty interior, then a continuous function $f:X \rightarrow A$ can be extended to a continuous function $g:T\rightarrow {\cal E}^n$. Additionally we show that a subset $A$ is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex compact subset of ${\cal E}^n$ with a non-empty interior. This article is based on \cite{DUDA:BM61}; \cite{Engelking:1989} and \cite{Engelking:1978} can also serve as reference books.   },
    MSC2010 = {54A05 03B35},
    SECTION1 = {Closed Hypercube},
    SECTION2 = {Properties of the Product of Closed Hypercube},
    SECTION3 = {Tietze Extension Theorem},
    EXTERNALREFS = {DUDA:BM61;Engelking:1989;Engelking:1978;},
    INTERNALREFS = {BINOP_1;BORSUK_1;BORSUK_2;BROUWER;CARD_1;COMPTS_1;CONVEX1;EUCLID;EUCLID_9;FINSEQ_1;FINSEQ_2;FINSET_1;FUNCOP_1;FUNCT_1;
FUNCT_2;FUNCT_6;METRIZTS;ORDINAL1;PARTFUN1;PARTFUN3;PRE_TOPC;RCOMP_1;RELAT_1;RELSET_1;RVSUM_1;SUBSET_1;TOPMETR;
TOPREAL9;TOPREALC;TOPS_1;TOPS_2;T_0TOPSP;VALUED_2;ZFMISC_1;},
    KEYWORDS = {Tietze extension; hypercube; },
    SUBMITTED = {February 11, 2014})

@ARTICLE(BROUWER3.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 1}},
    PAGES = {21--28},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0003},
    VERSION = {8.1.02 5.22.1199},
    TITLE = {{B}rouwer Invariance of Domain Theorem},
    ANNOTE = {The paper has been financed by the resources of the Polish National Science Centre granted by decision no DEC-2012/07/N/ST6/02147.},
    AUTHOR = {P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
    SUMMARY = {In this article we focus on a special case of the Brouwer invariance of domain theorem. Let us $A$, $B$ be a subsets of ${\cal E}^n$, and $f:A \rightarrow B$ be a homeomorphic. We prove that, if $A$ is closed then $f$ transform the boundary of $A$ to the boundary of $B$; and if $B$ is closed then $f$ transform the interior of $A$ to the interior of $B$. These two cases are sufficient to prove the topological invariance of dimension, which is used to prove basic properties of the $n$-dimensional manifolds, and also to prove basic properties of the boundary and the interior of manifolds, e.g. the boundary of an $n$-dimension manifold with boundary is an $(n-1)$-dimension manifold. This article is based on \cite{DUDA:BM61}; \cite{Engelking:1989} and \cite{Engelking:1978} can also serve as reference books.   },
    MSC2010 = {54A05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {A Distribution of Sphere},
    SECTION3 = {A Characterization of Open Sets in Euclidean Space in Terms of Continuous Transformations},
    SECTION4 = {Brouwer Invariance of Domain Theorem -- Special Case},
    SECTION5 = {Topological Invariance of Dimension -- An Introduction to Manifolds},
    EXTERNALREFS = {DUDA:BM61;Engelking:1989;Engelking:1978;},
    INTERNALREFS = {BROUWER;CARD_1;EUCLID;FINSEQ_1;FINSEQ_2;FINSET_1;FUNCOP_1;FUNCT_1;FUNCT_2;INT_1;JORDAN2B;MATRTOP3;MEMBERED;
NAT_1;ORDINAL1;PARTFUN1;PCOMPS_1;PRE_TOPC;PSCOMP_1;RELAT_1;RELSET_1;RLTOPSP1;RLVECT_1;SUBSET_1;TIETZE_2;
TOPDIM_1;TOPMETR;TOPREAL9;TOPREALB;TOPREALC;TOPS_1;TOPS_2;T_0TOPSP;VALUED_2;VECTSP_1;WAYBEL23;ZFMISC_1;},
    KEYWORDS = {continuous transformations; topological dimension; },
    SUBMITTED = {February 11, 2014})

@ARTICLE(PETRI_DF.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 1}},
    PAGES = {29--35},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0004},
    VERSION = {8.1.02 5.22.1199},
    TITLE = {{T}he Formalization of Decision-Free {P}etri Net},
    AUTHOR = {Shah, Pratima K. and Kawamoto, Pauline N. and Giero, Mariusz},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    NOTE2 = {This work was supported in part by JSPS Kakenhi grant number 2230028502.},
    ADDRESS3 = {University of Bia{\l}ystok\\Poland},
    NOTE3 = {This work was supported in part by the University of Bialystok grant BST225 {\it Database of mathematical texts checked by computer}.},
    SUMMARY = {In this article we formalize the definition of Decision-Free Petri Net (DFPN) presented in \cite{Wang:1998}. Then we formalize the concept of directed path and directed circuit nets in Petri nets to prove properties of DFPN. We also present the definition of firing transitions and transition sequences with natural numbers marking that always check whether transition is enabled or not and after firing it only removes the available tokens (i.e., it does not remove from zero number of tokens). At the end of this article, we show that the total number of tokens in a circuit of decision-free Petri net always remains the same after firing any sequences of the transition.   },
    MSC2010 = {68Q60 68Q85 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {The Number of Tokens in a Circuit},
    SECTION3 = {Decision-Free Petri Net Concept and Properties of Circuits in Petri Nets},
    SECTION4 = {Firable and Firing Conditions for Transitions and Transition Sequences with Natural Marking},
    SECTION5 = {The Theorem Stating that the Number of Tokens in a Circuit Remains the Same After any Firing Sequences},
    EXTERNALREFS = {Wang:1998;},
    INTERNALREFS = {CARD_1;FINSEQ_1;FINSEQ_4;FINSEQ_6;FINSET_1;FUNCT_1;FUNCT_2;INT_1;JORDAN23;ORDINAL1;PETRI;RAT_1;
RELAT_1;RFINSEQ;RLAFFIN3;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {specification and verification of discrete systems; Petri net; },
    SUBMITTED = {March 31, 2014})

@ARTICLE(ABSRED_0.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 1}},
    PAGES = {37--56},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0005},
    VERSION = {8.1.02 5.22.1199},
    TITLE = {{A}bstract Reduction Systems and Idea of {K}nuth-{B}endix Completion Algorithm},
    AUTHOR = {Bancerek, Grzegorz},
    ADDRESS1 = {Association of Mizar Users\\Bia\l ystok, Poland},
    SUMMARY = {Educational content for abstract reduction systems concerning reduction, convertibility, normal forms, divergence and convergence, Church-Rosser property, term rewriting systems, and the idea of the Knuth-Bendix Completion Algorithm. The theory is based on \cite{KlopTRS}.   },
    MSC2010 = {68Q42 03B35},
    SECTION1 = {Reduction and Convertibility},
    SECTION2 = {Examples of an Abstract Reduction System},
    SECTION3 = {Normal Forms},
    SECTION4 = {Divergence and Convergence},
    SECTION5 = {Church-Rosser Property},
    SECTION6 = {Term Rewriting Systems},
    SECTION7 = {Idea of Knuth-Bendix Algorithm},
    EXTERNALREFS = {KlopTRS;},
    INTERNALREFS = {CARD_1;ENUMSET1;EQREL_1;FINSEQ_1;FINSEQ_2;FINSEQ_4;FINSET_1;FUNCOP_1;FUNCT_1;FUNCT_2;FUNCT_7;MARGREL1;NAT_1;
ORDINAL1;PARTFUN1;PUA2MSS1;RELAT_1;RELSET_1;REWRITE1;SUBSET_1;UNIALG_1;ZFMISC_1;},
    KEYWORDS = {abstract reduction systems; Knuth-Bendix algorithm; },
    SUBMITTED = {March 31, 2014})

@ARTICLE(DBLSEQ_2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 1}},
    PAGES = {57--68},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0006},
    VERSION = {8.1.03 5.23.1204},
    TITLE = {{D}ouble Series and Sums},
    ANNOTE = {This work was supported by JSPS KAKENHI 23500029.},
    AUTHOR = {Endou, Noboru},
    ADDRESS1 = {Gifu National College of Technology\\Gifu, Japan},
    SUMMARY = {In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper \cite{DBLSEQ_1.ABS}. In section 1 we introduce double series and their convergence. Then we show the relationship between Pringsheim-type convergence and iterated convergence. In section 2 we study double series having non-negative terms. As a result, we have equality of three type sums of non-negative double sequence. In section 3 we show that if a non-negative sequence is summable, then the squence of rearrangement of terms is summable and it has the same sums. In the last section two basic relations between double sequences and matrices are introduced.   },
    MSC2010 = {40A05 40B05 03B35},
    SECTION1 = {Double Series and their Convergence},
    SECTION2 = {Double Series of Non-Negative Double Sequence},
    SECTION3 = {Summability for Rearrangements of Non-Negative Real Sequence},
    SECTION4 = {Basic Relations between Double Sequences and Matrices},
    INTERNALREFS = {BINOP_1;CARD_1;CLASSES1;DBLSEQ_1;FINSEQ_1;FINSEQ_2;FINSET_1;FUNCT_1;FUNCT_2;GLIB_003;INT_1;
MATRIX_0;MATRPROB;MESFUNC9;NAT_1;ORDINAL1;PARTFUN1;PARTFUN3;RAT_1;RELAT_1;RELSET_1;RVSUM_1;
SEQ_2;SERIES_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {double series; },
    SUBMITTED = {March 31, 2014})

@ARTICLE(DUALSP01.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
      VOLUME = {22},
    NUMBER = {{\bf 1}},
    PAGES = {69--77},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0007},
    VERSION = {8.1.03 5.23.1204},
    TITLE = {{D}ual Spaces and {H}ahn-{B}anach Theorem},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285, 23500029.},
    AUTHOR = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
    ADDRESS1 = {Hirosaki-city\\Aomori, Japan},
    ADDRESS2 = {Gifu National College of Technology\\Gifu, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article \cite{LOPBAN_1.ABS}. Finally, applying theorems of the second section, we proved the Hahn-Banach extension theorem in real normed spaces. We have used extensively used \cite{HAHNBAN.ABS}.   },
    MSC2010 = {46A22 46E15 03B35},
    SECTION1 = {Dual Spaces of Real Linear Spaces},
    SECTION2 = {Dual Spaces of Real Normed Spaces},
    SECTION3 = {Hahn-Banach Extension Theorem},
    INTERNALREFS = {COMPLEX1;COMSEQ_2;FUNCOP_1;FUNCSDOM;FUNCT_1;FUNCT_2;HAHNBAN;HAHNBAN1;LOPBAN_1;MEMBERED;MONOID_0;NAT_1;NORMSP_1;
ORDINAL1;PARTFUN1;REALSET1;RELAT_1;RLSUB_1;RLVECT_1;RSSPACE;RSSPACE3;SEQ_2;SEQ_4;SUBSET_1;VECTSP_1;ZFMISC_1;},
    KEYWORDS = {dual space; Hahn-Banach extension; },
    SUBMITTED = {March 31, 2014})

@ARTICLE(SRINGS_1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 1}},
    PAGES = {79--84},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0008},
    VERSION = {8.1.03 5.23.1204},
    TITLE = {{S}emiring of Sets},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
    SUMMARY = {Schmets \cite{Schmets:2004} has developed a measure theory from a generalized notion of a semiring of sets. Goguadze \cite{GOGUADZE:2003} has introduced another generalized notion of semiring of sets and proved that all known properties that semiring have according to the old definitions are preserved. We show that this two notions are almost equivalent. We note that Patriota \cite{2011arXiv1103.6166P} has defined this quasi-semiring. We propose the formalization of some properties developed by the authors.   },
    MSC2010 = {28A05 03E02 03E30 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {The Existence of Partitions},
    SECTION3 = {Partitions in a Difference of Sets},
    SECTION4 = {Countable Covers},
    SECTION5 = {Semiring of Sets},
    SECTION6 = {A Ring of Sets},
    EXTERNALREFS = {Schmets:2004;GOGUADZE:2003;2011arXiv1103.6166P;},
    INTERNALREFS = {CARD_1;CARD_3;CLASSES1;EQREL_1;FINSEQ_1;FINSET_1;FINSUB_1;FUNCT_1;FUNCT_2;INT_1;LATTICE7;
MEMBERED;NAT_LAT;ORDINAL1;PARTIT1;RAT_1;RELAT_1;RELSET_1;SETFAM_1;SUBSET_1;TAXONOM2;TEX_1;ZFMISC_1;},
    KEYWORDS = {sets; set partitions; distributive lattice; },
    SUBMITTED = {March 31, 2014})

@ARTICLE(SRINGS_2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 1}},
    PAGES = {85--88},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0009},
    VERSION = {8.1.03 5.23.1204},
    TITLE = {{S}emiring of Sets: Examples},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
    SUMMARY = {This article proposes the formalization of some examples of semiring of sets proposed by Goguadze \cite{GOGUADZE:2003} and Schmets \cite{Schmets:2004}.   },
    MSC2010 = {28A05 03E02 03E30 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Ordinary Examples of Semirings of Sets},
    SECTION3 = {Numerical Example},
    EXTERNALREFS = {GOGUADZE:2003;Schmets:2004;},
    INTERNALREFS = {CARD_3;ENUMSET1;FINSET_1;FINSUB_1;MCART_1;MEASURE5;MEMBERED;PROB_1;RCOMP_1;RELAT_1;SETFAM_1;SRINGS_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {semiring of sets; },
    SUBMITTED = {March 31, 2014})

@ARTICLE(ROUGHS_4.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 1}},
    PAGES = {89--97},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0010},
    VERSION = {8.1.03 5.23.1204},
    TITLE = {{T}opological Interpretation of Rough Sets},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia\l ystok\\Poland},
    SUMMARY = {Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of \cite{ROUGHS_1.ABS}, we characterize rough sets in terms of topological closure and interior, as the approximations have the properties of the Kuratowski operators. We decided to merge topological spaces with tolerance approximation spaces. As a testbed for our developed approach, we restated the results of Isomichi \cite{ISOMICHI} (formalized in Mizar in \cite{ISOMICHI.ABS}) and about fourteen sets of Kuratowski \cite{KURAT:4} (encoded with the help of Mizar adjectives and clusters' registrations in \cite{KURATO_1.ABS}) in terms of rough approximations. The upper bounds which were 14 and 7 in the original paper of Kuratowski, in our case are six and three, respectively. \par It turns out that within the classification given by Isomichi, $1^{\rm st}$ class subsets are precisely crisp sets, $2^{\rm nd}$ class subsets are proper rough sets, and there are no $3^{\rm rd}$ class subsets in topological spaces generated by approximations. Also the important results about these spaces is that they are extremally disconnected \cite{TDLAT_3.ABS}, hence lattices of their domains are Boolean. \par Furthermore, we develop the theory of abstract spaces equipped with maps possessing characteristic properties of rough approximations which enables us to freely use the notions from the theory of rough sets and topological spaces formalized in the Mizar Mathematical Library \cite{GrabowskiFI:2013}.   },
    MSC2010 = {54H10 68T37 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Ordinary Properties of Maps},
    SECTION3 = {Structural Part},
    SECTION4 = {Merging with Topologies},
    SECTION5 = {Introducing Rough Sets},
    SECTION6 = {On Sequential Closure and Frechet Spaces},
    SECTION7 = {Connections between Closures and Approximations},
    SECTION8 = {Isomichi Results Reuse},
    SECTION9 = {Reexamination of Kuratowski's 14 Sets for Approximation Spaces},
    EXTERNALREFS = {GrabowskiFI:2013;ISOMICHI;KURAT:4;},
    INTERNALREFS = {CARD_1;ENUMSET1;FINSEQ_1;FINSET_1;FINSUB_1;FRECHET;FRECHET2;FUNCT_1;FUNCT_2;ISOMICHI;KURATO_1;ORDERS_2;ORDINAL1;
PARTFUN1;PRE_TOPC;RELAT_1;ROUGHS_1;ROUGHS_2;SETFAM_1;SUBSET_1;TDLAT_3;TOPS_1;WAYBEL_9;ZFMISC_1;},
    KEYWORDS = {rough sets; rough approximations; Kuratowski closure-complement problem; topological spaces; },
    SUBMITTED = {March 31, 2014})
    
@ARTICLE(HILBERT4.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 2}},
    PAGES = {99--103},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0011},
    VERSION = {8.1.03 5.23.1207},
    TITLE = {{P}seudo-Canonical Formulae are Classical},
    AUTHOR = {Caminati, Marco B. and Korni{\l}owicz, Artur},
    ADDRESS1 = {School of Computer Science\\University of Birmingham\\Birmingham, B15 2TT\\United Kingdom},
    NOTE1 = {My work has been partially supported by EPSRC grant EP/J007498/1 and an LMS Computer Science Small Grant.},
    ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
    SUMMARY = {An original result about Hilbert Positive Propositional Calculus introduced in \cite{HILBERT1.ABS} is proven. That is, it is shown that the pseudo-canonical formulae of that calculus (and hence also the canonical ones, see \cite{HILBERT3.ABS}) are a subset of the classical tautologies.   },
    MSC2010 = {03B20 03B35},
    SECTION1 = {Preliminaries about Injectivity, Involutiveness, Fixed Points},
    SECTION2 = {Facts about Perm's Fixed Points},
    SECTION3 = {Axiom of Choice in Functional Form via the Fraenkel Operator},
    SECTION4 = {Building a Suitable Set Valuation and a Suitable Permutation of It},
    SECTION5 = {Classical Semantics via {\rm SetVal$_{0}$}, an Extension of {\rm SetVal}},
    INTERNALREFS = {ABIAN;CARD_1;DOMAIN_1;FINSET_1;FOMODEL0;FUNCOP_1;FUNCT_1;FUNCT_2;FUNCT_3;FUNCT_4;HILBERT1;HILBERT2;
HILBERT3;ORDINAL1;PARTFUN1;PARTIT_2;RELAT_1;RELAT_2;RELSET_1;SETFAM_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {Hilbert positive propositional calculus; classical logic; canonical formulae; },
    SUBMITTED = {May 25, 2014})

@ARTICLE(LAGRA4SQ.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 2}},
    PAGES = {105--110},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0012},
    VERSION = {8.1.03 5.23.1207},
    TITLE = {{L}agrange's Four-Square Theorem},
    AUTHOR = {Watase, Yasushige},
    ADDRESS1 = {Suginami-ku Matsunoki 6\\3-21 Tokyo, Japan},
    SUMMARY = {This article provides a formalized proof of the so-called ``the four-square theorem", namely any natural number can be expressed by a sum of four squares, which was proved by Lagrange in 1770. An informal proof of the theorem can be found in the number theory literature, e.g. in \cite{HardyWright}, \cite{Baker:1984} or \cite{Wada:1981}. \par This theorem is item {\tt{\#19}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} .   },
    MSC2010 = {11P99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Any Prime Number can be Expressed as a Sum of Four Squares},
    SECTION3 = {Proof of Lagrange's theorem},
    EXTERNALREFS = {HardyWright;Baker:1984;Wada:1981;},
    INTERNALREFS = {ABIAN;CARD_1;COMPLEX1;FINSEQ_1;FINSET_1;FUNCT_1;FUNCT_2;INT_1;INT_2;NAT_1;
NAT_3;NEWTON;ORDINAL1;PARTFUN1;PEPIN;RELAT_1;RELSET_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {Lagrange's four-square theorem; },
    SUBMITTED = {June 4, 2014})

@ARTICLE(NAT_6.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 2}},
    PAGES = {111--118},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0013},
    VERSION = {8.1.03 5.23.1207},
    TITLE = {{P}roth Numbers},
    AUTHOR = {Schwarzweller, Christoph},
    ADDRESS1 = {WSB Schools of Banking\\Gda\'nsk, Poland},
    SUMMARY = {In this article we introduce Proth numbers and prove two theorems on such numbers being prime \cite{Buchmann:1992}. We also give revised versions of Pocklington's theorem and of the Legendre symbol. Finally, we prove Pepin's theorem and that the fifth Fermat number is not prime.   },
    MSC2010 = {11A41 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Some Properties of Congruences and Prime Numbers},
    SECTION3 = {Some basic properties of relation ``>"},
    SECTION4 = {Pocklington's Theorem Revisited},
    SECTION5 = {Euler's Criterion},
    SECTION6 = {Proth Numbers},
    SECTION7 = {Fermat Numbers},
    SECTION8 = {Cullen Numbers},
    EXTERNALREFS = {Buchmann:1992;},
    INTERNALREFS = {ABIAN;EC_PF_2;GROUP_1;INT_1;INT_2;INT_5;INT_7;NAT_1;NEWTON;ORDINAL1;PEPIN;RAT_1;SQUARE_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {prime numbers; Pocklington's theorem; Proth's theorem; Pepin's theorem; },
    SUBMITTED = {June 9, 2014})

@ARTICLE(BALLOT_1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 2}},
    PAGES = {119--123},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0014},
    VERSION = {8.1.03 5.23.1210},
    TITLE = {{B}ertrand's Ballot Theorem},
    ANNOTE = {The paper has been financed by the resources of the Polish National Science Centre granted by decision no DEC-2012/07/N/ST6/02147.},
    AUTHOR = {P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
    SUMMARY = {In this article we formalize the Bertrand's Ballot Theorem based on \cite{BALLOT}. Suppose that in an election we have two candidates: $A$ that receives $n$ votes and $B$ that receives $k$ votes, and additionally $n\geq k$. Then this theorem states that the probability of the situation where $A$ maintains more votes than $B$ throughout the counting of the ballots is equal to $(n-k)/(n+k)$. \par This theorem is item {\tt{\#30}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} .   },
    MSC2010 = {60C05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Properties of Elections},
    SECTION3 = {Properties of Dominated Elections},
    SECTION4 = {Main Theorem},
    EXTERNALREFS = {BALLOT;},
    INTERNALREFS = {AFINSQ_1;CARD_1;CARD_FIN;CATALAN2;DOMAIN_1;FINSEQ_1;FINSEQ_2;FINSET_1;FUNCOP_1;FUNCT_1;NEWTON;
ORDINAL1;PARTFUN3;RELAT_1;RELSET_1;RPR_1;RVSUM_1;SETFAM_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {ballot theorem; probability; },
    SUBMITTED = {June 13, 2014})

@ARTICLE(MSAFREE5.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 2}},
    PAGES = {125--155},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0015},
    VERSION = {8.1.03 5.23.1210},
    TITLE = {{T}erm Context},
    AUTHOR = {Bancerek, Grzegorz},
    ADDRESS1 = {Association of Mizar Users\\Bia{\l}ystok, Poland},
    SUMMARY = {Two construction functors: simple term with a variable and compound term with an operation and argument terms and schemes of term induction are introduced. The degree of construction as a number of used operation symbols is defined. Next, the term context is investigated. An $x$-context is a term which includes a variable $x$ once only. The compound term is $x$-context iff the argument terms include an $x$-context once only. The context induction is shown and used many times. As a key concept, the context substitution is introduced. Finally, the translations and endomorphisms are expressed by context substitution.   },
    MSC2010 = {08A35 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Constructing Terms},
    SECTION3 = {Construction Degree},
    SECTION4 = {Context},
    SECTION5 = {Context vs. Translations},
    SECTION6 = {Context vs. Endomorphism},
    INTERNALREFS = {ABCMIZ_1;CARD_1;CARD_2;CARD_3;CATALG_1;DOMAIN_1;DTCONSTR;FINSEQ_1;FINSEQ_2;FINSEQ_4;FINSET_1;FUNCOP_1;FUNCT_1;FUNCT_2;
FUNCT_4;FUNCT_6;FUNCT_7;INSTALG1;INT_1;MATRIX_7;MCART_1;MEMBERED;MSAFREE;MSAFREE1;MSAFREE3;MSAFREE4;MSATERM;MSUALG_1;
MSUALG_3;MSUALG_4;MSUALG_6;NAT_1;ORDINAL1;ORDINAL6;PARTFUN1;PBOOLE;RAT_1;RELAT_1;RELSET_1;REWRITE1;SEQ_4;SUBSET_1;
TREES_1;TREES_2;TREES_3;TREES_4;TREES_9;WSIERP_1;ZFMISC_1;},
    KEYWORDS = {construction degree; context; translation; endomorphism; },
    SUBMITTED = {June 13, 2014})

@ARTICLE(RVSUM_3.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 2}},
    PAGES = {157--166},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0016},
    VERSION = {8.1.03 5.23.1210},
    TITLE = {{C}auchy Mean Theorem},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia\l ystok\\Poland},
    SUMMARY = {The purpose of this paper was to prove formally, using the Mizar language, Arithmetic Mean/Geometric Mean theorem known maybe better under the name of AM-GM inequality or Cauchy mean theorem. It states that the arithmetic mean of a list of a non-negative real numbers is greater than or equal to the geometric mean of the same list. \par The formalization was tempting for at least two reasons: one of them, perhaps the strongest, was that the proof of this theorem seemed to be relatively easy to formalize (e.g. the weaker variant of this was proven in \cite{SERIES_3.ABS}). Also Jensen's inequality is already present in the Mizar Mathematical Library. We were impressed by the beauty and elegance of the simple proof by induction and so we decided to follow this specific way. \par The proof follows similar lines as that written in Isabelle \cite{Cauchy-AFP}; the comparison of both could be really interesting as it seems that in both systems the number of lines needed to prove this are really close. \par This theorem is item {\tt{\#38}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} .   },
    MSC2010 = {26A06 26A12 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Arithmetic Mean and Geometric Mean},
    SECTION3 = {Heterogeneity of a Finite Sequence},
    SECTION4 = {Auxiliary Replacement Function},
    SECTION5 = {Homogenization of a Finite Sequence},
    SECTION6 = {Cauchy Mean Theorem},
    EXTERNALREFS = {Cauchy-AFP;},
    INTERNALREFS = {CARD_1;CLASSES1;COMPLEX1;FINSEQ_1;FINSET_1;FUNCOP_1;FUNCT_1;FUNCT_7;INT_1;MEMBERED;NAT_1;
NEWTON;ORDINAL1;PARTFUN3;POWER;RAT_1;RELAT_1;RFINSEQ;RVSUM_1;SERIES_3;SQUARE_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {geometric mean; arithmetic mean; AM-GM inequality; Cauchy mean theorem; },
    SUBMITTED = {June 13, 2014})

@ARTICLE(GTARSKI1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 2}},
    PAGES = {167--176},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0017},
    VERSION = {8.1.03 5.23.1213},
    TITLE = {{T}arski Geometry Axioms},
    AUTHOR = {Richter, William and Grabowski, Adam and Alama, Jesse},
    ADDRESS1 = {Departament of Mathematics\\Nortwestern University\\Evanston, USA},
    ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia\l ystok\\Poland},
    ADDRESS3 = {Technical University of Vienna\\ Austria},
    SUMMARY = {This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski \cite{TarskiGivant}, and we hope to continue this work.\footnote{The first author ported the code to HOL Light (\url{http://www.cl.cam.ac.uk/~jrh13/hol-light/}), which can be found in any recent subversion of HOL Light as {\tt hol\_light/RichterHilbertAxiomGeometry/TarskiAxiomGeometry\_read.ml}} \par The article is an extension and upgrading of the source code written by the first author with the help of \verb!miz3! tool; his primary goal was to use proof checkers to help teach rigorous axiomatic geometry in high school using Hilbert's axioms. \par This is largely a Mizar port of Julien Narboux's Coq pseudo-code \cite{Narboux:2007}. We partially prove the theorem of \cite{Schwabhauser:1983} that Tarski's (extremely weak!) plane geometry axioms imply Hilbert's axioms. Specifically, we obtain Gupta's amazing proof which implies Hilbert's axiom \verb!I1! that two points determine a line. \par The primary Mizar coding was heavily influenced by \cite{INCSP_1.ABS} on axioms of incidence geometry. The original development was much improved using Mizar adjectives instead of predicates only, and to use this machinery in full extent, we have to construct some models of Tarski geometry. These are listed in the second section, together with appropriate registrations of clusters. Also models of Tarski's geometry related to real planes were constructed.   },
    MSC2010 = {51A05 51M04 03B35},
    SECTION1 = {Tarski's Geometry Axioms},
    SECTION2 = {Existence Proofs for Tarski Plane},
    SECTION3 = {Proofs of Basic Properties},
    SECTION4 = {Construction of the Euclidean Example},
    EXTERNALREFS = {TarskiGivant;Narboux:2007;Schwabhauser:1983;},
    INTERNALREFS = {FUNCT_1;FUNCT_2;INCSP_1;METRIC_1;ORDINAL1;RELAT_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {Tarski's geometry axioms; foundations of geometry; incidence geometry; },
    SUBMITTED = {June 16, 2014})

@ARTICLE(GRAPH_3A.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 2}},
    PAGES = {177--178},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0018},
    VERSION = {8.1.03 5.23.1213},
    TITLE = {{A} Note on the Seven Bridges of {K}\"onigsberg Problem},
    AUTHOR = {Naumowicz, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
    SUMMARY = {In this paper we account for the formalization of the seven bridges of K\"onigsberg puzzle. The problem originally posed and solved by Euler in 1735 is historically notable for having laid the foundations of graph theory, cf. \cite{Chartrand:1985}. Our formalization utilizes a simple set-theoretical graph representation with four distinct sets for the graph's vertices and another seven sets that represent the edges (bridges). The work appends the article by Nakamura and Rudnicki \cite{GRAPH_3.ABS} by introducing the classic example of a graph that does not contain an Eulerian path. \par This theorem is item {\tt{\#54}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} .   },
    MSC2010 = {05C45 05C62 03B35},
    EXTERNALREFS = {Chartrand:1985;},
    INTERNALREFS = {ENUMSET1;FINSEQ_1;FINSET_1;FUNCT_1;FUNCT_2;GRAPH_1;GRAPH_3;MSSCYC_1;ORDINAL1;RELAT_1;SUBSET_1;ZFMISC_1;},
    KEYWORDS = {Eulerian paths; Eulerian cycles; K\"onigsberg bridges problem; },
    SUBMITTED = {June 16, 2014})

@ARTICLE(MFOLD_0.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 2}},
    PAGES = {179--186},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0019},
    VERSION = {8.1.03 5.23.1213},
    TITLE = {{T}opological Manifolds},
    AUTHOR = {P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
    SUMMARY = {Let  us recall that a topological space $M$ is a topological manifold if $M$ is second-countable Hausdorff and locally Euclidean, i.e. each point has a neighborhood that is homeomorphic to an open ball of ${\cal E}^n$ for some $n$. However, if we would like to consider a topological manifold with a boundary, we have to extend this definition. Therefore, we introduce here the concept of a locally Euclidean space that covers both cases (with and without a boundary), i.e. where each point has a neighborhood that is homeomorphic to a closed ball of ${\cal E}^n$ for some $n$. \par Our purpose is to prove, using the Mizar formalism, a number of properties of such locally Euclidean spaces and use them to demonstrate basic properties of a manifold. Let $T$ be a locally Euclidean space. We prove that every interior point of $T$ has a neighborhood homeomorphic to an open ball and that every boundary point of $T$ has a neighborhood homeomorphic to a closed ball, where additionally this point is transformed into a point of the boundary of this ball. When $T$ is $n$-dimensional, i.e. each point of $T$ has a neighborhood that is homeomorphic to a closed ball of ${\cal E}^n$, we show that the interior of $T$ is a locally Euclidean space without boundary of dimension $n$ and the boundary of $T$ is a locally Euclidean space without boundary of dimension $n-1$. Additionally, we show that every connected component of a compact locally Euclidean space is a locally Euclidean space of some dimension. We prove also that the Cartesian product of locally Euclidean spaces also forms a locally Euclidean space. We determine the interior and boundary of this product and show that its dimension is the sum of the dimensions of its factors. At the end, we present several consequences of these results for topological manifolds. This article is based on \cite{ENGEL:BM51}.   },
    MSC2010 = {57N15 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Locally Euclidean Spaces},
    SECTION3 = {Locally Euclidean Spaces With and Without a Boundary},
    SECTION4 = {Interior and Boundary of Locally Euclidean Spaces},
    SECTION5 = {Cartesian Product of Locally Euclidean Spaces},
    SECTION6 = {Fixed Dimension Locally Euclidean Spaces},
    SECTION7 = {Connected Components of Locally Euclidean Spaces},
    SECTION8 = {Topological Manifold},
    EXTERNALREFS = {ENGEL:BM51;},
    INTERNALREFS = {BORSUK_1;BORSUK_3;BROUWER;CARD_1;COMPTS_1;CONNSP_1;CONNSP_2;EQREL_1;EUCLID;FINSEQ_1;FINSET_1;FUNCT_1;FUNCT_2;
MFOLD_1;NAT_1;ORDINAL1;PARTFUN1;PRE_TOPC;RELAT_1;SETFAM_1;SUBSET_1;TOPREAL9;TOPS_1;TOPS_2;WAYBEL23;ZFMISC_1;},
    KEYWORDS = {locally Euclidean spaces; interior; boundary; Cartesian product; },
    SUBMITTED = {June 16, 2014})

@MISC(PREFACE_22_2_2014,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 2}},
    PAGES = {i--iv},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0020},
    TITLE = {Preface},
    AUTHOR = {Grabowski, Adam and Shidama, Yasunari},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia\l ystok\\Poland},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    KEYWORDS = {Mizar; Mizar Mathematical Library; },
    SUBMITTED = {June 1, 2014})

@ARTICLE(ZMODUL05.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 3}},
    PAGES = {189--198},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0021},
    VERSION = {8.1.03 5.24.1215},
    TITLE = {{R}ank of Submodule, Linear Transformations and Linearly Independent Subsets of {$\mathbb{Z}$}-module},
    ANNOTE = {This work was supported by JSPS KAKENHI 21240001 and 22300285.},
    AUTHOR = {Nakasho, Kazuhisa and Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
      ADDRESS2 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      ADDRESS4 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, we formalize some basic facts of $\mathbb Z$-module. In the first section, we discuss the rank of submodule of $\mathbb Z$-module and its properties. Especially, we formally prove that the rank of any $\mathbb Z$-module is equal to or more than that of its submodules, and vice versa, and that there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts of linear transformations between two $\mathbb Z$-modules. In this section, we define homomorphism between two $\mathbb Z$-modules and deal with kernel and image of homomorphism. In the last section, we formally prove some basic facts about linearly independent subsets and linear combinations. These formalizations are based on \cite{BourbakiAlgI}(p.191-242), \cite{lang-algebra}(p.117-172) and \cite{atiyah1969introduction}(p.17-35).   },
    MSC2010 = {13C10 15A04 03B35},
    SECTION1 = {Rank of Submodule of $\mathbb Z$-module},
    SECTION2 = {Basic Facts of Linear Transformations},
    SECTION3 = {Some Basic Facts about Linearly Independent Subsets and Linear Combinations},
    EXTERNALREFS = {BourbakiAlgI; lang-algebra; atiyah1969introduction; },
    INTERNALREFS = {CARD_1.ABS;DOMAIN_1.ABS;FINSEQOP.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_4.ABS;
FUNCT_7.ABS;GR_CY_1.ABS;INT_1.ABS;MEMBERED.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RELAT_1.ABS;RELSET_1.ABS;RLVECT_1.ABS;RVSUM_1.ABS;
SUBSET_1.ABS;VECTSP_1.ABS;ZFMISC_1.ABS;ZMODUL01.ABS;ZMODUL02.ABS;ZMODUL03.ABS;},
    KEYWORDS = {free Z-module; rank of Z-module; homomorphism of Z-module; linearly independent; linear combination; },
    SUBMITTED = {July 10, 2014})
    
@ARTICLE(FINANCE2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 3}},
    PAGES = {199--204},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0022},
    VERSION = {8.1.03 5.24.1215},
    TITLE = {{E}vents of {B}orel Sets, Construction of {B}orel Sets and Random Variables for Stochastic Finance},
    AUTHOR = {Jaeger, Peter},
    ADDRESS1 = {Siegmund-Schacky-Str. 18a\\80993 Munich, Germany},
    SUMMARY = {We consider special events of Borel sets with the aim to prove, that the set of the irrational numbers is an event of the Borel sets. The set of the natural numbers, the set of the integer numbers and the set of the rational numbers are countable, so we can use the literature \cite{analysis1:2001} (pp. 78-81) as a basis for the similar construction of the proof. Next we prove, that different sets can construct the Borel sets \cite{klenke:2006} (pp. 9-10). Literature \cite{klenke:2006} (pp. 9-10) and \cite{georgii:2004} (pp. 11-12) gives an overview, that there exists some other sets for this construction. Last we define special functions as random variables for stochastic finance in discrete time. The relevant functions are implemented in the article \cite{FINANCE1.ABS}, see \cite{follmerschied:2004} (p. 4). The aim is to construct events and random variables, which can easily be used with a probability measure. See as an example theorems (10) and (14) in \cite{RANDOM_3.ABS}. Then the formalization is more similar to the presentation used in the book \cite{follmerschied:2004}. As a background, further literatures is \cite{bosch:2008} (pp. 9-12), \cite{analysisheuser:2003} (pp. 17-20), and \cite{linalgfischer:2002} (pp.32-35).   },
    MSC2010 = {28A05 03E30 03B35},
    SECTION1 = {Events of Borel Sets},
    SECTION2 = {Construction of Borel Sets},
    SECTION3 = {Random Variables for Stochastic Finance in Discrete Time},
    EXTERNALREFS = {analysis1:2001; klenke:2006; georgii:2004; follmerschied:2004; bosch:2008; analysisheuser:2003; 
    linalgfischer:2002; },
    INTERNALREFS = {BORSUK_5.ABS;FINANCE1.ABS;FINSUB_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;INT_1.ABS;MEMBERED.ABS;NAT_1.ABS;ORDINAL1.ABS;
PARTFUN1.ABS;POWER.ABS;PROB_1.ABS;PROB_3.ABS;RANDOM_3.ABS;RAT_1.ABS;RCOMP_1.ABS;REAL_1.ABS;RELAT_1.ABS;SEQ_4.ABS;SERIES_1.ABS;
SETFAM_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {event; Borel set; random variable; },
    SUBMITTED = {July 10, 2014})
    
@ARTICLE(NEWTON01.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 3}},
    PAGES = {205--208},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0023},
    VERSION = {8.1.03 5.25.1215},
    TITLE = {{S}ome Remarkable Identities Involving Numbers},
    AUTHOR = {Ziobro, Rafa{\l}},
    ADDRESS1 = {Department of Carbohydrate Technology\\University of Agriculture\\Krakow, Poland},
    ACKNOWLEDGEMENT = {Ad Maiorem Dei Gloriam.},
    SUMMARY = {The article focuses on simple identities found for binomials, their divisibility, and basic inequalities. A general formula allowing factorization of the sum of like powers is introduced and used to prove elementary theorems for natural numbers. \par Formulas for short multiplication are sometimes referred in English or French as remarkable identities. The same formulas could be found in works concerning polynomial factorization, where there exists no single term for various identities. Their usability is not questionable, and they have been successfully utilized since for ages. For example, in his books published in 1731 (p. 385), Edward Hatton \cite{Hatton1731} wrote: ``Note, that the differences of any two like powers of two quantities, will always be divided by the difference of the quantities without any remainer...". \par Despite of its conceptual simplicity, the problem of factorization of sums/diffe\-ren\-ces of two like powers could still be analyzed \cite{Nowak1998}, giving new and possibly interesting results \cite{Mostafa2005}.   },
    MSC2010 = {11A67 03B35},
    EXTERNALREFS = {Hatton1731; Nowak1998; Mostafa2005; },
    INTERNALREFS = {ABIAN.ABS;INT_1.ABS;INT_2.ABS;NEWTON.ABS;ORDINAL1.ABS;},
    KEYWORDS = {identity; divisibility; inequations; powers; },
    SUBMITTED = {September 5, 2014})

@ARTICLE(NORMSP_3.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 3}},
    PAGES = {209--223},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0024},
    VERSION = {8.1.03 5.25.1220},
    TITLE = {{T}opological Properties of Real Normed Space},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285.},
    AUTHOR = {Nakasho, Kazuhisa and Futa, Yuichi and Shidama, Yasunari},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
    ADDRESS2 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered here. Then we argue the condition when real normed subspaces become Banach's spaces. We also formalize quotient vector space. In the last session, we argue the properties of the closure of real normed space. These formalizations are based on \cite{rudin1991functional}(p.3-41), \cite{bourbaki1987elements} and \cite{yoshida:1980}(p.3-67).   },
    MSC2010 = {46B20 46A19 03B35},
    SECTION1 = {Open and Closed},
    SECTION2 = {Density},
    SECTION3 = {Separability},
    SECTION4 = {Sequence and Convergence},
    SECTION5 = {Subspace},
    SECTION6 = {Linear Functions},
    SECTION7 = {Banach Space},
    SECTION8 = {Quotient Vector Space},
    SECTION9 = {Closure},
    EXTERNALREFS = {rudin1991functional; bourbaki1987elements; yoshida:1980; },
    INTERNALREFS = {COMPLEX1.ABS;DOMAIN_1.ABS;DUALSP01.ABS;FUNCT_1.ABS;FUNCT_2.ABS;LOPBAN_1.ABS;MEMBERED.ABS;NFCONT_1.ABS;NORMSP_1.ABS;
NORMSP_2.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PRE_TOPC.ABS;RELAT_1.ABS;RELSET_1.ABS;RLSUB_1.ABS;RLVECT_1.ABS;RLVECT_3.ABS;RSSPACE.ABS;
RSSPACE3.ABS;SEQ_4.ABS;SETFAM_1.ABS;SUBSET_1.ABS;TOPGEN_1.ABS;TOPS_1.ABS;VECTSP10.ABS;VECTSP_1.ABS;VECTSP_4.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {functional analysis; normed linear space; topological vector space; },
    SUBMITTED = {September 15, 2014})

@ARTICLE(AOFA_L00.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 3}},
    PAGES = {225--255},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0025},
    VERSION = {7.11.07 4.160.1126},
    TITLE = {{A}lgebraic Approach to Algorithmic Logic},
    AUTHOR = {Bancerek, Grzegorz},
    ADDRESS1 = {Association of Mizar Users\\Bia\l ystok, Poland},
    SUMMARY = {We introduce algorithmic logic -- an algebraic approach according to \cite{Salwicki}. It is done in three stages: propositional calculus, quantifier calculus with equality, and finally proper algorithmic logic. For each stage appropriate signature and theory are defined. Propositional calculus and quantifier calculus with equality are explored according to \cite{Mendelson}. A language is introduced with language signature including free variables, substitution, and equality. Algorithmic logic requires a bialgebra structure which is an extension of language signature and program algebra. While-if algebra of generator set and algebraic signature is bialgebra with appropriate properties and is used as basic type of algebraic logic.   },
    MSC2010 = {03B05 03B10 03B35},
    SECTION1 = {Algorithmic Langugage Signature},
    SECTION2 = {Language},
    SECTION3 = {Algorithmic Theory},
    SECTION4 = {Propositional Calculus},
    SECTION5 = {Quantifier Calculus},
    SECTION6 = {Algorithmic Logic},
    EXTERNALREFS = {Salwicki; Mendelson; },
    INTERNALREFS = {AOFA_000.ABS;AOFA_A00.ABS;BINOP_1.ABS;CARD_1.ABS;CARD_3.ABS;CATALG_1.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSET_1.ABS;
FREEALG.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_7.ABS;INSTALG1.ABS;MARGREL1.ABS;MSAFREE.ABS;MSUALG_1.ABS;ORDINAL1.ABS;
PARTFUN1.ABS;PBOOLE.ABS;RELAT_1.ABS;SUBSET_1.ABS;UNIALG_1.ABS;UNIALG_2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {propsitional calcus; quantifier calcus; algorithmic logic; },
    SUBMITTED = {September 15, 2014})

@ARTICLE(LATTAD_1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 3}},
    PAGES = {257--267},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0026},
    VERSION = {8.1.03 5.25.1220},
    TITLE = {{F}ormalization of Generalized Almost Distributive Lattices},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok\\Poland},
    ACKNOWLEDGEMENT = {The author wants to express his gratitude to the anonymous referee for his/her work on the last section of this article; although I did not want to add more concrete examples than the simplest ones, these additional constructions proposed by the referee complete the Mizar article as a faithful translation of the Rao's results, at the same time suggesting possible improvements of the Mizar Mathematical Library.},
    SUMMARY = {Almost Distributive Lattices (ADL) are structures defined by Swamy and Rao \cite{SwamyRao1981} as a common abstraction of some generalizations of the Boolean algebra. In our paper, we deal with a certain further generalization of ADLs, namely the Generalized Almost Distributive Lattices (GADL). Our main aim was to give the formal counterpart of this structure and we succeeded formalizing all items from the Section 3 of {R}ao et al.'s paper \cite{GADL:2009}. Essentially among GADLs we can find structures which are neither $\vee$-commutative nor $\wedge$-commutative (resp., $\wedge$-commutative); consequently not all forms of absorption identities hold.\par We characterized some necessary and sufficient conditions for commutativity and distributivity, we also defined the class of GADLs with zero element. We tried to use as much attributes and cluster registrations as possible, hence many identities are expressed in terms of adjectives; also some generalizations of well-known notions from lattice theory \cite{Gratzer2011} formalized within the Mizar Mathematical Library were proposed. Finally, some important examples from Rao's paper were introduced. We construct the example of GADL which is not an ADL. Mechanization of proofs in this specific area could be a good starting point towards further generalization of lattice theory \cite{Gratzer} with the help of automated theorem provers \cite{GrabMo2004}.   },
    MSC2010 = {03G10 06B75 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Almost Distributive Lattices},
    SECTION3 = {Properties of Almost Distributive Lattices},
    SECTION4 = {Generalization of Almost Distributive Lattices},
    SECTION5 = {Order Properties of the Generated Relation on GADLs},
    SECTION6 = {Formalization of \cite{GADL:2009} paper},
    SECTION7 = {Generalized Almost Distributive Lattices with Zero},
    SECTION8 = {Constructing Examples of Almost Distributive Lattices},
    EXTERNALREFS = {SwamyRao1981; GADL:2009; },
    INTERNALREFS = {BINOP_1.ABS;ENUMSET1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;LATTICES.ABS;ORDERS_1.ABS;ORDERS_2.ABS;PARTFUN1.ABS;REALSET1.ABS;
RELAT_1.ABS;RELAT_2.ABS;ROBBINS3.ABS;SUBSET_1.ABS;WAYBEL_0.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {almost distributive lattices; generalized almost distributive lattices; lattice identities; },
    SUBMITTED = {September 26, 2014})

@ARTICLE(VSDIFF_1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 3}},
    PAGES = {269--275},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0027},
    VERSION = {8.1.03 5.25.1220},
    TITLE = {{D}ifference of Function on~Vector~Space~over~\hbox{$\mathbb F$}},
    ANNOTE = {This work was supported by JSPS KAKENHI Grant Number 26730067.},
    AUTHOR = {Arai, Kenichi and Wakabayashi, Ken and Okazaki, Hiroyuki},
    ADDRESS1 = {Tokyo University of Science\\Chiba, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    ACKNOWLEDGEMENT = {We sincerely thank Professor Yasunari Shidama for his helpful advices.},
    SUMMARY = {In \cite{DIFF_1.ABS}, the definitions of forward difference, backward difference, and central difference as difference operations for functions on $\mathbb{R}$ were formalized. However, the definitions of forward difference, backward difference, and central difference for functions on vector spaces over $\mathbb{F}$ have not been formalized. In cryptology, these definitions are very important in evaluating the security of cryptographic systems \cite{BA1991}, \cite{Lai1994}. Differential cryptanalysis \cite{BA1993} that undertakes a general purpose attack against block ciphers \cite{DESCIP_1.ABS} can be formalized using these definitions. In this article, we formalize the definitions of forward difference, backward difference, and central difference for functions on vector spaces over $\mathbb{F}$. Moreover, we formalize some facts about these definitions.   },
    MSC2010 = {39A70 15A03 03B35},
    EXTERNALREFS = {BA1991; Lai1994; BA1993; },
    INTERNALREFS = {ALGSTR_1.ABS;BINOM.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GROUP_1.ABS;NAT_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RELAT_1.ABS;
RELSET_1.ABS;RLVECT_1.ABS;SEQFUNC.ABS;SUBSET_1.ABS;VECTSP_1.ABS;VFUNCT_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Mizar formalization; difference of function on vector space over $\mathbb{F}$; },
    SUBMITTED = {September 26, 2014})     
    
@ARTICLE(ZMODUL06.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 4}},
    PAGES = {277--289},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0028},
    VERSION = {8.1.03 5.26.1224},
    TITLE = {{T}orsion {$\mathbb{Z}$}-module and Torsion-free {$\mathbb{Z}$}-module},
    ANNOTE = {This work was supported by JSPS KAKENHI 21240001 and 22300285.},
    AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Nakasho, Kazuhisa and Shidama, Yasunari},
    ADDRESS1 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    ADDRESS4 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we formalize a torsion $\mathbb Z$-module and a torsion-free $\mathbb Z$-module. Especially, we prove formally that finitely generated torsion-free $\mathbb Z$-modules are finite rank free. We also formalize properties related to rank of finite rank free $\mathbb Z$-modules. The notion of $\mathbb Z$-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov\'asz) base reduction algorithm \cite{LLL}, cryptographic systems with lattice \cite{LATTICE2002}, and coding theory \cite{LANDC}.   },
    MSC2010 = {13C10 15A04 03B35},
    SECTION1 = {Torsion $\mathbb{Z}$-module and Torsion-free $\mathbb{Z}$-module},
    SECTION2 = {Rank of Finite Rank Free $\mathbb{Z}$-module},
    EXTERNALREFS = {LLL; LATTICE2002; LANDC; },
    INTERNALREFS = {BINOM.ABS;BINOP_1.ABS;CARD_1.ABS;DOMAIN_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GAUSSINT.ABS;GROUP_1.ABS;
INT_1.ABS;INT_3.ABS;NAT_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RAT_1.ABS;RELAT_1.ABS;RELSET_1.ABS;RLVECT_1.ABS;SUBSET_1.ABS;VECTSP_1.ABS;
VECTSP_7.ABS;ZFMISC_1.ABS;ZMODUL01.ABS;ZMODUL02.ABS;ZMODUL03.ABS;ZMODUL05.ABS;},
    KEYWORDS = {free Z-module; rank of Z-module; homomorphism of Z-module; linearly independent; linear combination; },
    SUBMITTED = {November 29, 2014})
 
 @ARTICLE(RING_2.ABS,
     JOURNAL = {Formalized Mathematics},
     ISSN = {1426-2630},
     EISSN = {1898-9934},
     VOLUME = {22},
     NUMBER = {{\bf 4}},
     PAGES = {291--301},
     YEAR = {2014},
     DOI = {10.2478/forma-2014-0029},
     VERSION = {8.1.03 5.26.1224},
     TITLE = {{T}he First Isomorphism Theorem and Other Properties of Rings},
     AUTHOR = {Korni{\l}owicz, Artur and Schwarzweller, Christoph},
     ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Sosnowa 64, 15-887 Bia{\l}ystok\\Poland},
     ADDRESS2 = {Institute of Computer Science\\University of Gda{\'n}sk\\Wita Stwosza 57, 80-952 Gda{\'n}sk\\Poland},
     SUMMARY = {Different properties of rings and fields are discussed \cite{Jacobson2009}, \cite{Waerden2003} and \cite{Luneburg1999}. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism $f: R\longrightarrow S$ we have $R/_{\mbox{ker}(f)} \cong {\mbox{Im}(f)}$. Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial.   },
     MSC2010 = {13A05 13A15 03B35},
     SECTION1 = {Preliminaries},
     SECTION2 = {Homomorphisms, Kernel and Image},
     SECTION3 = {Units and Non Units},
     SECTION4 = {Prime and Irreducible Elements},
     SECTION5 = {Principal Ideal Domains and Factorial Rings},
     SECTION6 = {Polynomial Rings over Fields},
     EXTERNALREFS = {Jacobson2009; Waerden2003; Luneburg1999; },
     INTERNALREFS = {ALGSTR_1.ABS; C0SP1.ABS; CARD_1.ABS; DOMAIN_1.ABS; EQREL_1.ABS; FINSEQ_1.ABS; FINSET_1.ABS; FUNCOP_1.ABS;
 FUNCT_1.ABS; FUNCT_2.ABS; GCD_1.ABS; GROUP_1.ABS; GROUP_4.ABS; GROUP_6.ABS; HURWITZ.ABS; IDEAL_1.ABS; INT_1.ABS; INT_3.ABS;
 MEMBERED.ABS; MOD_4.ABS; NAT_1.ABS; ORDINAL1.ABS; PARTFUN1.ABS; POLYNOM1.ABS; POLYNOM3.ABS; QUOFIELD.ABS; RATFUNC1.ABS;
 RAT_1.ABS; REALSET1.ABS; RELAT_1.ABS; RELSET_1.ABS; RINGCAT1.ABS; RING_1.ABS; RLVECT_1.ABS; SETFAM_1.ABS; SUBSET_1.ABS; VECTSP10.ABS;
 VECTSP_1.ABS; VECTSP_2.ABS; ZFMISC_1.ABS; },
     KEYWORDS = {commutative algebra; ring theory; first isomorphism theorem; },
     SUBMITTED = {November 29, 2014})

@ARTICLE(DUALSP02.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 4}},
    PAGES = {303--311},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0030},
    VERSION = {8.1.03 5.26.1224},
    TITLE = {{B}idual Spaces and Reflexivity of Real Normed Spaces},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285 and 23500029.},
    AUTHOR = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
    ADDRESS1 = {Hirosaki-city\\Aomori, Japan},
    ADDRESS2 = {Gifu National College of Technology\\Gifu, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of {$\mathbb R$}, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from \cite{LOPBAN_5.ABS}. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on \cite{ReedSimon1972}, \cite{rudin1991functional}, \cite{Dax2002} and \cite{Brezis2011}.   },
    MSC2010 = {46B10 46A25 03B35},
    SECTION1 = {The Application of Hahn-Banach Theorem},
    SECTION2 = {Bidual Spaces of Real Normed Spaces},
    SECTION3 = {Uniform Boundedness Theorem for Linear Functionals},
    SECTION4 = {Reflexivity of Real Normed Spaces},
    EXTERNALREFS = {ReedSimon1972; rudin1991functional; Dax2002; Brezis2011; },
    INTERNALREFS = {COMPLEX1.ABS;DUALSP01.ABS;EUCLID.ABS;FUNCT_1.ABS;FUNCT_2.ABS;HAHNBAN.ABS;LOPBAN_1.ABS;MEMBERED.ABS;NFCONT_1.ABS;
NORMSP_1.ABS;NORMSP_3.ABS;PARTFUN1.ABS;REAL_1.ABS;RELAT_1.ABS;RLSUB_1.ABS;RLVECT_1.ABS;SEQ_4.ABS;SUBSET_1.ABS;VECTSP_1.ABS;
ZFMISC_1.ABS;},
    KEYWORDS = {continuous dual space; topological duality; reflexivity; },
    SUBMITTED = {November 29, 2014})

@ARTICLE(EUCLID10.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 4}},
    PAGES = {313--319},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0031},
    VERSION = {8.1.03 5.26.1224},
    TITLE = {{S}ome Facts about Trigonometry and {E}uclidean Geometry},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
    SUMMARY = {We calculate the values of the trigonometric functions for angles: $\frac{\pi}{3}$ and $\frac{\pi}{6}$, by \cite{hartshorne2000geometry}. After defining some trigonometric identities, we demonstrate conventional trigonometric formulas in the triangle, and the geometric property, by \cite{efimov1981geometrie}, of the triangle inscribed in a semicircle, by the proposition 3.31 in \cite{fitzpatrick2007euclid}. Then we define the diameter of the circumscribed circle of a triangle using the definition of the area of a triangle and prove some identities of a triangle \cite{Coxeter:1967}. We conclude by indicating that the diameter of a circle is twice the length of the radius.   },
    MSC2010 = {51M04 03B35},
    SECTION1 = {Values of the Trigonometric Functions for Angles: $\frac{\pi}{3}$ and $\frac{\pi}{6}$},
    SECTION2 = {Some Trigonometric Identities},
    SECTION3 = {Trigonometric Functions and Right Triangle},
    SECTION4 = {Triangle Inscribed in a Semicircle is a Right Triangle},
    SECTION5 = {Diameter of the Circumcircle of a Triangle},
    SECTION6 = {Some Identities of a Triangle},
    SECTION7 = {Diameter of a Circle},
    EXTERNALREFS = {hartshorne2000geometry; efimov1981geometrie; fitzpatrick2007euclid; Coxeter:1967; },
    INTERNALREFS = {CARD_1.ABS;COMPTS_1.ABS;EUCLID.ABS;EUCLID_3.ABS;EUCLID_6.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;JGRAPH_6.ABS;
ORDINAL1.ABS;RELAT_1.ABS;RLTOPSP1.ABS;RVSUM_1.ABS;SETFAM_1.ABS;SIN_COS.ABS;SIN_COS4.ABS;SIN_COS6.ABS;SQUARE_1.ABS;SUBSET_1.ABS;
TBSP_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Euclidean geometry; trigonometry; circumcircle; right-angled; },
    SUBMITTED = {September 29, 2014})

@ARTICLE(FUZNUM_1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {22},
    NUMBER = {{\bf 4}},
    PAGES = {321--327},
    YEAR = {2014},
    DOI = {10.2478/forma-2014-0032},
    VERSION = {8.1.03 5.29.1227},
    TITLE = {{T}he Formal Construction of Fuzzy Numbers},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok\\Poland},
    SUMMARY = {In this article, we continue the development of the theory of fuzzy sets \cite{Zadeh:1965}, started with \cite{FUZZY_1.ABS} with the future aim to provide the formalization of fuzzy numbers \cite{DuboisPrade:1978} in terms reflecting the current state of the Mizar Mathematical Library. Note that in order to have more usable approach in \cite{FUZZY_1.ABS}, we revised that article as well; some of the ideas were described in \cite{GrabowskiFuzzy:2013}. As we can actually understand fuzzy sets just as their membership functions (via the equality of membership function and their set-theoretic counterpart), all the calculations are much simpler. To test our newly proposed approach, we give the notions of (normal) triangular and trapezoidal fuzzy sets as the examples of concrete fuzzy objects. Also $\alpha$-cuts, the core of a fuzzy set, and normalized fuzzy sets were defined. Main technical obstacle was to prove continuity of the glued maps, and in fact we did this not through its topological counterpart, but extensively reusing properties of the real line (with loss of generality of the approach, though), because we aim at formalizing fuzzy numbers in our future submissions, as well as merging with rough set approach as introduced in \cite{ROUGHS_1.ABS} and \cite{GrabowskiFI:2014}. Our base for formalization was \cite{DuboisPrade:1980} and \cite{DuboisPrade:1990}.   },
    MSC2010 = {03E72 94D99 03B35},
    SECTION1 = {Preliminaries: Affine Maps},
    SECTION2 = {Towards Development of Fuzzy Numbers},
    SECTION3 = {Convexity and the Height of a Fuzzy Set},
    SECTION4 = {Pasting aka Glueing Lemmas},
    SECTION5 = {Triangular and Trapezoidal Fuzzy Sets},
    EXTERNALREFS = {Zadeh:1965; DuboisPrade:1978; GrabowskiFuzzy:2013; GrabowskiFI:2014; DuboisPrade:1980; DuboisPrade:1990; },
    INTERNALREFS = {FCONT_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_4.ABS;FUZZY_1.ABS;MEASURE5.ABS;MEMBERED.ABS;ORDINAL1.ABS;PARTFUN1.ABS;
RCOMP_1.ABS;RELAT_1.ABS;RELSET_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {fuzzy sets; formal models of fuzzy sets; triangular fuzzy numbers; },
    SUBMITTED = {December 31, 2014})

@ARTICLE{CAT_7.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 1}},
    PAGES = {1--14},
    YEAR = {2015},
    DOI = {10.2478/forma-2015-0001},
    VERSION = {8.1.03 5.29.1227},
    TITLE = {{C}ategorical Pullbacks},
    AUTHOR = {Riccardi, Marco},
    ADDRESS1 = {Via del Pero 102\\54038 Montignoso\\Italy},
    SUMMARY = {The main purpose of this article is to introduce the categorical concept of pullback in Mizar. In the first part of this article we redefine hom-sets, monomorphisms, epimorpshisms and isomorphisms \cite{Borceaux} within a free-object category \cite{Adamek:2009} and it is shown there that ordinal numbers can be considered as categories. Then the pullback is introduced in terms of its universal property and the Pullback Lemma is formalized \cite{MacLane:1}. In the last part of the article we formalize the pullback of functors \cite{Lawvere:1963} and it is also shown that it is not possible to write an equivalent definition in the context of the previous Mizar formalization of category theory \cite{CAT_1.ABS}.   },
    MSC2010 = {18A30 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Hom-sets},
    SECTION3 = {Monomorphisms, Epimorphisms and Isomorphisms},
    SECTION4 = {Ordinal Numbers as Categories},
    SECTION5 = {Pullbacks},
    SECTION6 = {Pullbacks of Functors},
    EXTERNALREFS = {Borceaux; Lawvere:1963; },
    INTERNALREFS = {CARD_1.ABS;CAT_1.ABS;CAT_6.ABS;ENUMSET1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;ORDINAL1.ABS;
PARTFUN1.ABS;RELAT_1.ABS;RELSET_1.ABS;SETFAM_1.ABS;SUBSET_1.ABS;WELLORD1.ABS;WELLORD2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {category pullback; pullback lemma; },
    SUBMITTED = {December 31, 2014}}

@ARTICLE{GROUP_19.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 1}},
    PAGES = {15--27},
    YEAR = {2015},
    DOI = {10.2478/forma-2015-0002},
    VERSION = {8.1.03 5.29.1227},
    TITLE = {{D}efinition and Properties of Direct Sum Decomposition of Groups},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285.},
    AUTHOR = {Nakasho, Kazuhisa and Yamazaki, Hiroshi and Okazaki, Hiroyuki and Shidama, Yasunari},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    ADDRESS4 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to \cite{rotman1995introduction}, \cite{robinson2012course}, \cite{BourbakiAlgI} and \cite{lang-algebra} in the formalization.   },
    MSC2010 = {20E34 03B35},
    SECTION1 = {Miscellanies},
    SECTION2 = {Support of Element of Direct Product Group},
    SECTION3 = {Product Map and Sum Map},
    SECTION4 = {Definition of Internal and External Direct Sum Decomposition},
    SECTION5 = {Equivalent Expression of Internal Direct Sum Decomposition},
    EXTERNALREFS = {rotman1995introduction; robinson2012course; BourbakiAlgI; lang-algebra; },
    INTERNALREFS = {CARD_1.ABS;CARD_3.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_3.ABS;FUNCT_4.ABS;
FUNCT_7.ABS;GROUP_1.ABS;GROUP_12.ABS;GROUP_17.ABS;GROUP_2.ABS;GROUP_4.ABS;GROUP_6.ABS;GROUP_7.ABS;GRSOLV_1.ABS;INT_1.ABS;
MONOID_0.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PRALG_1.ABS;RELAT_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {group theory; direct sum decomposition; },
    SUBMITTED = {December 31, 2014}}
   
@ARTICLE{ZMATRLIN.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 1}},
    PAGES = {29--49},
    YEAR = {2015},
    DOI = {10.2478/forma-2015-0003},
    VERSION = {8.1.04 5.31.1231},
    TITLE = {{M}atrix of {$\mathbb Z$}-module},
    ANNOTE = {This work was supported by JSPS KAKENHI 21240001 and 22300285.},
    AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
    ADDRESS1 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we formalize a matrix of $\mathbb Z$-module and its properties. Specially, we formalize a matrix of a linear transformation of $\mathbb Z$-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free $\mathbb Z$-module $V$, determinant of its Gramian matrix is constant regardless of selection of its basis. $\mathbb Z$-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm and cryptographic systems with lattices \cite{LATTICE2002} and coding theory \cite{LANDC}. Some theorems in this article are described by translating theorems in \cite{MATRLIN.ABS}, \cite{HAHNBAN.ABS} and \cite{BILINEAR.ABS} into theorems of $\mathbb Z$-module.   },
    MSC2010 = {11E39 13C10 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Sequences and Matrices Concerning Linear Transformations},
    SECTION3 = {Decomposition of a Vector in Basis},
    SECTION4 = {Matrices of Linear Transformations},
    SECTION5 = {Real-valued Function of $\mathbb Z$-Module},
    SECTION6 = {Bilinear Form of $\mathbb Z$-Module},
    SECTION7 = {Matrix of Bilinear Form},
    EXTERNALREFS = {LATTICE2002; },
    INTERNALREFS = {BINOP_1.ABS;CARD_1.ABS;FINSEQOP.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSET_1.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;
FUNCT_2.ABS;FUNCT_5.ABS;FVSUM_1.ABS;GROUP_1.ABS;HAHNBAN1.ABS;INT_1.ABS;INT_3.ABS;LAPLACE.ABS;MATRIX_1.ABS;MATRIX_3.ABS;
MATRLIN.ABS;MOD_2.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RAT_1.ABS;RELAT_1.ABS;RELSET_1.ABS;RLVECT_1.ABS;SUBSET_1.ABS;VECTSP_1.ABS;
VECTSP_6.ABS;VECTSP_7.ABS;ZFMISC_1.ABS;ZMODUL01.ABS;ZMODUL03.ABS;},
    KEYWORDS = {matrix of Z-module; matrix of linear transformation; bilinear form; },
    SUBMITTED = {February 18, 2015}}

@ARTICLE{SRINGS_3.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 1}},
    PAGES = {51--57},
    YEAR = {2015},
    DOI = {10.2478/forma-2015-0004},
    VERSION = {8.1.04 5.31.1231},
    TITLE = {{$\sigma$}-ring and {$\sigma$}-algebra of Sets},
    ANNOTE = {This work was supported by JSPS KAKENHI 23500029 and 22300285.},
    AUTHOR = {Endou, Noboru and Nakasho, Kazuhisa and Shidama, Yasunari},
    ADDRESS1 = {Gifu National College of Technology\\Gifu, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in \cite{SRINGS_1.ABS}, that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades \cite{GOGUADZE:2003}. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set \cite{FINSUB_1.ABS} and field of subsets \cite{PROB_1.ABS}, respectively. In the second section, definitions of a ring and a $\sigma$-ring of sets, which are based on a semiring and a ring of sets respectively, are formalized and their related theorems are proved. In the third section, definitions of an algebra and a $\sigma$-algebra of sets, which are based on a semialgebra and an algebra of sets respectively, are formalized and their related theorems are proved. In the last section, mutual relationships between $\sigma$-ring and $\sigma$-algebra of sets are formalized and some related examples are given. The formalization is based on \cite{GOGUADZE:2003}, and also referred to \cite{bogachev2007measure} and \cite{Halmos74}.   },
    MSC2010 = {03E30 28A05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Classical Semiring, Ring and $\sigma$-ring of Sets},
    SECTION3 = {Semialgebra, Algebra and $\sigma$-algebra of Sets},
    SECTION4 = {Mutual Relationships between $\sigma$-ring and $\sigma$-algebra of Sets},
    EXTERNALREFS = {GOGUADZE:2003; bogachev2007measure; Halmos74; },
    INTERNALREFS = {CARD_3.ABS;CLASSES1.ABS;COHSP_1.ABS;COH_SP.ABS;EQREL_1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FINSUB_1.ABS;FUNCT_1.ABS;
FUNCT_2.ABS;MEASURE1.ABS;MEASURE5.ABS;MEMBERED.ABS;NAT_1.ABS;ORDINAL1.ABS;PROB_1.ABS;PROB_2.ABS;RELAT_1.ABS;RELSET_1.ABS;
SETFAM_1.ABS;SRINGS_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {semiring of sets; $\sigma$-ring of sets; $\sigma$-algebra of sets; },
    SUBMITTED = {February 18, 2015}}

@ARTICLE{NORMSP_4.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 1}},
    PAGES = {59--65},
    YEAR = {2015},
    DOI = {10.2478/forma-2015-0005},
    VERSION = {8.1.04 5.31.1231},
    TITLE = {{S}eparability of Real Normed Spaces and Its Basic Properties},
    AUTHOR = {Nakasho, Kazuhisa and Endou, Noboru},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
    ADDRESS2 = {Gifu National College of Technology\\Gifu, Japan},
    ACKNOWLEDGEMENT = {We would like to thank Keiko Narita and Yasunari Shidama.},
    SUMMARY = {In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the separable normed spaces are discussed. It is applied to isomorphic spaces via bounded linear operators and double dual spaces. In the last section, it is proved that the completeness and reflexivity are transferred to sublinear normed spaces. The formalization is based on \cite{yoshida:1980}, and also referred to \cite{BOURBAKI:1-5}, \cite{Dunford:1958} and \cite{kolmogorov2012}.   },
    MSC2010 = {46B20 46A19 03B35},
    SECTION1 = {Separability of Real Normed Space},
    SECTION2 = {Basic Properties of Separable Spaces},
    SECTION3 = {Completeness and Reflexivity of Sublinear Normed Spaces},
    EXTERNALREFS = {yoshida:1980; BOURBAKI:1-5; Dunford:1958; kolmogorov2012; },
    INTERNALREFS = {CARD_1.ABS;CARD_3.ABS;COMPLEX1.ABS;DOMAIN_1.ABS;DUALSP01.ABS;DUALSP02.ABS;FINSET_1.ABS;FUNCT_1.ABS;
FUNCT_2.ABS;HAHNBAN.ABS;IDEAL_1.ABS;LOPBAN_1.ABS;NORMSP_1.ABS;NORMSP_2.ABS;NORMSP_3.ABS;ORDINAL1.ABS;PARTFUN1.ABS;
PRE_TOPC.ABS;RAT_1.ABS;RELAT_1.ABS;RELSET_1.ABS;RLSUB_1.ABS;RLVECT_1.ABS;RLVECT_2.ABS;RLVECT_3.ABS;SUBSET_1.ABS;VECTSP_1.ABS;
ZFMISC_1.ABS;},
    KEYWORDS = {functional analysis; normed linear space; topological vector space; },
    SUBMITTED = {February 26, 2015}}

@ARTICLE{GROUP_20.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 1}},
    PAGES = {67--73},
    YEAR = {2015},
    DOI = {10.2478/forma-2015-0006},
    VERSION = {8.1.04 5.31.1231},
    TITLE = {{E}quivalent Expressions of Direct Sum Decomposition of Groups},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285.},
    AUTHOR = {Nakasho, Kazuhisa and Okazaki, Hiroyuki and Yamazaki, Hiroshi and Shidama, Yasunari},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    ADDRESS4 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of commutative groups. In the last section, we formalize that the external direct sum leads an internal direct sum. We referred to \cite{rotman1995introduction}, \cite{robinson2012course} \cite{BourbakiAlgI} and \cite{lang-algebra} in the formalization.   },
    MSC2010 = {20E34 03B35},
    SECTION1 = {Internal Direct Sum Decomposition into Normal Subgroups},
    SECTION2 = {Internal Direct Sum Decomposition for Commutative Group},
    SECTION3 = {Equivalence between Internal and External Direct Sum},
    EXTERNALREFS = {rotman1995introduction; robinson2012course; BourbakiAlgI; lang-algebra; },
    INTERNALREFS = {CARD_3.ABS;DOMAIN_1.ABS;FINSEQ_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_7.ABS;GROUP_1.ABS;GROUP_12.ABS;GROUP_19.ABS;
GROUP_2.ABS;GROUP_3.ABS;GROUP_4.ABS;GROUP_6.ABS;GROUP_7.ABS;MONOID_0.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PRALG_1.ABS;RELAT_1.ABS;
RELSET_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {group theory; direct sum decomposition; },
    SUBMITTED = {February 26, 2015}}

@ARTICLE{EUCLID11.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 2}},
    PAGES = {75--79},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0007},
    VERSION = {8.1.04 5.32.1234},
    TITLE = {{M}orley's Trisector Theorem},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {Morley's trisector theorem states that ``The points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle" \cite{Coxeter:1967}. \par There are many proofs of Morley's trisector theorem \cite{donolato2013vector, maor2014beautiful, MAI1, MAI2, connes1998new, stonebridge2009simple, CTK, oakley1978morley}. We follow the proof given by A. Letac in \cite{letac}.   },
    MSC2010 = {51M04 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Morley's Theorem},
    EXTERNALREFS = {Coxeter:1967; donolato2013vector; maor2014beautiful; MAI1; MAI2; connes1998new; stonebridge2009simple; CTK; oakley1978morley; letac; },
    INTERNALREFS = {CARD_1.ABS;EUCLID.ABS;EUCLID10.ABS;EUCLID_3.ABS;EUCLID_6.ABS;FINSEQ_1.ABS;FUNCT_1.ABS;RELAT_1.ABS;
RLTOPSP1.ABS;SIN_COS.ABS;SQUARE_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Euclidean geometry; Morley's trisector theorem; equilateral triangle; },
    SUBMITTED = {March 26, 2015}}
    
    
    @ARTICLE{FLEXARY1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 2}},
    PAGES = {81--92},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0008},
    VERSION = {8.1.04 5.32.1237},
    TITLE = {{F}lexary Operations},
    ANNOTE = {This work has been financed by the resources of the Polish National Science Centre granted by decision no. DEC-2012/07/N/ST6/02147.},
    AUTHOR = {P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
      SUMMARY = {In this article we introduce necessary notation and definitions to prove the Euler's Partition Theorem according to H.S. Wilf's lecture notes \cite{Wilf}. Our aim is to create an environment which allows to formalize the theorem in a way that is as similar as possible to the original informal proof. \par {E}uler's {P}artition {T}heorem is listed as item {\tt{\#45}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} \cite{Freek-100-theorems}.   },
    MSC2010 = {11B99 03B35},
    SECTION1 = {Auxiliary Facts about Finite Sequences Concatenation},
    SECTION2 = {Flexary Plus},
    SECTION3 = {Power Function},
    SECTION4 = {Value-based Function (Re)Organization},
    EXTERNALREFS = {Wilf; Freek-100-theorems; },
    INTERNALREFS = {CARD_1.ABS;CLASSES1.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSET_1.ABS;FINSOP_1.ABS;FOMODEL0.ABS;FOMODEL2.ABS;
FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_7.ABS;INT_1.ABS;MEMBERED.ABS;MONOID_0.ABS;NAT_1.ABS;NEWTON.ABS;ORDINAL1.ABS;
POWER.ABS;RELAT_1.ABS;RELSET_1.ABS;RVSUM_1.ABS;SETFAM_1.ABS;SUBSET_1.ABS;VALUED_2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {summation method; flexary plus; matrix generalization; },
    SUBMITTED = {March 26, 2015}}
    
    
    @ARTICLE{EULRPART.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 2}},
    PAGES = {93--99},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0009},
    VERSION = {8.1.04 5.32.1237},
    TITLE = {{E}uler's {P}artition {T}heorem},
    ANNOTE = {This work has been financed by the resources of the Polish National Science Centre granted by decision no. DEC-2012/07/N/ST6/02147.},
    AUTHOR = {P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
      SUMMARY = {In this article we prove the Euler's Partition Theorem which states that the number of integer partitions with odd parts equals the number of partitions with distinct parts. The formalization follows H.S. Wilf's lecture notes \cite{Wilf} (see also \cite{Andrews}). \par {E}uler's {P}artition {T}heorem is listed as item {\tt{\#45}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} \cite{Freek-100-theorems}.   },
    MSC2010 = {05A17 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Euler Transformation},
    SECTION3 = {Main Theorem},
    EXTERNALREFS = {Wilf; Andrews; Freek-100-theorems; },
    INTERNALREFS = {ABIAN.ABS;CARD_1.ABS;CARD_3.ABS;FIB_NUM2.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FLEXARY1.ABS;FOMODEL0.ABS;
FOMODEL2.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;INT_1.ABS;NAT_1.ABS;NEWTON.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RELAT_1.ABS;
RVSUM_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {partition theorem; },
    SUBMITTED = {March 26, 2015}}
    
    
    @ARTICLE{DIOPHAN1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 2}},
    PAGES = {101--106},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0010},
    VERSION = {8.1.04 5.32.1237},
    TITLE = {{I}ntroduction to {D}iophantine Approximation},
    AUTHOR = {Watase, Yasushige},
    ADDRESS1 = {Suginami-ku Matsunoki 6\\3-21 Tokyo, Japan},
      SUMMARY = {In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution $(x,y)$ of the inequality $| x\theta - y | \leq 1/x$, where $\theta$ is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet's proof (1842) of existence of the solution \cite{HardyWright}, \cite{Baker:1984}.   },
    MSC2010 = {11A55 11J68 03B35},
    SECTION1 = {Irrational Numbers and Continued Fractions},
    SECTION2 = {Integer Solution of $| x\theta - y | \leq 1/x$},
    SECTION3 = {Proof of Dirichlet's Theorem},
    EXTERNALREFS = {HardyWright; Baker:1984; },
    INTERNALREFS = {ABIAN.ABS;CARD_1.ABS;COMPLEX1.ABS;COMSEQ_2.ABS;FINANCE1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;
FUNCT_2.ABS;INT_1.ABS;INT_2.ABS;MEMBERED.ABS;NAT_1.ABS;NAT_6.ABS;NEWTON.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PROB_3.ABS;
RAT_1.ABS;REAL_1.ABS;REAL_3.ABS;RELAT_1.ABS;RELSET_1.ABS;SEQ_2.ABS;SETFAM_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {irrational number; approximation; continued fraction; rational number; Dirichlet's proof; },
    SUBMITTED = {April 19, 2015}}
    
    
    @ARTICLE{SRINGS_4.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 2}},
    PAGES = {107--114},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0011},
    VERSION = {8.1.04 5.32.1237},
    TITLE = {{F}inite Product of Semiring of Sets},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {We formalize that the image of a semiring of sets \cite{GOGUADZE:2003} by an injective function is a semiring of sets. We offer a non-trivial example of a semiring of sets in a topological space \cite{Schmets:2004}. Finally, we show that the finite product of a semiring of sets is also a semiring of sets \cite{Schmets:2004} and that the finite product of a classical semiring of sets \cite{bogachev2007measure} is a classical semiring of sets. In this case, we use here the notation from the book of Aliprantis and Border \cite{aliprantis2006infinite}.   },
    MSC2010 = {28A05 03E02 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {The Direct Image of a Semiring of Sets by an Injective Function},
    SECTION3 = {The Set of Set Differences of All Elements of a Semiring of Sets},
    SECTION4 = {The Collection of All Locally Closed Sets $LC(X,\tau)$ of a Topological Space $(X,\tau)$},
    SECTION5 = {The Finite Product of Semirings of Sets},
    SECTION6 = {The Finite Product of Classical Semirings of Sets},
    SECTION7 = {Measurable Rectangle},
    EXTERNALREFS = {GOGUADZE:2003; Schmets:2004; bogachev2007measure; aliprantis2006infinite; },
    INTERNALREFS = {BINOP_1.ABS;CARD_1.ABS;CARD_3.ABS;CLASSES1.ABS;ENUMSET1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FINSUB_1.ABS;
FUNCT_1.ABS;FUNCT_2.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PRE_TOPC.ABS;RELAT_1.ABS;RELSET_1.ABS;SETFAM_1.ABS;SRINGS_1.ABS;
SRINGS_3.ABS;SUBSET_1.ABS;TEX_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {set partitions; semiring of sets; },
    ACKNOWLEDGEMENT = {I would like to thank Artur Korni\l owicz for suggestions.},
    SUBMITTED = {April 19, 2015}}
    
    
    @ARTICLE{NELSON_1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 2}},
    PAGES = {115--125},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0012},
    VERSION = {8.1.04 5.32.1237},
    TITLE = {{T}wo Axiomatizations of {N}elson Algebras},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\ University of Bia{\l}ystok\\ Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
      SUMMARY = {Nelson algebras were first studied by Rasiowa and Bia\l ynicki-Birula \cite{RasiowaBirula} under the name N-lattices or quasi-pseudo-Boolean algebras. Later, in investigations by Monteiro and Brignole \cite{BrignoleI:1967, BrignoleII:1967}, and \cite{Brignole} the name ``Nelson algebras" was adopted -- which is now commonly used to show the correspondence with Nelson's paper \cite{Nelson} on constructive logic with strong negation.\par By a Nelson algebra we mean an abstract algebra $$\langle L, \top, -, \neg, \rightarrow, \Rightarrow, \sqcup, \sqcap \rangle$$ \noindent where $L$ is the carrier, $-$ is a quasi-complementation (Rasiowa used the sign $\sim$, but in Mizar ``$-$" should be used to follow the approach described in \cite{GrabMo2004} and \cite{GrabowskiJAR40}), $\neg$ is a weak pseudo-complementation, $\rightarrow$ is weak relative pseudo-complementation and $\Rightarrow$ is implicative operation. $\sqcup$ and $\sqcap$ are ordinary lattice binary operations of supremum and infimum. \par In this article we give the definition and basic properties of these algebras according to \cite{RasiowaNonClassical} and \cite{Rasiowa:2001}. We start with preliminary section on quasi-Boolean algebras (i.e. de Morgan bounded lattices). Later we give the axioms in the form of Mizar adjectives with names corresponding with those in \cite{Rasiowa:2001}. As our main result we give two axiomatizations (non-equational and equational) and the full formal proof of their equivalence. \par The second set of equations is rather long but it shows the logical essence of Nelson lattices. This formalization aims at the construction of algebraic model of rough sets \cite{GrabowskiFI:2013} in our future submissions. Section 4 contains all items from Th. 1.2 and 1.3 (and the itemization is given in the text). In the fifth section we provide full formal proof of Th. 2.1 p. 75 \cite{RasiowaNonClassical}.   },
    MSC2010 = {06D30 08A05 03B35},
    SECTION1 = {De Morgan and Quasi-Boolean Lattices},
    SECTION2 = {The Structure and Operators in Nelson Algebras},
    SECTION3 = {The Non-Equational Axiomatization},
    SECTION4 = {Properties of Nelson Algebras},
    SECTION5 = {Alternative Equational Axiomatics by Rasiowa},
    EXTERNALREFS = {RasiowaBirula; BrignoleI:1967; BrignoleII:1967; Brignole; Nelson; GrabMo2004; GrabowskiJAR40; RasiowaNonClassical; Rasiowa:2001; GrabowskiFI:2013; },
    INTERNALREFS = {BINOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;LATTICES.ABS;ROBBINS1.ABS;ROBBINS3.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {quasi-pseudo-Boolean algebras; Nelson lattices; de Morgan lattices; },
    SUBMITTED = {April 19, 2015}}
    
    
    @ARTICLE{GROUP_1A.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},    
    VOLUME = {23},
    NUMBER = {{\bf 2}},
    PAGES = {127--160},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0013},
    VERSION = {8.1.04 5.32.1240},
    TITLE = {{G}roups -- Additive Notation},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec \cite{GROUP_1.ABS, GROUP_2.ABS, GROUP_3.ABS} and Artur Korni{\l}owicz \cite{TOPGRP_1.ABS}. \par In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange's theorem and some other theorems concerning these notions \cite{blahut2014cryptography, inui2012group, hewitt2012abstract} are presented. \par Note that ``The term {$\mathbb Z$}-module is simply another name for an additive abelian group" \cite{NC2012}. We take an approach different than that used by Futa et al. \cite{ZMODUL01.ABS} to use in a future article the results obtained by Artur Korni{\l}owicz \cite{TOPGRP_1.ABS}. Indeed, H{\"o}lzl et al. showed that it was possible to build ``a generic theory of limits based on filters" in Isabelle/HOL \cite{holzl2013type, boldo2014formalization}. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group \cite{bourbaki2013general} using the notion of filters.   },
    MSC2010 = {20A05 20K27 03B35},
    SECTION1 = {Additive Notation for Groups -- {\tt GROUP\_1}},
    SECTION2 = {Subgroups and Lagrange Theorem -- {\tt GROUP\_2}},
    SECTION3 = {Classes of Conjugation and Normal Subgroups -- {\tt GROUP\_3}},
    SECTION4 = {Topological Groups -- {\tt TOPGRP\_1}},
    EXTERNALREFS = {blahut2014cryptography; inui2012group; hewitt2012abstract; NC2012; holzl2013type; boldo2014formalization; bourbaki2013general; },
    INTERNALREFS = {BINOP_1.ABS;BORSUK_1.ABS;CANTOR_1.ABS;CARD_1.ABS;COMPTS_1.ABS;CONNSP_2.ABS;DOMAIN_1.ABS;FINSEQOP.ABS;
FINSET_1.ABS;FINSUB_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;IDEAL_1.ABS;INT_1.ABS;INT_2.ABS;MCART_1.ABS;MEMBERED.ABS;NAT_1.ABS;
ORDINAL1.ABS;PARTFUN1.ABS;PRE_TOPC.ABS;REALSET1.ABS;RELAT_1.ABS;RELSET_1.ABS;RLVECT_1.ABS;SETFAM_1.ABS;SETWISEO.ABS;
SUBSET_1.ABS;TOPGRP_1.ABS;TOPS_1.ABS;TOPS_2.ABS;WELLORD2.ABS;YELLOW_8.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {additive group; subgroup; Lagrange theorem; conjugation; normal subgroup; index; additive topological group; basis; neighborhood; additive abelian group; Z-module; },
    ACKNOWLEDGEMENT = {The author wants to express his gratitude to the anonymous referee for his/her work on merging the three initial articles.},
    SUBMITTED = {April 30, 2015}}
    
    
@ARTICLE{POLNOT_1.ABS,
        JOURNAL = {Formalized Mathematics},
        ISSN = {1426-2630},
        EISSN = {1898-9934},
        VOLUME = {23},
        NUMBER = {{\bf 3}},
        PAGES = {161--176},
        YEAR = {2015},
        DOI = {10.1515/forma-2015-0014},
        VERSION = {8.1.04 5.32.1240},
        TITLE = {{P}olish Notation},
        AUTHOR = {Huuskonen, Taneli},
        ADDRESS1 = {Department of Mathematics and Statistics\\University of Helsinki\\Finland},
        NOTE1 = {Work supported by Polish National Science Center (NCN) grant ``Logic of language experience" nr 2011/03/B/HS1/04580.},
        SUMMARY = {This article is the first in a series formalizing some results in my joint work with Prof.\ Joanna Goli\'nska-Pilarek (\cite{GPH:2012} and \cite{GPH:2015}) concerning a logic proposed by Prof. Andrzej Grzegorczyk (\cite{Grzegorczyk:2012}).\par We present some {\em mathematical folklore} about representing formulas in ``Polish notation", that is, with operators of fixed arity prepended to their arguments. This notation, which was published by Jan {\L}ukasiewicz in~\cite{Lukasiewicz:1931}, eliminates the need for parentheses and is generally well suited for rigorous reasoning about syntactic properties of formulas.   },
        MSC2010 = {68R15 03B35},
        SECTION1 = {Preliminaries},
        SECTION2 = {The Language},
        SECTION3 = {Parsing},
        EXTERNALREFS = {GPH:2012; GPH:2015; Grzegorczyk:2012; Lukasiewicz:1931; },
        INTERNALREFS = {BINOP_1.ABS;CARD_1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;ORDINAL1.ABS;PARTFUN1.ABS;
    PROB_2.ABS;RELAT_1.ABS;SETFAM_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
        KEYWORDS = {Polish notation; syntax; well-formed formula; },
        SUBMITTED = {April 30, 2015}}
        
        
    @ARTICLE{GRZLOG_1.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 3}},
    PAGES = {177--187},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0015},
    VERSION = {8.1.04 5.32.1240},
    TITLE = {{G}rzegorczyk's Logics. {P}art {I}},
    AUTHOR = {Huuskonen, Taneli},
    ADDRESS1 = {Department of Mathematics and Statistics\\University of Helsinki\\Finland},
    NOTE1 = {Work supported by Polish National Science Center (NCN) grant ``Logic of language experience" nr 2011/03/B/HS1/04580.},
    SUMMARY = {This article is the second in a series formalizing some results in my joint work with Prof.\ Joanna Goli\'nska-Pilarek (\cite{GPH:2012} and \cite{GPH:2015}) concerning a logic proposed by Prof. Andrzej Grzegorczyk (\cite{Grzegorczyk:2012}).\par This part presents the syntax and axioms of Grzegorczyk's {\em Logic of Descriptions} (LD) as originally proposed by him, as well as some theorems not depending on any semantic constructions. There are both some clear similarities and fundamental differences between LD and the non-Fregean logics introduced by Roman Suszko in~\cite{Suszko:1968}. In particular, we were inspired by Suszko's semantics for his non-Fregean logic~SCI, presented in~\cite{Suszko:1971}.   },
    MSC2010 = {03B60 03B35},
    SECTION1 = {The Construction of Grzegorczyk's LD Language},
    SECTION2 = {Axioms and Rules},
    SECTION3 = {Provability},
    EXTERNALREFS = {GPH:2012; GPH:2015; Grzegorczyk:2012; Suszko:1968; Suszko:1971; },
    INTERNALREFS = {BINOP_1.ABS;ENUMSET1.ABS;EQREL_1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;ORDINAL1.ABS;
PARTFUN1.ABS;POLNOT_1.ABS;RELAT_1.ABS;RELAT_2.ABS;RELSET_1.ABS;SETFAM_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {non-Fregean logic; logic of descriptions; non-classical propositional logic; equimeaning connective; },
    SUBMITTED = {April 30, 2015}}
    
    
    @ARTICLE{CARDFIL2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 3}},
    PAGES = {189--203},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0016},
    VERSION = {8.1.04 5.32.1246},
    TITLE = {{C}onvergent Filter Bases},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {We are inspired by the work of Henri Cartan \cite{cartan1937a}, Bourbaki \cite{bourbaki2013general} (TG. I Filtres) and Claude Wagschal \cite{wagschal}. We define the base of filter, image filter, convergent filter bases, limit filter and the filter base of tails (fr: {\it filtre des sections}).   },
    MSC2010 = {54A20 03B35},
    SECTION1 = {Filters -- Set-Theoretical Approach},
    SECTION2 = {Filters -- Lattice-Theoretical Approach},
    SECTION3 = {Limit of a Filter},
    SECTION4 = {Nets},
    EXTERNALREFS = {cartan1937a; bourbaki2013general; wagschal; },
    INTERNALREFS = {CANTOR_1.ABS;CARD_1.ABS;CARD_3.ABS;CARD_FIL.ABS;DICKSON.ABS;FINSET_1.ABS;FINSUB_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;
FUNCT_3.ABS;INT_1.ABS;LATTICE3.ABS;LATTICES.ABS;MCART_1.ABS;MEMBERED.ABS;NAT_1.ABS;NAT_LAT.ABS;ORDERS_2.ABS;ORDINAL1.ABS;
PARTFUN1.ABS;PRE_TOPC.ABS;RELAT_1.ABS;RELSET_1.ABS;SETFAM_1.ABS;SUBSET_1.ABS;WAYBEL_0.ABS;WAYBEL_7.ABS;YELLOW19.ABS;YELLOW_1.ABS;
YELLOW_6.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {convergence; filter; filter base; Frechet filter; limit; net; sequence; },
    SUBMITTED = {June 30, 2015}}
    
    
    @ARTICLE{ASYMPT_2.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 3}},
    PAGES = {205--213},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0017},
    VERSION = {8.1.04 5.32.1246},
    TITLE = {{P}olynomially Bounded Sequences and Polynomial Sequences},
    AUTHOR = {Okazaki, Hiroyuki and Futa, Yuichi},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
      ADDRESS2 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
      SUMMARY = {In this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems \cite{KleinbergTardos2005}, \cite{Knuth1997}. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences \cite{Barbeau2003}.   },
    MSC2010 = {03D15 68Q15 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Polynomially Bounded Sequences},
    SECTION3 = {Polynomial Sequences},
    EXTERNALREFS = {KleinbergTardos2005, Knuth1997; Barbeau2003; },
    INTERNALREFS = {AFINSQ_1.ABS;AFINSQ_2.ABS;ASYMPT_0.ABS;ASYMPT_1.ABS;COMPLEX1.ABS;FDIFF_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;
FUNCT_2.ABS;INT_1.ABS;LIMFUNC1.ABS;NAT_1.ABS;ORDINAL1.ABS;ORDINAL4.ABS;PARTFUN3.ABS;POWER.ABS;RELAT_1.ABS;RELSET_1.ABS;
SERIES_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {computational complexity; polynomial time; },
    ACKNOWLEDGEMENT = {The authors would also like to express their gratitude to Prof. Yasunari Shidama for his support and encouragement.},
    SUBMITTED = {June 30, 2015}}
    
    
    @ARTICLE{NEWTON02.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 3}},
    PAGES = {215--229},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0018},
    VERSION = {8.1.04 5.32.1246},
    TITLE = {{F}ermat's {L}ittle {T}heorem via Divisibility of {N}ewton's Binomial},
    AUTHOR = {Ziobro, Rafa{\l}},
    ADDRESS1 = {Department of Carbohydrate Technology\\University of Agriculture\\Krakow, Poland},
      SUMMARY = {Solving equations in integers is an important part of the number theory \cite{sierpinski1956}. In many cases it can be conducted by the factorization of equation's elements, such as the Newton's binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books \cite{sierpinski1950}, \cite{gancarzewicz2000}. \par In the second section of the article, Fermat's Little Theorem is proved in a classical way, on the basis of divisibility of Newton's binomial. Although slightly redundant in its content (another proof of the theorem has earlier been included in \cite{EULER_2.ABS}), the article provides a good example, how the application of registrations could shorten the length of Mizar proofs \cite{caminati2013custom}, \cite{kornilowicz2013rewriting}.   },
    MSC2010 = {11A51 11Y55 03B35},
    SECTION1 = {Divisibility of Newton's Binomial},
    SECTION2 = {Fermat's Little Theorem Revisited},
    EXTERNALREFS = {sierpinski1956; sierpinski1950; gancarzewicz2000; caminati2013custom; kornilowicz2013rewriting; },
    INTERNALREFS = {ABIAN.ABS;CARD_1.ABS;COMPLEX1.ABS;EULER_2.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;INT_1.ABS;INT_2.ABS;
NEWTON.ABS;ORDINAL1.ABS;RECDEF_1.ABS;RELAT_1.ABS;RFINSEQ.ABS;RVSUM_1.ABS;SETFAM_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {factorization; primes; Fermat; },
    ACKNOWLEDGEMENT = {Ad Maiorem Dei Gloriam},
    SUBMITTED = {June 30, 2015}}
    
    
    @ARTICLE{DUALSP03.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 3}},
    PAGES = {231--241},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0019},
    VERSION = {8.1.04 5.32.1246},
    TITLE = {{W}eak Convergence and Weak* Convergence},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285 and 23500029.},
    AUTHOR = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
    ADDRESS1 = {Hirosaki-city\\Aomori, Japan},
      ADDRESS2 = {Gifu National College of Technology\\Gifu, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By \verb!RNS_Real! Mizar functor, real normed spaces as real number spaces already defined in the article \cite{DUALSP02.ABS}, we regarded sequences of real numbers as sequences of \verb!RNS_Real!. So we proved the last theorem in this section using the theorem (8) from \cite{LOPBAN_5.ABS}. In Section 3, we defined weak sequential compactness of real normed spaces. We showed some lemmas for the proof and proved the theorem of weak sequential compactness of reflexive real Banach spaces. We referred to \cite{yoshida:1980}, \cite{ReedSimon1972}, \cite{rudin1991functional} and \cite{Brezis2011} in the formalization.   },
    MSC2010 = {46E15 46B10 03B35},
    SECTION1 = {Some Properties about Dual Spaces of Real Normed Spaces},
    SECTION2 = {Weak Convergence and Weak* Convergence},
    SECTION3 = {Weak Sequential Compactness of Real Banach Spaces},
    EXTERNALREFS = {yoshida:1980; ReedSimon1972; rudin1991functional; Brezis2011; },
    INTERNALREFS = {COMPLEX1.ABS;COMSEQ_2.ABS;DUALSP01.ABS;DUALSP02.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;HAHNBAN.ABS;
INT_1.ABS;LOPBAN_1.ABS;LOPBAN_5.ABS;MEMBERED.ABS;NAT_1.ABS;NORMSP_1.ABS;NORMSP_3.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RELAT_1.ABS;
RELSET_1.ABS;RINFSUP1.ABS;RLVECT_1.ABS;RSSPACE3.ABS;SEQ_2.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {normed linear spaces; Banach spaces; duality and reflexivity; weak topologies; weak* topologies; },
    SUBMITTED = {July 1, 2015}}
    
    
    @ARTICLE{DUALSP04.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 3}},
    PAGES = {243--252},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0020},
    VERSION = {8.1.04 5.32.1246},
    TITLE = {{T}he Orthogonal Projection and the {R}iesz Representation Theorem},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285 and 23500029.},
    AUTHOR = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
    ADDRESS1 = {Hirosaki-city\\Aomori, Japan},
      ADDRESS2 = {Gifu National College of Technology\\Gifu, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor \verb!RUSp2RNSp!, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals on real Hilbert spaces. \par Referring to the article \cite{DUALSP01.ABS}, we also defined some definitions on real Hilbert spaces and proved some theorems for defining dual spaces of real Hilbert spaces. As to the properties of all definitions, we proved that they are equivalent properties of functionals on real normed spaces. In Sec. 2, by the definitions \cite{RUSUB_5.ABS}, we showed properties of the orthogonal complement. Then we proved theorems on the orthogonal decomposition of elements of real Hilbert spaces. They are the last two theorems of existence and uniqueness. In the third and final section, we defined the kernel of linear functionals on real Hilbert spaces. By the last three theorems, we showed the Riesz representation theorem, existence, uniqueness, and the property of the norm of bounded linear functionals on real Hilbert spaces. We referred to \cite{yoshida:1980}, \cite{Dax2002}, \cite{rudin1991functional} and \cite{Brezis2011} in the formalization.   },
    MSC2010 = {46E20 46C15 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {The Orthogonal Projection},
    SECTION3 = {Riesz Representation Theorem},
    EXTERNALREFS = {yoshida:1980; Dax2002; rudin1991functional; Brezis2011; },
    INTERNALREFS = {BHSP_1.ABS;BHSP_2.ABS;BHSP_3.ABS;BHSP_6.ABS;COMPLEX1.ABS;COMSEQ_2.ABS;DUALSP01.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;
FUNCT_2.ABS;HAHNBAN.ABS;LOPBAN_1.ABS;MEMBERED.ABS;NFCONT_1.ABS;NORMSP_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RELAT_1.ABS;RELSET_1.ABS;
RLSUB_1.ABS;RLVECT_1.ABS;RSSPACE.ABS;RSSPACE3.ABS;RUSUB_1.ABS;RUSUB_5.ABS;SEQ_2.ABS;SEQ_4.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {normed linear spaces; Banach spaces; duality; orthogonal projection; Riesz representation; },
    SUBMITTED = {July 1, 2015}}
    
    
    @ARTICLE{DBLSEQ_3.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 3}},
    PAGES = {253--277},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0021},
    VERSION = {8.1.04 5.32.1246},
    TITLE = {{E}xtended Real-Valued Double Sequence and Its Convergence},
    ANNOTE = {This work was supported by JSPS KAKENHI 23500029.},
    AUTHOR = {Endou, Noboru},
    ADDRESS1 = {Gifu National College of Technology\\Gifu, Japan},
      SUMMARY = {In this article we introduce the convergence of extended real-valued double sequences \cite{FOLLAND}, \cite{GARLING:1}. It is similar to our previous articles \cite{DBLSEQ_1.ABS}, \cite{DBLSEQ_2.ABS}. In addition, we also prove Fatou's lemma and the monotone convergence theorem for double sequences.   },
    MSC2010 = {40A05 40B05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Subsequences of Convergent Extended Real-Valued Sequences},
    SECTION3 = {Convergency for Extended Real-Valued Double Sequences},
    SECTION4 = {Non-Negative Extended Real-Valued Double Sequences},
    SECTION5 = {Pringsheim Sense Convergence for Extended Real-Valued Double Sequences},
    EXTERNALREFS = {FOLLAND; GARLING:1; },
    INTERNALREFS = {BINOP_1.ABS;COMSEQ_2.ABS;DBLSEQ_1.ABS;DBLSEQ_2.ABS;EXTREAL1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;INT_1.ABS;
MESFUNC1.ABS;MESFUNC5.ABS;MESFUNC9.ABS;NAT_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RAT_1.ABS;RELAT_1.ABS;RELSET_1.ABS;RINFSUP2.ABS;
SEQ_2.ABS;SUBSET_1.ABS;SUPINF_1.ABS;SUPINF_2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {double sequence; Fatou's lemma for double sequence; monotone convergence theorem for double sequence; },
    SUBMITTED = {July 1, 2015}}
            

@ARTICLE{CARDFIL3.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 4}},
    PAGES = {279--288},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0022},
    VERSION = {8.1.04 5.34.1256},
    TITLE = {{S}ummable Family in a Commutative Group},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
    SUMMARY = {H{\"o}lzl et al. showed that it was possible to build ``a generic theory of limits based on filters" in Isabelle/HOL \cite{holzl2013type}, \cite{boldo2014formalization}. In this paper we present our formalization of this theory in Mizar \cite{Mizar-State-2015}. \par First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space \cite{FRECHET.ABS}, a real normed linear space \cite{NORMSP_1.ABS} and a linear topological space \cite{RLTOPSP1.ABS} with the concept of the limit of an image filter \cite{CARDFIL2.ABS}. \par Then, following Bourbaki \cite{bourbaki2007topologie}, \cite{bourbaki2013general} (TG.III, \S 5.1 {\it Familles sommables dans un groupe commutatif}), we conclude by defining the summable families in a commutative group (``additive notation" in \cite{GROUP_1A.ABS}), using the notion of filters.   },
    MSC2010 = {54A20 54H11 22A05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Convergence in Metric Spaces},
    SECTION3 = {Filter and Limit of a Sequence in Real Normed Space},
    SECTION4 = {Filter and Limit of a Sequence in Linear Topological Space},
    SECTION5 = {Series in Abelian Group: a Definition},
    SECTION6 = {Product of Family as Limit in Commutative Topological Group},
    SECTION7 = {Summable Family in Commutative Topological Group},
    EXTERNALREFS = {holzl2013type; boldo2014formalization; Mizar-State-2015; bourbaki2007topologie; bourbaki2013general; },
    INTERNALREFS = {ALGSTR_1.ABS;CARDFIL2.ABS;CARD_1.ABS;CONNSP_2.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FINSOP_1.ABS;FINSUB_1.ABS;
FRECHET.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GROUP_1.ABS;GROUP_1A.ABS;METRIC_1.ABS;NORMSP_1.ABS;NORMSP_2.ABS;ORDERS_2.ABS;ORDINAL1.ABS;
PARTFUN1.ABS;PCOMPS_1.ABS;PRE_TOPC.ABS;RELAT_1.ABS;RLTOPSP1.ABS;RLVECT_1.ABS;RUSUB_4.ABS;SETWISEO.ABS;SUBSET_1.ABS;TOPGRP_1.ABS;
WAYBEL_0.ABS;YELLOW13.ABS;YELLOW19.ABS;YELLOW_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {limits; filters; topological group; summable family; convergence series; linear topological space; },
    SUBMITTED = {August 14, 2015}}
    
    
    @ARTICLE{FINTOPO7.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 4}},
    PAGES = {289--296},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0023},
    VERSION = {8.1.04 5.34.1256},
    TITLE = {{T}opology from Neighbourhoods},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
    SUMMARY = {Using Mizar \cite{Mizar-State-2015}, and the formal topological space structure ({\tt FMT\_Space\_Str}) \cite{FINTOPO2.ABS}, we introduce the three {\tt U-FMT} conditions ({\tt U-FMT filter}, {\tt U-FMT with point} and {\tt U-FMT local}) similar to those $V_I$, $V_{II}$, $V_{III}$ and $V_{IV}$ of the proposition 2 in \cite{Bourbaki2013general}: \begin{quote} If to each element $x$ of a set $X$ there corresponds a set ${\cal{B}}(x)$ of subsets of $X$ such that the properties $V_I$, $V_{II}$, $V_{III}$ and $V_{IV}$ are satisfied, then there is a unique topological structure on $X$ such that, for each $x\in X$, ${\cal{B}}(x)$ is the set of neighborhoods of $x$ in this topology. \end{quote} We present a correspondence between a topological space and a space defined with the formal topological space structure with the three {\tt U-FMT} conditions called the topology from neighbourhoods. For the formalization, we were inspired by the works of Bourbaki \cite{bourbaki2007topologie} and Claude Wagschal \cite{wagschal}.   },
    MSC2010 = {54A05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Open, Neighborhood and Conditions for Topological Space from Neighborhoods},
    SECTION3 = {Topology from Neighborhoods: a Definition},
    SECTION4 = {Basis},
    SECTION5 = {Correspondence between Topological Space and Topology from Neighborhoods},
    EXTERNALREFS = {Mizar-State-2015; Bourbaki2013general; bourbaki2007topologie; wagschal; },
    INTERNALREFS = {CANTOR_1.ABS;CARDFIL2.ABS;CARD_1.ABS;CARD_FIL.ABS;FINSET_1.ABS;FINTOPO2.ABS;FUNCT_1.ABS;FUNCT_2.ABS;
INT_1.ABS;LATTICE3.ABS;LATTICES.ABS;MEMBERED.ABS;ORDERS_2.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PRE_TOPC.ABS;RELAT_1.ABS;RELSET_1.ABS;
SETFAM_1.ABS;SUBSET_1.ABS;WAYBEL_0.ABS;WAYBEL_7.ABS;YELLOW19.ABS;YELLOW_1.ABS;YELLOW_6.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {filter; topological space; neighbourhoods system; },
    SUBMITTED = {August 14, 2015}}
    
    
    @ARTICLE{ZMODUL07.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 4}},
    PAGES = {297--307},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0024},
    VERSION = {8.1.04 5.33.1254},
    TITLE = {{T}orsion {P}art of {$\mathbb{Z}$}-module},
    AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
    ADDRESS1 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    ADDRESS3 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we formalize in Mizar \cite{Mizar-State-2015} the definition of ``torsion part" of $\mathbb Z$-module and its properties. We show $\mathbb Z$-module generated by the field of rational numbers as an example of torsion-free non free $\mathbb Z$-modules. We also formalize the rank-nullity theorem over finite-rank free $\mathbb Z$-modules (previously formalized in \cite{RANKNULL.ABS}). $\mathbb Z$-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm \cite{LLL} and cryptographic systems with lattices \cite{LATTICE2002}.   },
    MSC2010 = {15A03 13C12 03B35},
    SECTION1 = {Torsion Part of $\mathbb{Z}$-module},
    SECTION2 = {$\mathbb Z$-module Generated by the Field of Rational Numbers},
    SECTION3 = {The Rank-Nullity Theorem},
    EXTERNALREFS = {Mizar-State-2015; LLL; LATTICE2002; },
    INTERNALREFS = {BINOM.ABS;BINOP_1.ABS;CARD_1.ABS;DOMAIN_1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;
GAUSSINT.ABS;INT_1.ABS;INT_2.ABS;INT_3.ABS;MOD_2.ABS;NAT_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RAT_1.ABS;RELAT_1.ABS;
RELSET_1.ABS;RLVECT_1.ABS;SUBSET_1.ABS;VECTSP_1.ABS;VECTSP_4.ABS;VECTSP_5.ABS;VECTSP_6.ABS;VECTSP_7.ABS;ZFMISC_1.ABS;
ZMODUL01.ABS;ZMODUL02.ABS;ZMODUL03.ABS;ZMODUL05.ABS;ZMODUL06.ABS;},
    KEYWORDS = {torsion part of $\mathbb{Z}$-module; torsion-free non free $\mathbb{Z}$-module; },
    SUBMITTED = {August 14, 2015}}
    
    
    @ARTICLE{MEASURE9.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 4}},
    PAGES = {309--323},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0025},
    VERSION = {8.1.04 5.33.1254},
    TITLE = {{C}onstruction of Measure from Semialgebra of Sets},
    ANNOTE = {This work was supported by JSPS KAKENHI 23500029.},
    AUTHOR = {Endou, Noboru},
    ADDRESS1 = {Gifu National College of Technology\\Gifu, Japan},
    SUMMARY = {In our previous article \cite{MEASURE8.ABS}, we showed complete additivity as a condition for extension of a measure. However, this condition premised the existence of a $\sigma$-field and the measure on it. In general, the existence of the measure on $\sigma$-field is not obvious. On the other hand, the proof of existence of a measure on a semialgebra is easier than in the case of a $\sigma$-field. Therefore, in this article we define a measure ({\tt pre-measure}) on a semialgebra and extend it to a measure on a $\sigma$-field. Furthermore, we give a $\sigma$-measure as an extension of the measure on a $\sigma$-field. We follow \cite{Halmos74}, \cite{Bogachev2006}, and \cite{Rao2004}.   },
    MSC2010 = {28A12 03B35},
    SECTION1 = {Joining Finite Sequences},
    SECTION2 = {Extended Real-Valued Matrix},
    SECTION3 = {Definition of Pre-Measure},
    SECTION4 = {Pre-Measure on Semialgebra and Construction of Measure},
    EXTERNALREFS = {Halmos74; Bogachev2006; Rao2004; },
    INTERNALREFS = {CARD_1.ABS;CARD_3.ABS;EXTREAL1.ABS;FINSEQOP.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSET_1.ABS;FINSUB_1.ABS;
FUNCT_1.ABS;FUNCT_2.ABS;MATRLIN.ABS;MEASURE1.ABS;MEASURE4.ABS;MEASURE8.ABS;MESFUNC1.ABS;MESFUNC5.ABS;MESFUNC9.ABS;
NAT_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PROB_1.ABS;PROB_2.ABS;RELAT_1.ABS;RELSET_1.ABS;SERIES_1.ABS;SETFAM_1.ABS;
SRINGS_3.ABS;SUBSET_1.ABS;SUPINF_1.ABS;SUPINF_2.ABS;WSIERP_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {measure theory; pre-measure; },
    SUBMITTED = {August 14, 2015}}
    
    
    @ARTICLE{PETERSON.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 4}},
    PAGES = {325--331},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0026},
    VERSION = {8.1.04 5.33.1254},
    TITLE = {{E}vent-Based Proof of the Mutual Exclusion Property of {P}eterson's Algorithm},
    AUTHOR = {Ivanov, Ievgen and Nikitchenko, Mykola and Abraham, Uri},
    ADDRESS1 = {Taras Shevchenko National University\\Kyiv, Ukraine},
    ADDRESS2 = {Taras Shevchenko National University\\Kyiv, Ukraine},
    ADDRESS3 = {Ben-Gurion University\\Beer-Sheva, Israel},
    SUMMARY = {Proving properties of distributed algorithms is still a highly challenging problem and various approaches that have been proposed to tackle it \cite{Abraham1999} can be roughly divided into state-based and event-based proofs. Informally speaking, state-based approaches define the behavior of a distributed algorithm as a set of sequences of memory states during its executions, while event-based approaches treat the behaviors by means of events which are produced by the executions of an algorithm. Of course, combined approaches are also possible. \par Analysis of the literature \cite{Abraham1999}, \cite{chandy1988}, \cite{Pratt1986}, \cite{lamport1986}, \cite{raynal991}, \cite{Ridge2006}, \cite{Ridge2007} shows that state-based approaches are more widely used than event-based approaches for proving properties of algorithms, and the difficulties in the event-based approach are often emphasized. We believe, however, that there is a certain naturalness and intuitive content in event-based proofs of correctness of distributed algorithms that makes this approach worthwhile. Besides, state-based proofs of correctness of distributed algorithms are usually applicable only to discrete-time models of distributed systems and cannot be easily adapted to the continuous time case which is important in the domain of cyber-physical systems. On the other hand, event-based proofs can be readily applied to continuous-time / hybrid models of distributed systems. \par In the paper \cite{AbrIvNikitch2011} we presented a compositional approach to reasoning about behavior of distributed systems in terms of events. Compositionality here means (informally) that semantics and properties of a program is determined by semantics of processes and process communication mechanisms. We demonstrated the proposed approach on a proof of the mutual exclusion property of the Peterson's algorithm \cite{Peterson1981}. We have also demonstrated an application of this approach for proving the mutual exclusion property in the setting of continuous-time models of cyber-physical systems in \cite{IvanovNikitchAbr2014}. \par Using Mizar \cite{Mizar-State-2015}, in this paper we give a formal proof of the mutual exclusion property of the Peterson's algorithm in Mizar on the basis of the event-based approach proposed in \cite{AbrIvNikitch2011}. Firstly, we define an event-based model of a shared-memory distributed system as a multi-sorted algebraic structure in which sorts are events, processes, locations (i.e. addresses in the shared memory), traces (of the system). The operations of this structure include a binary precedence relation $\leq$ on the set of events which turns it into a linear preorder (events are considered simultaneous, if ${e_1}\leq{e_2}$ and ${e_2}\leq{e_1}$), special predicates which check if an event occurs in a given process or trace, predicates which check if an event causes the system to read from or write to a given memory location, and a special partial function ``{\tt val of}" on events which gives the value associated with a memory read or write event (i.e. a value which is written or is read in this event) \cite{AbrIvNikitch2011}. Then we define several natural consistency requirements (axioms) for this structure which must hold in every distributed system, e.g. each event occurs in some process, etc. (details are given in \cite{AbrIvNikitch2011}). \par After this we formulate and prove the main theorem about the mutual exclusion property of the Peterson's algorithm in an arbitrary consistent algebraic structure of events. Informally, the main theorem states that if a system consists of two processes, and in some trace there occur two events $e_1$ and $e_2$ in different processes and each of these events is preceded by a series of three special events (in the same process) guaranteed by execution of the Peterson's algorithm (setting the flag of the current process, writing the identifier of the opposite process to the ``turn" shared variable, and reading zero from the flag of the opposite process or reading the identifier of the current process from the ``turn" variable), and moreover, if neither process writes to the flag of the opposite process or writes its own identifier to the ``turn" variable, then either the events $e_1$ and $e_2$ coincide, or they are not simultaneous (mutual exclusion property).   },
    MSC2010 = {68M14 68W15 68N30 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Peterson's Algorithm},
    EXTERNALREFS = {Abraham1999; chandy1988; Pratt1986; lamport1986; raynal991; Ridge2006; Ridge2007; AbrIvNikitch2011; Peterson1981; IvanovNikitchAbr2014; Mizar-State-2015; },
    INTERNALREFS = {FUNCT_1.ABS;FUNCT_2.ABS;ORDERS_2.ABS;RELAT_1.ABS;RELAT_2.ABS;SETFAM_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {distributed system; parallel computing; algorithm; verification; mathematical model; },
    SUBMITTED = {August 14, 2015}}
    
    
    @ARTICLE{RING_3.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 4}},
    PAGES = {333--349},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0027},
    VERSION = {8.1.04 5.33.1254},
    TITLE = {{C}haracteristic of Rings. {P}rime Fields},
    AUTHOR = {Schwarzweller, Christoph and Korni{\l}owicz, Artur},
    ADDRESS1 = {Institute of Computer Science\\University of Gda{\'n}sk\\Poland},
    ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    SUMMARY = {The notion of the characteristic of rings and its basic properties are formalized \cite{Jacobson2009}, \cite{Waerden2003}, \cite{Luneburg1999}. Classification of prime fields in terms of isomorphisms with appropriate fields ($\mathbb{Q}$ or $\mathbb{Z}/p$) are presented. To facilitate reasonings within the field of rational numbers, values of numerators and denominators of basic operations over rationals are computed.   },
    MSC2010 = {13A35 12E05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Properties of Fractions},
    SECTION3 = {Preliminaries about Rings and Fields},
    SECTION4 = {Embedding the Integers in Rings},
    SECTION5 = {Mono- and Isomorphisms of Rings},
    SECTION6 = {Characteristic of Rings},
    SECTION7 = {Prime Fields},
    EXTERNALREFS = {Jacobson2009; Waerden2003; Luneburg1999; },
    INTERNALREFS = {ALGSTR_1.ABS;BINOM.ABS;BINOP_1.ABS;C0SP1.ABS;CARD_1.ABS;COMPLFLD.ABS;DOMAIN_1.ABS;EC_PF_1.ABS;FINSET_1.ABS;
FUNCT_1.ABS;FUNCT_2.ABS;GAUSSINT.ABS;GCD_1.ABS;GROUP_1.ABS;GROUP_6.ABS;IDEAL_1.ABS;INT_1.ABS;INT_2.ABS;INT_3.ABS;
MATRLIN2.ABS;MEMBERED.ABS;MOD_4.ABS;NAT_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;QUOFIELD.ABS;RAT_1.ABS;REALSET1.ABS;RELAT_1.ABS;
RINGCAT1.ABS;RING_2.ABS;RLVECT_1.ABS;SUBSET_1.ABS;VECTSP10.ABS;VECTSP_1.ABS;VECTSP_2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {commutative algebra; characteristic of rings; prime field; },
    SUBMITTED = {August 14, 2015}}
    
    
    @ARTICLE{CAT_8.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 4}},
    PAGES = {351--369},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0028},
    VERSION = {8.1.04 5.33.1254},
    TITLE = {{E}xponential Objects},
    AUTHOR = {Riccardi, Marco},
    ADDRESS1 = {Via del Pero 102\\54038 Montignoso\\Italy},
    SUMMARY = {In the first part of this article we formalize the concepts of terminal and initial object, categorical product \cite{Borceaux} and natural transformation within a free-object category \cite{Adamek:2009}. In particular, we show that this definition of natural transformation is equivalent to the standard definition \cite{MacLane:1}. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category\cite{Lawvere:1963}.   },
    MSC2010 = {18A99 18A25 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Terminal Objects},
    SECTION3 = {Initial Objects},
    SECTION4 = {Categorical Products},
    SECTION5 = {Natural Transformations},
    SECTION6 = {Exponential Objects},
    EXTERNALREFS = {Borceaux; Adamek:2009; MacLane:1; Lawvere:1963; },
    INTERNALREFS = {CARD_1.ABS;CAT_1.ABS;CAT_6.ABS;CAT_7.ABS;ENUMSET1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GRAPH_1.ABS;
ISOCAT_1.ABS;NATTRA_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RELAT_1.ABS;RELSET_1.ABS;SETFAM_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {exponential objects; functor category; natural transformation; },
    SUBMITTED = {August 15, 2015}}
    
    
    @ARTICLE{ASYMPT_3.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 4}},
    PAGES = {371--378},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0029},
    VERSION = {8.1.04 5.33.1254},
    TITLE = {{A}lgebra of Polynomially Bounded Sequences and Negligible Functions},
    AUTHOR = {Okazaki, Hiroyuki},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article we formalize negligible functions that play an essential role in cryptology \cite{GOLDREICH}, \cite{BELLARE}. Generally, a cryptosystem is secure if the probability of succeeding any attacks against the cryptosystem is negligible. First, we formalize the algebra of polynomially bounded sequences \cite{ASYMPT_2.ABS}. Next, we formalize negligible functions and prove the set of negligible functions is a subset of the algebra of polynomially bounded sequences. Moreover, we then introduce equivalence relation between polynomially bounded sequences, using negligible functions.   },
    MSC2010 = {68Q25 94A60 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Algebra of Polynomially Bounded Sequences},
    SECTION3 = {Negligible Functions},
    EXTERNALREFS = {GOLDREICH; BELLARE; },
    INTERNALREFS = {AFINSQ_1.ABS;ASYMPT_0.ABS;ASYMPT_1.ABS;ASYMPT_2.ABS;COMPLEX1.ABS;COMSEQ_2.ABS;FINSET_1.ABS;FUNCOP_1.ABS;
FUNCSDOM.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GROUP_1.ABS;INT_1.ABS;NAT_1.ABS;PARTFUN1.ABS;PARTFUN3.ABS;POWER.ABS;RAT_1.ABS;
REALSET1.ABS;REAL_1.ABS;RELAT_1.ABS;RLVECT_1.ABS;SEQ_1.ABS;SEQ_2.ABS;SERIES_1.ABS;SUBSET_1.ABS;VECTSP_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {polynomially bounded function; negligible functions; },
    ACKNOWLEDGEMENT = {The author would like to express his gratitude to Prof. Yuichi Futa and Prof. Yasunari Shidama for their support and encouragement.},
    SUBMITTED = {August 15, 2015}}
    
    
    @ARTICLE{LTLAXIO5.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 4}},
    PAGES = {379--386},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0030},
    VERSION = {8.1.04 5.34.1256},
    TITLE = {{P}ropositional Linear Temporal Logic with~Initial Validity Semantics},
    ANNOTE = {This work was supported by the University of Bialystok grants: BST447 {\it Formalization of temporal logics in a proof-assistant. Application to System Verification}, and BST225 {\it Database of mathematical texts checked by computer}.},
    AUTHOR = {Giero, Mariusz},
    ADDRESS1 = {Faculty of Economics and Informatics\\ University of Bia{\l}ystok\\ Kalvariju 135, LT-08221 Vilnius\\ Lithuania},
    SUMMARY = {In the article \cite{LTLAXIO1.ABS} a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of ``until" operator in a very strict version. The very strict ``until" operator enables to express all other temporal operators.\par In this article we construct a formal system for LTLB with the initial semantics \cite{KroMer}. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article \cite{LTLAXIO1.ABS} and the one introduced in this article. \par Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in \cite{LTLAXIO1.ABS} and in this article can be used to carry out such verifications in Mizar \cite{Mizar-State-2015}.   },
    MSC2010 = {03B70 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Initial Validity Semantics - Definitions},
    SECTION3 = {The Connections between Normal Semantics and Initial Semantics},
    SECTION4 = {A Formal System (Hilbert-like) for LTLB with Initial Semantics},
    SECTION5 = {Soundness Theorem for LTLB with Initial Semantics},
    SECTION6 = {Weak Completeness Theorem for LTLB with Initial Semantics},
    SECTION7 = {Deduction Theorem},
    SECTION8 = {The Connections between Derivability in the Formal System for LTLB with Normal Semantics and the Formal System for LTLB with Initial Semantics},
    EXTERNALREFS = {KroMer; Mizar-State-2015; },
    INTERNALREFS = {DOMAIN_1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;HILBERT1.ABS;HILBERT2.ABS;LTLAXIO1.ABS;
MARGREL1.ABS;NAT_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RELAT_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {temporal logic; very strict until operator; completeness; },
    SUBMITTED = {October 22, 2015}}
    
    
    @ARTICLE{LATSTONE.ABS,
    JOURNAL = {Formalized Mathematics},
    ISSN = {1426-2630},
    EISSN = {1898-9934},
    VOLUME = {23},
    NUMBER = {{\bf 4}},
    PAGES = {387--396},
    YEAR = {2015},
    DOI = {10.1515/forma-2015-0031},
    VERSION = {8.1.04 5.34.1256},
    TITLE = {{S}tone Lattices},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\ University of Bia{\l}ystok\\ Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
    SUMMARY = {The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense. \par The core of the paper is of course the idea of Stone identity $$a^{\star} \sqcup a^{\star\star} = \top,$$ which is fundamental for us: Stone lattices are those lattices $L$, which are distributive, bounded, and satisfy Stone identity for all elements $a\in L.$ Stone algebras were introduced by Gr\"atzer and Schmidt in \cite{Gratzer:1957}. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined $a^{\star}$ as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices. \par All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of $p^2$ for arbitrary prime number $p$ (Section 6). At the end we formalize the notion of the Stone lattice $B^{[2]}$ (of pairs of elements $a,b$ of $B$ such that $a\leq b$) constructed as a sublattice of $B^2$, where $B$ is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices. \par Our formalization was inspired by \cite{Gratzer2011}, and is an important step in formalizing Jouni J{\"a}rvinen \emph{Lattice theory for rough sets} \cite{Jarvinen:2007}, so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423--426. The description of handling complemented structures in Mizar \cite{Mizar-State-2015} can be found in \cite{GrabowskiJAR40}. The current article together with \cite{NELSON_1.ABS} establishes the formal background for algebraic structures which are important for \cite{GrabowskiAssisted:2005}, \cite{GrabowskiPerspective:2007} by means of mechanisms of merging theories as described in \cite{GrabowskiFI:2014}.   },
    MSC2010 = {06D15 06E75 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Pseudocomplements in Lattices},
    SECTION3 = {Skeleton of a Pseudocomplemented Lattice},
    SECTION4 = {Stone Identity},
    SECTION5 = {Dense Elements in Lattices},
    SECTION6 = {An Example: Lattice of Natural Divisors},
    SECTION7 = {Products of Pseudocomplemented Lattices},
    SECTION8 = {Special Construction: ${B}^{[2]}$},
    EXTERNALREFS = {Gratzer:1957; Gratzer2011; Jarvinen:2007; Mizar-State-2015; GrabowskiJAR40; GrabowskiAssisted:2005; GrabowskiPerspective:2007; GrabowskiFI:2014; },
    INTERNALREFS = {CARD_1.ABS;ENUMSET1.ABS;FILTER_1.ABS;FILTER_2.ABS;FINSET_1.ABS;INT_1.ABS;INT_2.ABS;LATTICE3.ABS;
LATTICEA.ABS;LATTICES.ABS;MOEBIUS1.ABS;MOEBIUS2.ABS;NAT_LAT.ABS;ORDINAL1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {pseudocomplemented lattices; Stone lattices; Boolean lattices; lattice of natural divisors; },
    SUBMITTED = {October 22, 2015}}
    
    
    @ARTICLE{FINANCE3.ABS,
        JOURNAL = {Formalized Mathematics},
        ISSN = {1426-2630},
        EISSN = {1898-9934},
        VOLUME = {24},
        NUMBER = {{\bf 1}},
        PAGES = {1--16},
        YEAR = {2016},
        DOI = {10.1515/forma-2016-0001},
        VERSION = {8.1.04 5.36.1267},
        TITLE = {{M}odelling Real World Using Stochastic Processes and Filtration},
        AUTHOR = {Jaeger, Peter},
        ADDRESS1 = {Siegmund-Schacky-Str. 18a\\80993 Munich, Germany},
        SUMMARY = {First we give an implementation in Mizar \cite{Mizar-State-2015} basic important definitions of stochastic finance, i.e. filtration (\cite{klenke:2006}, pp. 183 and 185), adapted stochastic process (\cite{klenke:2006}, p. 185) and predictable stochastic process (\cite{follmerschied:2004}, p. 224). Second we give some concrete formalization and verification to real world examples. \par In article \cite{FINANCE2.ABS} we started to define random variables for a similar presentation to the book \cite{follmerschied:2004}. Here we continue this study. Next we define the stochastic process. For further definitions based on stochastic process we implement the definition of filtration. \par To get a better understanding we give a real world example and connect the statements to the theorems. Other similar examples are given in \cite{kremer:2006}, pp. 143--159 and in \cite{sandmann:2001}, pp. 110--124. First we introduce sets which give informations referring to today ($\Omega _{now}$, Def.6), tomorrow ($\Omega _{fut_{1}}$, Def.7) and the day after tomorrow ($\Omega _{fut_{2}}$, Def.8). We give an overview for some events in the $\sigma$-algebras $\Omega _{now} , \Omega _{fut1}$ and $\Omega _{fut2}$, see theorems (22) and (23).\par The given events are necessary for creating our next functions. The implementations take the form of: $\Omega _{now} \subset \Omega _{fut1} \subset \Omega _{fut2}$ see theorem (24). This tells us growing informations from now to the future 1$=$now, 2$=$tomorrow, 3$=$the day after tomorrow.\par We install functions $f: \lbrace 1,2,3,4 \rbrace \rightarrow \mathbb{R}$ as following:\par $f_1:x \rightarrow 100, \forall x \in$ dom $f$, see theorem (36),\par $f_2:x \rightarrow 80 $, for $ x=1$ or $x=2$ and\par $f_2:x \rightarrow 120 $, for $ x=3$ or $x=4$, see theorem (37),\par $f_3:x \rightarrow 60$, for $x=1, \, f_3:x \rightarrow 80$, for $ x=2$ and\par $f_3:x \rightarrow 100$, for $ x=3, \, f_3:x \rightarrow 120$, for $ x=4$ see theorem (38).\par These functions are real random variable: $f_{1}$ over $\Omega _{now}$, $f_{2}$ over $\Omega _{fut1}$, $f_{3}$ over $\Omega _{fut2}$, see theorems (46), (43) and (40). We can prove that these functions can be used for giving an example for an adapted stochastic process. See theorem (49).\par We want to give an interpretation to these functions: suppose you have an equity $A$ which has now $(=w _{1})$ the value 100. Tomorrow $A$ changes depending which scenario occurs $-$ e.g. another marketing strategy. In scenario 1 $(=w _{11})$ it has the value 80, in scenario 2 $(=w _{12})$ it has the value 120. The day after tomorrow $A$ changes again. In scenario 1 $(=w _{111})$ it has the value 60, in scenario 2 $(=w _{112})$ the value 80, in scenario 3 $(=w _{121})$ the value 100 and in scenario 4 $(=w _{122})$ it has the value 120. For a visualization refer to the tree: \par\par {\begin{array}{lll} Now & tomorrow & the~day~after~tomorrow\\ &&w_{111}=\lbrace 1 \rbrace \\ &w_{11}=\lbrace 1,2 \rbrace \ \ \ <\\ &&w_{112}=\lbrace 2 \rbrace \\ w_{1} = \lbrace 1,2,3,4 \rbrace \ \ \ <&&\\ &&w_{121}=\lbrace 3 \rbrace \\ &w_{12}=\lbrace 3,4 \rbrace \ \ \ <\\ &&w_{122}=\lbrace 4 \rbrace \end{array}} \par\par The sets $w _{1}$,$w _{11}$,$w _{12}$,$w _{111}$,$w _{112}$,$w _{121}$,$w _{122}$ which are subsets of $\lbrace 1,2,3,4 \rbrace$, see (22), tell us which market scenario occurs. The functions tell us the values to the relevant market scenario:\par\par {\begin{array}{lll} Now & tomorrow & the~day~after~tomorrow\\ &&f_{3}(w_{i})= 60, ~w_{i} {\rm ~in~} w_{111} \\ &f_{2}(w_{i})= 80 \ \ \ <\\ &w_{i} {\rm ~in~} w_{11}&f_{3}(w_{i})= 80, ~w_{i} {\rm ~in~} w_{112} \\ f_{1}(w_{i})= 100 \ \ \ <&&\\ w_{i} {\rm ~in~} w_{1}&&f_{3}(w_{i})= 100, ~w_{i} {\rm ~in~} w_{121} \\ &f_{2}(w_{i})= 120 \ \ \ <\\ &w_{i} {\rm ~in~} w_{12}&f_{3}(w_{i})= 120, ~w_{i} {\rm ~in~} w_{122} \end{array}} \par\par For a better understanding of the definition of the random variable and the relation to the functions refer to \cite{georgii:2004}, p. 20. For the proof of certain sets as $\sigma$-fields refer to \cite{georgii:2004}, pp. 10--11 and \cite{klenke:2006}, pp. 1--2.\par This article is the next step to the {\it arbitrage opportunity}. If you use for example a simple probability measure, refer, for example to literature \cite{caratheodorycommunity:2004}, pp. 28--34, \cite{follmerschied:2004}, p. 6 and p. 232 you can calculate whether an {\it arbitrage} exists or not. Note, that the example given in literature \cite{caratheodorycommunity:2004} needs 8 instead of 4 informations as in our model. If we want to code the first 3 given time points into our model we would have the following graph, see theorems (47), (44) and (41):\par\par {\begin{array}{lll} Now & tomorrow & the~day~after~tomorrow\\ &&f_{3}(w_{i})= 180, ~w_{i} {\rm ~in~} w_{111} \\ &f_{2}(w_{i})= 150 \ \ \ <\\ &w_{i} {\rm ~in~} w_{11}&f_{3}(w_{i})= 120, ~w_{i} {\rm ~in~} w_{112} \\ f_{1}(w_{i})= 125 \ \ \ <&&\\ w_{i} {\rm ~in~} w_{1}&&f_{3}(w_{i})= 120, ~w_{i} {\rm ~in~} w_{121} \\ &f_{2}(w_{i})= 100 \ \ \ <\\ &w_{i} {\rm ~in~} w_{12}&f_{3}(w_{i})= 80, ~w_{i} {\rm ~in~} w_{122} \end{array}} \par\par The function for the ``Call-Option" is given in literature \cite{caratheodorycommunity:2004}, p. 28. The function is realized in Def.5. As a background, more examples for using the definition of filtration are given in \cite{klenke:2006}, pp. 185--188.   },
        MSC2010 = {60G05 03B35},
        SECTION1 = {Preliminaries},
        SECTION2 = {Special Random Variables},
        SECTION3 = {Special $\sigma$-Fields},
        SECTION4 = {Construction of Filtration and Examples},
        SECTION5 = {Stochastic Process: Adapted and Predictable},
        SECTION6 = {Example for a Stochastic Process},
        EXTERNALREFS = {Mizar-State-2015; klenke:2006; follmerschied:2004; kremer:2006; sandmann:2001; georgii:2004; 
        caratheodorycommunity:2004; },
        INTERNALREFS = {FINANCE2.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;NAT_1.ABS;PROB_1.ABS;},
        KEYWORDS = {stochastic process; random variable; },
        SUBMITTED = {December 30, 2015}}
        
        
        @ARTICLE{EUCLID12.ABS,
        JOURNAL = {Formalized Mathematics},
        ISSN = {1426-2630},
        EISSN = {1898-9934},
        VOLUME = {24},
        NUMBER = {{\bf 1}},
        PAGES = {17--26},
        YEAR = {2016},
        DOI = {10.1515/forma-2016-0002},
        VERSION = {8.1.04 5.36.1267},
        TITLE = {{C}ircumcenter, Circumcircle and Centroid of a Triangle},
        AUTHOR = {Coghetto, Roland},
        ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
        SUMMARY = {We introduce, using the Mizar system \cite{Mizar-State-2015}, some basic concepts of Euclidean geometry: the half length and the midpoint of a segment, the perpendicular bisector of a segment, the medians (the cevians that join the vertices of a triangle to the midpoints of the opposite sides) of a triangle. \par We prove the existence and uniqueness of the circumcenter of a triangle (the intersection of the three perpendicular bisectors of the sides of the triangle). The extended law of sines and the formula of the radius of the Morley's trisector triangle are formalized \cite{Coxeter:1967}. \par Using the generalized Ceva's Theorem, we prove the existence and uniqueness of the centroid (the common point of the medians \cite{hartshorne2000geometry}) of a triangle.   },
        MSC2010 = {51M04 03B35},
        SECTION1 = {Preliminaries},
        SECTION2 = {Betweenness},
        SECTION3 = {Real Plane},
        SECTION4 = {The Midpoint of a Segment},
        SECTION5 = {Perpendicularity},
        SECTION6 = {The Perpendicular Bisector of a Segment},
        SECTION7 = {The Circumcircle of a Triangle},
        SECTION8 = {Extended Law of Sines},
        SECTION9 = {The Centroid of a Triangle},
        EXTERNALREFS = {Mizar-State-2015; Coxeter:1967; hartshorne2000geometry; },
        INTERNALREFS = {EUCLIDLP.ABS;EUCLID_6.ABS;RLTOPSP1.ABS;},
        KEYWORDS = {Euclidean geometry; perpendicular bisector; circumcenter; circumcircle; centroid; extended law of sines; },
        SUBMITTED = {December 30, 2015}}
        
        
        @ARTICLE{EUCLID13.ABS,
        JOURNAL = {Formalized Mathematics},
        ISSN = {1426-2630},
        EISSN = {1898-9934},
        VOLUME = {24},
        NUMBER = {{\bf 1}},
        PAGES = {27--36},
        YEAR = {2016},
        DOI = {10.1515/forma-2016-0003},
        VERSION = {8.1.04 5.36.1267},
        TITLE = {{A}ltitude, Orthocenter of a Triangle and Triangulation},
        AUTHOR = {Coghetto, Roland},
        ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
        SUMMARY = {We introduce the altitudes of a triangle (the cevians perpendicular to the opposite sides). Using the generalized Ceva's Theorem, we prove the existence and uniqueness of the orthocenter of a triangle \cite{Coxeter:1967}. Finally, we formalize in Mizar \cite{Mizar-State-2015} some formulas \cite{campbell1956trigonometrie} to calculate distance using triangulation.   },
        MSC2010 = {51M04 03B35},
        SECTION1 = {Preliminaries},
        SECTION2 = {Orthocenter},
        SECTION3 = {Triangulation},
        EXTERNALREFS = {Coxeter:1967; Mizar-State-2015; campbell1956trigonometrie; },
        INTERNALREFS = {COMPLEX2.ABS;EUCLID10.ABS;EUCLID11.ABS;EUCLID12.ABS;EUCLIDLP.ABS;EUCLID_3.ABS;EUCLID_4.ABS;EUCLID_6.ABS;
        MENELAUS.ABS;SIN_COS.ABS;SQUARE_1.ABS;},
        KEYWORDS = {Euclidean geometry; trigonometry; altitude; orthocenter; triangulation; distance; },
        SUBMITTED = {December 30, 2015}}
        
        
        @ARTICLE{ZMODUL08.ABS,
        JOURNAL = {Formalized Mathematics},
        ISSN = {1426-2630},
        EISSN = {1898-9934},
        VOLUME = {24},
        NUMBER = {{\bf 1}},
        PAGES = {37--47},
        YEAR = {2016},
        DOI = {10.1515/forma-2016-0004},
        VERSION = {8.1.04 5.36.1267},
        TITLE = {{D}ivisible {$\mathbb{Z}$}-modules},
        AUTHOR = {Futa, Yuichi and Shidama, Yasunari},
        ADDRESS1 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
        ADDRESS2 = {Shinshu University\\Nagano, Japan},
        SUMMARY = {In this article, we formalize the definition of divisible $\mathbb Z$-module and its properties in the Mizar system \cite{Mizar-State-2015}. We formally prove that any non-trivial divisible $\mathbb Z$-modules are not finitely-generated. We introduce a divisible $\mathbb Z$-module, equivalent to a vector space of a torsion-free $\mathbb Z$-module with a coefficient ring $\mathbb Q$. $\mathbb Z$-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm \cite{LLL}, cryptographic systems with lattices \cite{Micci:2002} and coding theory \cite{LANDC}.   },
        MSC2010 = {15A03 16D20 13C13 03B35},
        SECTION1 = {Divisible Module},
        SECTION2 = {Divisible Module for Torsion-free $\mathbb Z$-module},
        EXTERNALREFS = {Mizar-State-2015; LLL; Micci:2002; LANDC; },
        INTERNALREFS = {ABSVALUE.ABS;CARD_2.ABS;COMPLEX1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;INT_1.ABS;NAT_1.ABS;RELAT_1.ABS;RLVECT_1.ABS;
        VECTSP_1.ABS;ZFMISC_1.ABS;ZMODUL01.ABS;ZMODUL02.ABS;ZMODUL03.ABS;ZMODUL06.ABS;ZMODUL07.ABS;},
        KEYWORDS = {divisible vector; divisible $\mathbb{Z}$-module; },
        SUBMITTED = {December 30, 2015}}
        
        
        @ARTICLE{ZMODLAT1.ABS,
        JOURNAL = {Formalized Mathematics},
        ISSN = {1426-2630},
        EISSN = {1898-9934},
        VOLUME = {24},
        NUMBER = {{\bf 1}},
        PAGES = {49--68},
        YEAR = {2016},
        DOI = {10.1515/forma-2016-0005},
        VERSION = {8.1.04 5.36.1267},
        TITLE = {{L}attice of {$\mathbb{Z}$}-module},
        AUTHOR = {Futa, Yuichi and Shidama, Yasunari},
        ADDRESS1 = {Japan Advanced Institute\\of Science and Technology\\Ishikawa, Japan},
        ADDRESS2 = {Shinshu University\\Nagano, Japan},
        SUMMARY = {In this article, we formalize the definition of lattice of $\mathbb Z$-module and its properties in the Mizar system \cite{Mizar-State-2015}. We formally prove that scalar products in lattices are bilinear forms over the field of real numbers $\mathbb R$. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of $\mathbb Z$-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm \cite{LLL}, and cryptographic systems with lattices \cite{Micci:2002} and coding theory \cite{LANDC}.   },
        MSC2010 = {15A03 15A63 11E39 03B35},
        SECTION1 = {Definition of Lattices of $\mathbb Z$-module},
        SECTION2 = {Bilinear Forms over Field of Reals and Their Properties},
        SECTION3 = {Matrices of Bilinear Form over Field of Real Numbers},
        EXTERNALREFS = {Mizar-State-2015; LLL; Micci:2002; LANDC; },
        INTERNALREFS = {CARD_2.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_5.ABS;NAT_1.ABS;RLVECT_1.ABS;
        ZFMISC_1.ABS;ZMATRLIN.ABS;ZMODUL01.ABS;ZMODUL02.ABS;ZMODUL06.ABS;},
        KEYWORDS = {$\mathbb{Z}$-lattice; Gram matrix; integral $\mathbb{Z}$-lattice; positive definite $\mathbb{Z}$-lattice; },
        SUBMITTED = {December 30, 2015}}
        
        
        @ARTICLE{MEASUR10.ABS,
        JOURNAL = {Formalized Mathematics},
        ISSN = {1426-2630},
        EISSN = {1898-9934},
        VOLUME = {24},
        NUMBER = {{\bf 1}},
        PAGES = {69--79},
        YEAR = {2016},
        DOI = {10.1515/forma-2016-0006},
        VERSION = {8.1.04 5.36.1267},
        TITLE = {{P}roduct Pre-Measure},
        AUTHOR = {Endou, Noboru},
        ADDRESS1 = {Gifu National College of Technology\\Gifu, Japan},
        SUMMARY = {In this article we formalize in Mizar \cite{Mizar-State-2015} product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure \cite{Halmos74}, \cite{Bauer:2002}, \cite{Bogachev2007measure}, \cite{FOLLAND}, \cite{Rao2004}, we start it from $\sigma$-measure because existence of $\sigma$-measure on any semialgebras has been proved in \cite{MEASURE9.ABS}. In this approach, we use some theorems for integrals.   },
        MSC2010 = {28A35 03B35},
        SECTION1 = {Preliminaries},
        SECTION2 = {Family of Semialgebras, Fields and Measures},
        SECTION3 = {Product of Two Measures},
        EXTERNALREFS = {Mizar-State-2015; Halmos74; Bauer:2002; Bogachev2007measure; FOLLAND; Rao2004; },
        INTERNALREFS = {ABCMIZ_1.ABS;FINSEQ_1.ABS;FINSEQ_3.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_3.ABS;FUNCT_5.ABS;
        MEASURE1.ABS;MEASURE8.ABS;MEASURE9.ABS;MESFUNC1.ABS;MESFUNC2.ABS;MESFUNC5.ABS;MESFUNC9.ABS;NAT_1.ABS;PARTFUN1.ABS;
        PROB_1.ABS;RELAT_1.ABS;RINFSUP2.ABS;SUPINF_2.ABS;ZFMISC_1.ABS;},
        KEYWORDS = {product measure; pre-measure; },
        SUBMITTED = {December 31, 2015}}
        
        
        @ARTICLE{GROUP_21.ABS,
        JOURNAL = {Formalized Mathematics},
        ISSN = {1426-2630},
        EISSN = {1898-9934},
        VOLUME = {24},
        NUMBER = {{\bf 1}},
        PAGES = {81--94},
        YEAR = {2016},
        DOI = {10.1515/forma-2016-0007},
        VERSION = {8.1.04 5.36.1267},
        TITLE = {{C}onservation Rules of Direct Sum Decomposition of Groups},
        AUTHOR = {Nakasho, Kazuhisa and Yamazaki, Hiroshi and Okazaki, Hiroyuki and Shidama, Yasunari},
        ADDRESS1 = {Shinshu University\\Nagano, Japan},
        ADDRESS2 = {Shinshu University\\Nagano, Japan},
        ADDRESS3 = {Shinshu University\\Nagano, Japan},
        ADDRESS4 = {Shinshu University\\Nagano, Japan},
        SUMMARY = {In this article, conservation rules of the direct sum decomposition of groups are mainly discussed. In the first section, we prepare miscellaneous definitions and theorems for further formalization in Mizar \cite{Mizar-State-2015}. In the next three sections, we formalized the fact that the property of direct sum decomposition is preserved against the substitutions of the subscript set, flattening of direct sum, and layering of direct sum, respectively. We referred to \cite{rotman1995introduction}, \cite{robinson2012course} \cite{BourbakiAlgI} and \cite{lang-algebra} in the formalization.   },
        MSC2010 = {20E34 03B35},
        SECTION1 = {Preliminaries},
        SECTION2 = {Conservation Rule of Direct Sum Decomposition for Substitution of Subscript Set},
        SECTION3 = {Conservation Rule of Direct Sum Decomposition for Flattening},
        SECTION4 = {Conservation Rule of Direct Sum Decomposition for Layering},
        EXTERNALREFS = {Mizar-State-2015; rotman1995introduction; robinson2012course; BourbakiAlgI; lang-algebra; },
        INTERNALREFS = {CARD_1.ABS;CARD_2.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_7.ABS;FINSEQ_1.ABS;GROUP_19.ABS;
        GROUP_2.ABS;GROUP_6.ABS;GROUP_7.ABS;RELAT_1.ABS;ZFMISC_1.ABS;},
        KEYWORDS = {group theory; direct sum decomposition; },
        SUBMITTED = {December 31, 2015}}
        
        
        @ARTICLE{BAGORD_2.ABS,
            JOURNAL = {Formalized Mathematics},
        	ISSN = {1426-2630},
        	EISSN = {1898-9934},
            VOLUME = {24},
            NUMBER = {{\bf 2}},
            PAGES = {95--106},
            YEAR = {2016},
            DOI = {10.1515/forma-2016-0008},
            VERSION = {8.1.04 5.36.1267},
            TITLE = {{O}n Multiset Ordering},
            AUTHOR = {Bancerek, Grzegorz},
            ADDRESS1 = {Association of Mizar Users\\Bia\l ystok, Poland},
            SUMMARY = {Formalization of a part of \cite{JouannaudLescanne}. Unfortunately, not all is possible to be formalized. Namely, in the paper there is a mistake in the proof of Lemma 3. It states that there exists $x\in M_1$ such that $M_1(x)>N_1(x)$ and $(\forall y\in N_1)x\not\prec y$. It should be $M_1(x)\geq N_1(x)$. Nevertheless we do not know whether $x\in N_1$ or not and cannot prove the contradiction. In the article we referred to \cite{DershowitzTCS}, \cite{DershowitzManna1979} and \cite{HuetOppen1980}.   },
            MSC2010 = {06F05 03B35},
            SECTION1 = {Preliminaries},
            SECTION2 = {Relational Extension},
            SECTION3 = {Dershowitz-Manna Order},
            SECTION4 = {Monoidal Order},
            EXTERNALREFS = {JouannaudLescanne; DershowitzTCS; DershowitzManna1979; HuetOppen1980; },
            INTERNALREFS = {FINSEQ_1.ABS;FINSEQ_2.ABS;FINSEQ_3.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;NAT_1.ABS;NAT_2.ABS;ORDERS_2.ABS;
            PREFER_1.ABS;REWRITE1.ABS;TREES_2.ABS;ZFMISC_1.ABS;},
            KEYWORDS = {ordering; Dershowitz-Manna ordering; },
            SUBMITTED = {December 31, 2015}}
            
            
            @ARTICLE{COUSIN.ABS,
            JOURNAL = {Formalized Mathematics},
        	ISSN = {1426-2630},
        	EISSN = {1898-9934},
            VOLUME = {24},
            NUMBER = {{\bf 2}},
            PAGES = {107--119},
            YEAR = {2016},
            DOI = {10.1515/forma-2016-0009},
            VERSION = {8.1.04 5.36.1267},
            TITLE = {{C}ousin's Lemma},
            AUTHOR = {Coghetto, Roland},
            ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
            SUMMARY = { We formalize, in two different ways, that ``the $n$-dimensional Euclidean metric space is a 
            complete metric space" (version 1. with the results obtained in \cite{REAL_NS1.ABS}, \cite{RSSPACE3.ABS}, 
            \cite{LOPBAN_1.ABS} and version 2., the results obtained in \cite{REAL_NS1.ABS}, \cite{NORMSP_2.ABS}, 
            (\textit{registrations}) \cite{LOPBAN_5.ABS}). \par With the Cantor's theorem - in complete metric space 
            (proof by Karol P\k{a}k in \cite{COMPL_SP.ABS}), we formalize ``The Nested Intervals Theorem in 
            1-dimensional Euclidean metric space". \par Pierre Cousin's proof in 1892 \cite{Maurey2005} the lemma, 
            published in 1895 \cite{Cousin1895} states that: \begin{quote} ``Soit, sur le plan YOX, une aire 
            connexe $S$ limit\'ee par un contour ferm\'e simple ou complexe; on suppose qu'\`a chaque point 
            de $S$ ou de son p\'erim\`etre correspond un cercle, de rayon non nul, ayant ce point pour centre : 
            il est alors toujours possible de subdiviser $S$ en r\'egions, en nombre fini et assez petites pour 
            que chacune d'elles soit compl\'etement int\'erieure au cercle correspondant \`a un point convenablement 
            choisi dans $S$ ou sur son p\'erim\`etre." \end{quote} (In the plane YOX let $S$ be a connected area 
            bounded by a closed contour, simple or complex; one supposes that at each point of $S$ or its perimeter 
            there is a circle, of non-zero radius, having this point as its centre; it is then always possible to 
            subdivide $S$ into regions, finite in number and sufficiently small for each one of them to be entirely 
            inside a circle corresponding to a suitably chosen point in $S$ or on its perimeter) \cite{Raman2015}. \par 
            Cousin's Lemma, used in Henstock and Kurzweil integral \cite{yee2000integral} (generalized Riemann integral), 
            state that: ``for any gauge $\delta$, there exists at least one $\delta$-fine tagged partition". 
            In the last section, we formalize this theorem. We use the suggestions given to the Cousin's Theorem 
            p.11 in \cite{bartle2001modern} and with notations: \cite{bartle1996return}, \cite{yee2000integral}, 
            \cite{mawhin2001eternel}, \cite{peng2008integral} and \cite{INTEGRA1.ABS}.   },
            MSC2010 = {54D30 03B35},
            SECTION1 = {Preliminaries},
            SECTION2 = {${\cal E}^{n}$ is Complete - Proof Version 1},
            SECTION3 = {${\cal E}^{n}$ is Complete - Proof Version 2},
            SECTION4 = {The Nested Intervals Theorem (1-dimensional Euclidean Space)},
            SECTION5 = {Tagged Partition},
            SECTION6 = {Partition Composition},
            SECTION7 = {Examples of Partitions},
            SECTION8 = {Cousin's Lemma},
            EXTERNALREFS = {Maurey2005; Cousin1895; Raman2015; yee2000integral; bartle2001modern; bartle1996return; 
            yee2000integral; mawhin2001eternel; peng2008integral; },
            INTERNALREFS = {CARD_1.ABS;CARD_3.ABS;COMPLEX1.ABS;COMPL_SP.ABS;COMSEQ_2.ABS;EUCLID.ABS;FINSEQ_1.ABS;
            FINSEQ_2.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;INTEGRA1.ABS;INT_1.ABS;LOPBAN_1.ABS;LOPBAN_5.ABS;
            MATRPROB.ABS;MEASURE5.ABS;MEMBERED.ABS;METRIC_1.ABS;NAT_1.ABS;NEWTON.ABS;NORMSP_1.ABS;NORMSP_2.ABS;
            ORDINAL1.ABS;PARTFUN1.ABS;PARTFUN3.ABS;POWER.ABS;PROB_1.ABS;RCOMP_1.ABS;RCOMP_3.ABS;REAL_NS1.ABS;REAL_NS.ABS;
            RELAT_1.ABS;RELSET_1.ABS;RFINSEQ.ABS;RSSPACE3.ABS;SEQ_2.ABS;SEQ_4.ABS;SETFAM_1.ABS;SUBSET_1.ABS;TBSP_1.ABS;
            TOPMETR.ABS;TOPS_2.ABS;ZFMISC_1.ABS;},
            KEYWORDS = {Cousin's lemma; Cousin's theorem; nested intervals theorem; },
            SUBMITTED = {December 31, 2015}}
            
            
            @ARTICLE{SRINGS_5.ABS,
            JOURNAL = {Formalized Mathematics},
        	ISSN = {1426-2630},
        	EISSN = {1898-9934},
              VOLUME = {24},
            NUMBER = {{\bf 2}},
            PAGES = {121--141},
            YEAR = {2016},
            DOI = {10.1515/forma-2016-0010},
            VERSION = {8.1.04 5.36.1267},
            TITLE = {{C}hebyshev Distance},
            AUTHOR = {Coghetto, Roland},
            ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
              SUMMARY = {In \cite{MFOLD_2.ABS}, Marco Riccardi formalized that $\mathbb{R}$N-basis $n$ is a basis (in the algebraic sense defined in \cite{RLVECT_3.ABS}) of ${\cal E}^{n}_{T}$ and in \cite{MFOLD_1.ABS} he has formalized that ${\cal E}^{n}_{T}$ is second-countable, we build (in the topological sense defined in \cite{CANTOR_1.ABS}) a denumerable base of ${\cal E}^{n}_{T}$. \par Then we introduce the $n$-dimensional intervals (interval in $n$-dimensional Euclidean space, \textit{pav\'e (born\'e) de $\mathbb{R}^n$} \cite{mawhin1992analyse}, \textit{semi-intervalle (born\'e) de $\mathbb{R}^n$} \cite{SchmetsAM:2004}). \par We conclude with the definition of Chebyshev distance \cite{deza2009encyclopedia}.   },
            MSC2010 = {54E35 03B35},
            SECTION1 = {Preliminaries},
            SECTION2 = {The Set of $n$-Tuples of Rational Numbers},
            SECTION3 = {A Countable Base of an $n$-Dimensional Euclidean Space},
            SECTION4 = {The Set of All Left Open Real Bounded Intervals},
            SECTION5 = {The Set of All Right Open Real Bounded Intervals},
            SECTION6 = {Finite Product of Left Open Intervals},
            SECTION7 = {Finite Product of Right Open Intervals},
            SECTION8 = {$n$-Dimensional Product of Subset Family},
            SECTION9 = {The Set of All Closed Real Bounded Intervals},
            SECTION10 = {The Set of All Open Real Bounded Intervals},
            SECTION11 = {$n$-Dimensional Subset Family of $\mathbb R$},
            SECTION12 = {Closed/Open/Left-Open/Right-Open -- Hyper Interval},
            SECTION13 = {Correspondance between Interval and 1-Dimensional Hyper Interval},
            SECTION14 = {Correspondance between Measurable Rectangle and Product},
            SECTION15 = {Chebyshev Distance},
            EXTERNALREFS = {mawhin1992analyse; SchmetsAM:2004; deza2009encyclopedia; },
            INTERNALREFS = {BINOP_1.ABS;CANTOR_1.ABS;CARD_1.ABS;CARD_3.ABS;COMPLEX1.ABS;EUCLID.ABS;EUCLID_9.ABS;FINSEQ_1.ABS;
            FINSEQ_2.ABS;FINSET_1.ABS;FINSUB_1.ABS;FUNCOP_1.ABS;FUNCSDOM.ABS;FUNCT_1.ABS;FUNCT_2.ABS;INT_1.ABS;MEASURE5.ABS;
            MEMBERED.ABS;METRIC_1.ABS;MFOLD_1.ABS;MFOLD_2.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PCOMPS_1.ABS;PRE_TOPC.ABS;RAT_1.ABS;
            RCOMP_1.ABS;RELAT_1.ABS;RELSET_1.ABS;RLTOPSP1.ABS;RLVECT_1.ABS;RLVECT_3.ABS;SETFAM_1.ABS;SQUARE_1.ABS;SRINGS_1.ABS;
            SRINGS_3.ABS;SRINGS_4.ABS;SUBSET_1.ABS;TIETZE_2.ABS;TOPMETR.ABS;TOPS_1.ABS;TOPS_2.ABS;YELLOW_8.ABS;ZFMISC_1.ABS;},
            KEYWORDS = {second-countable; intervals; Chebyshev distance; },
            SUBMITTED = {December 31, 2015}}
            
            
            @ARTICLE{ROUGHS_3.ABS,
            JOURNAL = {Formalized Mathematics},
        	ISSN = {1426-2630},
        	EISSN = {1898-9934},
            VOLUME = {24},
            NUMBER = {{\bf 2}},
            PAGES = {143--155},
            YEAR = {2016},
            DOI = {10.1515/forma-2016-0011},
            VERSION = {8.1.04 5.36.1267},
            TITLE = {{B}inary Relations-based Rough Sets -- an Automated Approach},
            AUTHOR = {Grabowski, Adam},
            ADDRESS1 = {Institute of Informatics\\ University of Bia{\l}ystok\\ Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
            SUMMARY = {Rough sets, developed by Zdzis{\l}aw Pawlak \cite{Pawlak1982}, are an important tool to describe the state of incomplete or partially unknown information. In this article, which is essentially the continuation of \cite{ROUGHS_2.ABS}, we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library \cite{FourDecades}). Here we drop the classical equivalence- and tolerance-based models of rough sets trying to formalize some parts of \cite{Zhu:2007}. \par The main aim of this Mizar article is to provide a formal counterpart for the rest of the paper of William Zhu \cite{Zhu:2007}. In order to do this, we recall also Theorem~3 from Y.Y. Yao's paper \cite{Yao96}. The first part of our formalization (covering first seven pages) is contained in \cite{ROUGHS_2.ABS}. Now we start from page 5003, sec. 3.4. \cite{Zhu:2007}. We formalized almost all numbered items (definitions, propositions, theorems, and corollaries), with the exception of Proposition 7, where we stated our theorem only in terms of singletons. We provided more thorough discussion of the property {\em positive alliance} and its connection with seriality and reflexivity (and also transitivity). Examples were not covered as a rule as we tried to construct a~more general mechanism of finding appropriate models for approximation spaces in Mizar providing more automatization than it is now \cite{GrabowskiPerspective:2007}. \par Of course, we can see some more general applications of some registrations of clusters, essentially not dealing with the notion of an approximation: the notions of an alliance binary relation were not defined in the Mizar Mathematical Library before, and we should think about other properties which are also absent but needed in the context of rough approximations \cite{GrabowskiJ10}, \cite{GrabowskiAssisted:2005}. Via theory merging, using mechanisms described in \cite{GrabowskiFI:2014} and \cite{GrabowskiJAR40}, such elementary constructions can be extended to other frameworks.   },
            MSC2010 = {03E70 03E99 03B35},
            SECTION1 = {Preliminaries},
            SECTION2 = {On the Union and the Intersection of Two Relational Structures},
            SECTION3 = {Ordinary Properties of Maps},
            SECTION4 = {On the Relational Structure Generated by Rough Approximation},
            SECTION5 = {Construction Revisited: Yao's \cite{Yao96} Theorem 3},
            SECTION6 = {Transitive Binary Relations},
            SECTION7 = {Mediate and Transitive Binary Relations},
            SECTION8 = {Alliance Binary Relations},
            SECTION9 = {Preparation for Translation of Proposition 10 (7H')},
            SECTION10 = {Translation Continued},
            SECTION11 = {On the Uniqueness of Binary Relations to Generate Rough Sets},
            EXTERNALREFS = {Pawlak1982; FourDecades; Zhu:2007; Yao96; GrabowskiPerspective:2007; GrabowskiJ10; 
            GrabowskiAssisted:2005; GrabowskiFI:2014; GrabowskiJAR40; },
            INTERNALREFS = {ALTCAT_2.ABS;CLASSES1.ABS;COHSP_1.ABS;COH_SP.ABS;DOMAIN_1.ABS;FINSET_1.ABS;FINSUB_1.ABS;FUNCT_1.ABS;
            FUNCT_2.ABS;ORDERS_2.ABS;ORDERS_3.ABS;PARTFUN1.ABS;RELAT_1.ABS;RELSET_1.ABS;ROUGHS_1.ABS;ROUGHS_2.ABS;ROUGHS_4.ABS;
            SETFAM_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
            KEYWORDS = {rough set; lower approximation; upper approximation; binary relation; },
            SUBMITTED = {February 15, 2016}}
            
            
            @ARTICLE{GTARSKI2.ABS,
            JOURNAL = {Formalized Mathematics},
        	ISSN = {1426-2630},
        	EISSN = {1898-9934},
            VOLUME = {24},
            NUMBER = {{\bf 2}},
            PAGES = {157--166},
            YEAR = {2016},
            DOI = {10.1515/forma-2016-0012},
            VERSION = {8.1.05 5.37.1275},
            TITLE = {{T}arski Geometry Axioms -- {P}art {II}},
            AUTHOR = {Coghetto, Roland and Grabowski, Adam},
            ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere\\Belgium},
            ADDRESS2 = {Institute of Informatics\\ University of Bia{\l}ystok\\ Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
            SUMMARY = {In our earlier article \cite{GTARSKI1.ABS}, the first part of axioms of geometry proposed by Alfred Tarski \cite{TarskiGivant} was formally introduced by means of Mizar proof assistant \cite{FourDecades}. We defined a structure \verb!TarskiPlane! with the following predicates: \begin{itemize} \item of betweenness \verb!between! (a ternary relation), \item of congruence of segments \verb!equiv! (quarternary relation), \end{itemize} \noindent which satisfy the following properties: \begin{itemize} \item congruence symmetry (A1), \item congruence equivalence relation (A2), \item congruence identity (A3), \item segment construction (A4), \item SAS (A5), \item betweenness identity (A6), \item Pasch (A7). \end{itemize} Also a simple model, which satisfies these axioms, was previously constructed, and described in \cite{GrabowskiFed2016}. In this paper, we deal with four remaining axioms, namely: \begin{itemize} \item the lower dimension axiom (A8), \item the upper dimension axiom (A9), \item the Euclid axiom (A10), \item the continuity axiom (A11). \end{itemize} They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles \verb!cong! is introduced via congruence of sides (SSS). \par In order to show that the structure which satisfies all eleven Tarski's axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural \cite{GrabowskiFI:2014} extension of ordinary metric structure \verb!Euclid 2! satisfies all these attributes. \par Although the tradition of the mechanization of Tarski's geometry in Mizar is not as long as in Coq \cite{Narboux:2007}, first approaches to this topic were done in Mizar in 1990 \cite{INCSP_1.ABS} (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time \cite{GrabowskiDuplication}). Connection with another proof assistant should be mentioned -- we had some doubts about the proof of the Euclid's axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle \cite{Makarios} clarified things a bit. Our development allows for the future faithful mechanization of \cite{Schwabhauser:1983} and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory \cite{GrabowskiJAR40}.   },
            MSC2010 = {51A05 51M04 03B35},
            SECTION1 = {Preliminaries},
            SECTION2 = {Basic Properties of the Euclidean Plane},
            SECTION3 = {Ordered Affine Space Generated by ${\cal E}^{2}_{\rm T}$},
            SECTION4 = {Eucldiean Plane Satisfies First 7 Tarski's Axioms},
            SECTION5 = {Preparation for The Rest of Tarski's Axioms},
            SECTION6 = {Four Remaining Axioms of Tarski},
            SECTION7 = {Axiom A11 -- Axiom Schema of Continuity},
            SECTION8 = {Corrolaries},
            EXTERNALREFS = {TarskiGivant; FourDecades; GrabowskiFed2016; GrabowskiFI:2014; Narboux; GrabowskiDuplication; 
            Makarios; Schwabhauser:1983; GrabowskiJAR40; },
            INTERNALREFS = {ANALOAF.ABS;BINOP_1.ABS;CARD_1.ABS;DIRAF.ABS;EUCLID.ABS;EUCLIDLP.ABS;EUCLID_3.ABS;FINSEQ_1.ABS;
            FINSEQ_2.ABS;FUNCT_1.ABS;GTARSKI1.ABS;INCSP_1.ABS;METRIC_1.ABS;ORDINAL1.ABS;RELAT_1.ABS;RLTOPSP1.ABS;RLVECT_1.ABS;
            SIN_COS.ABS;SQUARE_1.ABS;SUBSET_1.ABS;TOPREAL6.ABS;ZFMISC_1.ABS;},
            KEYWORDS = {Tarski's geometry axioms; foundations of geometry; Euclidean plane; },
            SUBMITTED = {June 30, 2016}}
            

@ARTICLE{TOPMETR4.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 3}},
    PAGES = {167--172},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0013},
    VERSION = {8.1.05 5.37.1275},
    TITLE = {{C}ompactness in Metric Spaces},
    AUTHOR = {Nakasho, Kazuhisa and Narita, Keiko and Shidama, Yasunari},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
      ADDRESS2 = {Hirosaki-city\\Aomori, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, we mainly formalize in Mizar \cite{Mizar-State-2015} the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness, sequential compactness, and totally boundedness with completeness in metric spaces. \par In the third section, we discuss compactness in norm spaces. We formalize the equivalence of compactness and sequential compactness in norm space. In the fourth section, we formalize topological properties of the real line in terms of convergence of real number sequences. In the last section, we formalize the equivalence of compactness and sequential compactness in the real line. These formalizations are based on \cite{yoshida:1980}, \cite{bourbaki1987elements}, \cite{rudin1991functional}, \cite{Kreyszig1989}, and \cite{bourbaki2013general}.   },
    MSC2010 = {46B50 54E45 03B35},
    SECTION1 = {Topological Properties of Metric Spaces},
    SECTION2 = {Compactness in Metric Spaces},
    SECTION3 = {Compactness in Norm Spaces},
    SECTION4 = {Topological Properties of the Real Line},
    SECTION5 = {Compactness in the Real Line},
    EXTERNALREFS = {Mizar-State-2015; yoshida:1980; bourbaki1987elements; rudin1991functional; Kreyszig1989; bourbaki2013general; },
    INTERNALREFS = {CARD_1.ABS;COMPLEX1.ABS;COMPL_SP.ABS;COMPTS_1.ABS;COMSEQ_2.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;
INT_1.ABS;MEMBERED.ABS;METRIC_1.ABS;NAT_1.ABS;NFCONT_1.ABS;NORMSP_1.ABS;NORMSP_2.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PCOMPS_1.ABS;
PRE_TOPC.ABS;RCOMP_1.ABS;RELAT_1.ABS;RELSET_1.ABS;RLVECT_1.ABS;SEQ_2.ABS;SETFAM_1.ABS;SUBSET_1.ABS;TBSP_1.ABS;
TOPGEN_1.ABS;TOPMETR.ABS;TOPS_2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {metric spaces; normed linear spaces; compactness; },
    SUBMITTED = {June 30, 2016}}

    
    @ARTICLE{CARDFIL4.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 3}},
    PAGES = {173--186},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0014},
    VERSION = {8.1.05 5.37.1275},
    TITLE = {{D}ouble Sequences and Iterated Limits in Regular Space},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {First, we define in Mizar \cite{Mizar-State-2015}, the Cartesian product of two filters bases and the Cartesian product of two filters. After comparing the product of two Fr{\'e}chet filters on $\mathbb N$ (${\cal F}_1$) with the Fr{\'e}chet filter on $\mathbb N\times \mathbb N$ (${\cal F}_2$), we compare $lim_{{\cal F}_1}$ and $lim_{{\cal F}_2}$ for all double sequences in a non empty topological space. \par Endou, Okazaki and Shidama formalized in \cite{DBLSEQ_1.ABS} the ``convergence in Pringsheim's sense'' for double sequence of real numbers. We show some basic correspondences between the $p$-convergence and the filter convergence in a topological space. Then we formalize that the double sequence $(x_{m,n}=\frac{1}{m+1})_{(m,n)}\in{\mathbb{N}\times\mathbb{N}}$ converges in ``Pringsheim's sense" but not in Frechet filter on $\mathbb{N}\times\mathbb{N}$ sense. \par In the next section, we generalize some definitions: ``is convergent in the first coordinate", ``is convergent in the second coordinate", ``the $lim$ in the first coordinate of", ``the $lim$ in the second coordinate of" according to \cite{DBLSEQ_1.ABS}, in Hausdorff space. \par Finally, we generalize two theorems: (3) and (4) from \cite{DBLSEQ_1.ABS} in the case of double sequences and we formalize the ``iterated limit" theorem (``Double limit" \cite{bourbaki2013general}, p. 81, par. 8.5 \textit{``Double limite"} \cite{bourbaki2007topologie} (TG I,57)), all in regular space. We were inspired by the exercises (2.11.4), (2.17.5) \cite{wagschal} and the corrections B.10 \cite{wagschaltopex}.   },
    MSC2010 = {54A20 40A05 40B05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Cartesian Product of Two Filters},
    SECTION3 = {Comparison between Cartesian Product of Frechet Filter on $\mathbb{N}$ and the Frechet Filter of $\mathbb{N}\times\mathbb{N}$},
    SECTION4 = {Topological Space and Double Sequence},
    SECTION5 = {Metric Space and Double Sequence},
    SECTION6 = {One-dimensional Euclidean Metric Space and Double Sequence},
    SECTION7 = {Basic Relations Convergence in Pringsheim's Sense and Filter Convergence},
    SECTION8 = {Example: Double Sequence Converges in Pringsheim's Sense but not in Frechet Filter of $\mathbb{N}\times\mathbb{N}$ Sense},
    SECTION9 = {Correspondence with some Definitions from \cite{DBLSEQ_1.ABS}},
    SECTION10 = {Regular Space, Double Limit and Iterated Limit},
    EXTERNALREFS = {Mizar-State-2015; bourbaki2013general; bourbaki2007topologie; wagschal; wagschaltopex; },
    INTERNALREFS = {BINOP_1.ABS;CARDFIL2.ABS;CARD_1.ABS;CARD_3.ABS;CARD_FIL.ABS;COMPLEX1.ABS;COMSEQ_2.ABS;CONNSP_2.ABS;
DBLSEQ_1.ABS;DICKSON.ABS;DOMAIN_1.ABS;EUCLID.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FRECHET.ABS;FUNCT_1.ABS;FUNCT_2.ABS;INT_1.ABS;
MCART_1.ABS;MEMBERED.ABS;MESFUNC9.ABS;METRIC_1.ABS;NAT_1.ABS;ORDINAL1.ABS;PCOMPS_1.ABS;PRE_TOPC.ABS;RAT_1.ABS;RELAT_1.ABS;
RELSET_1.ABS;SEQ_2.ABS;SETFAM_1.ABS;SRINGS_5.ABS;SUBSET_1.ABS;TOPMETR.ABS;TOPS_1.ABS;WAYBEL_0.ABS;WAYBEL_7.ABS;YELLOW19.ABS;
YELLOW_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {filter; double limits; Pringsheim convergence; iterated limits; regular space; },
    SUBMITTED = {June 30, 2016}}
 
 
    @ARTICLE{NEWTON03.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 3}},
    PAGES = {187--198},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0015},
    VERSION = {8.1.05 5.37.1275},
    TITLE = {{P}rime Factorization of Sums and Differences of Two Like Powers},
    AUTHOR = {Ziobro, Rafa{\l}},
    ADDRESS1 = {Department of Carbohydrate Technology\\University of Agriculture\\Krakow, Poland},
      SUMMARY = {Representation of a non zero integer as a signed product of primes is unique similarly to its representations in various types of positional notations \cite{gancarzewicz2000}, \cite{Erdos2003}. The study focuses on counting the prime factors of integers in the form of sums or differences of two equal powers (thus being represented by 1 and a series of zeroes in respective digital bases). \par Although the introduced theorems are not particularly important, they provide a couple of shortcuts useful for integer factorization, which could serve in further development of Mizar projects \cite{Mizar-State-2015}. This could be regarded as one of the important benefits of proof formalization \cite{Naumowicz2006396}.   },
    MSC2010 = {11A51 03B35},
    EXTERNALREFS = {gancarzewicz2000; Erdos2003; Mizar-State-2015; Naumowicz2006396; },
    INTERNALREFS = {},
    KEYWORDS = {integers; factorization; primes; },
    ACKNOWLEDGEMENT = {Ad Maiorem Dei Gloriam},
    SUBMITTED = {June 30, 2016}}
    
    
    @ARTICLE{INTEGR22.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 3}},
    PAGES = {199--204},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0016},
    VERSION = {8.1.05 5.37.1275},
    TITLE = {{R}iemann-{S}tieltjes Integral},
    ANNOTE = {This work was supported by JSPS KAKENHI 22300285.},
    AUTHOR = {Narita, Keiko and Nakasho, Kazuhisa and Shidama, Yasunari},
    ADDRESS1 = {Hirosaki-city\\Aomori, Japan},
      ADDRESS2 = {Akita Prefectural University\\Akita, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar \cite{Mizar-State-2015}. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties. \par In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article \cite{INTEGR18.ABS}, we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article \cite{INTEGR18.ABS}. These formalizations are based on \cite{Stroock1999}, \cite{Kestelman1960}, \cite{Gupta1986}, and \cite{Hille1974}.   },
    MSC2010 = {26A42 26A45 03B35},
    SECTION1 = {Properties of Real Finite Sequences},
    SECTION2 = {The Definitions of Bounded Variation},
    SECTION3 = {The Definitions of Riemann-Stieltjes Integral},
    SECTION4 = {Linearity of Riemann-Stieltjes Integral},
    EXTERNALREFS = {Mizar-State-2015; Stroock1999; Kestelman1960; Gupta1986; Hille1974; },
    INTERNALREFS = {BINOP_1.ABS;CARD_1.ABS;COMPLEX1.ABS;COMSEQ_2.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSET_1.ABS;FUNCT_1.ABS;
FUNCT_2.ABS;INTEGRA1.ABS;INTEGRA2.ABS;INTEGRA3.ABS;INTEGR18.ABS;INT_1.ABS;MEASURE5.ABS;MEMBERED.ABS;NAT_1.ABS;
ORDINAL1.ABS;PARTFUN1.ABS;RAT_1.ABS;RCOMP_1.ABS;RELAT_1.ABS;RELSET_1.ABS;RVSUM_1.ABS;SEQ_2.ABS;SEQ_4.ABS;SUBSET_1.ABS;
ZFMISC_1.ABS;},
    KEYWORDS = {Riemann-Stieltjes integral; bounded variation; linearity; },
    SUBMITTED = {June 30, 2016}}
    
    
    @ARTICLE{UNIFORM2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 3}},
    PAGES = {205--214},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0017},
    VERSION = {8.1.05 5.37.1275},
    TITLE = {{Q}uasi-uniform Space},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {In this article, using mostly Pervin \cite{pervin1962quasi}, Kunzi \cite{kunzi1993quasi}, \cite{kunzi1995bourbaki}, \cite{kunzi2009introduction}, Williams \cite{williams1972locally} and Bourbaki \cite{bourbaki2013general} works, we formalize in Mizar \cite{Mizar-State-2015} the notions of quasi-uniform space, semi-uniform space and locally uniform space. \par We define the topology induced by a quasi-uniform space. Finally we formalize from the sets of the form $((X\setminus\Omega) \times X) \cup (X\times \Omega)$, the Csaszar-Pervin quasi-uniform space induced by a topological space.   },
    MSC2010 = {54E15 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Uniform Space Structure},
    SECTION3 = {Axioms},
    SECTION4 = {Quasi-Uniform Space},
    SECTION5 = {Semi-Uniform Space},
    SECTION6 = {Locally Uniform Space},
    SECTION7 = {Topological Space Induced by a Uniform Space Structure},
    SECTION8 = {The Quasi-Uniform Pervin Space Induced by a Topological Space},
    EXTERNALREFS = {pervin1962quasi; kunzi1993quasi; kunzi1995bourbaki; kunzi2009introduction; williams1972locally; bourbaki2013general; Mizar-State-2015; },
    INTERNALREFS = {CANTOR_1.ABS;CARDFIL2.ABS;CARD_1.ABS;CARD_FIL.ABS;DOMAIN_1.ABS;EQREL_1.ABS;FINSET_1.ABS;FINSUB_1.ABS;
FINTOPO2.ABS;FINTOPO7.ABS;FUNCT_1.ABS;FUNCT_2.ABS;MCART_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PRE_TOPC.ABS;RELAT_1.ABS;
RELAT_2.ABS;RELSET_1.ABS;SETFAM_1.ABS;SUBSET_1.ABS;TEX_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {quasi-uniform space; quasi-uniformity; Pervin space; Csaszar-Pervin quasi-uniformity; },
    ACKNOWLEDGEMENT = {The author wants to express his gratitude to the anonymous referee for his/her work for the introduction of new notations and to make the presentation more readable.},
    SUBMITTED = {June 30, 2016}}
    
    
    @ARTICLE{UNIFORM3.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 3}},
    PAGES = {215--226},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0018},
    VERSION = {8.1.05 5.37.1275},
    TITLE = {{U}niform Space},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {In this article, we formalize in Mizar \cite{Mizar-State-2015} the notion of uniform space introduced by Andr\'e Weil using the concepts of entourages \cite{bourbaki2013general}. \par We present some results between uniform space and pseudo metric space. We introduce the concepts of left-uniformity and right-uniformity of a topological group. \par Next, we define the concept of the partition topology. Following the Vlach's works \cite{vlach2008topologies,vlach2008algebraic}, we define the semi-uniform space induced by a tolerance and the uniform space induced by an equivalence relation. \par Finally, using mostly Gehrke, Grigorieff and Pin \cite{gehrke2010topological} works, a Pervin uniform space defined from the sets of the form $((X\setminus A) \times (X\setminus A)) \cup (A\times A)$ is presented.   },
    MSC2010 = {54E15 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Uniform Space},
    SECTION3 = {Open Set and Uniform Space},
    SECTION4 = {Pseudo Metric Space and Uniform Space},
    SECTION5 = {Uniform Space and Topological Group},
    SECTION6 = {Function Uniformly Continuous},
    SECTION7 = {Partition Topology},
    SECTION8 = {Uniform Space and Partition Topology},
    SECTION9 = {Uniformity Induced by a Tolerance or by an Equivalence},
    SECTION10 = {Uniform Pervin Space},
    EXTERNALREFS = {Mizar-State-2015; bourbaki2013general; vlach2008algebraic; gehrke2010topological; },
    INTERNALREFS = {CANTOR_1.ABS;CARDFIL2.ABS;CARD_3.ABS;CONNSP_2.ABS;DOMAIN_1.ABS;DYNKIN.ABS;EQREL_1.ABS;FINSET_1.ABS;
FINSUB_1.ABS;FINTOPO7.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GROUP_1.ABS;GROUP_1A.ABS;GROUP_2.ABS;IDEAL_1.ABS;LATTICE7.ABS;
LOPCLSET.ABS;MCART_1.ABS;MEASURE1.ABS;METRIC_1.ABS;ORDERS_2.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PCOMPS_1.ABS;PRE_TOPC.ABS;
PROB_1.ABS;RELAT_1.ABS;RELAT_2.ABS;RELSET_1.ABS;RLVECT_1.ABS;ROUGHS_4.ABS;SETFAM_1.ABS;SRINGS_4.ABS;SUBSET_1.ABS;
TOPGEN_4.ABS;TOPGRP_1.ABS;UNIFORM2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {uniform space; uniformity; pseudo-metric space; topological group; partition topology; Pervin uniform space; },
    ACKNOWLEDGEMENT = {The author wants to express his gratitude to the anonymous referee for his/her work, to make the presentation more readable.},
    SUBMITTED = {June 30, 2016}}
    

@ARTICLE{RING_4.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 3}},
    PAGES = {227--237},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0019},
    VERSION = {8.1.05 5.37.1275},
    TITLE = {{S}ome Algebraic Properties of Polynomial Rings},
    AUTHOR = {Schwarzweller, Christoph and Korni{\l}owicz, Artur and Rowi{\'n}ska-Schwarzweller, Agnieszka},
    ADDRESS1 = {Institute of Computer Science\\University of Gda{\'n}sk\\Poland},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS3 = {Sopot, Poland},
      SUMMARY = {In this article we extend the algebraic theory of polynomial rings, formalized in Mizar \cite{Mizar-State-2015}, based on \cite{HH90}, \cite{WE2009}. After introducing constant and monic polynomials we present the canonical embedding of $R$ into $R[X]$ and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields $F$ and irreducible polynomials $p$ the field $F[X]\slash\!\!\!<\!\!\!p\!\!\!>$ is iso\-mor\-phic to the field of polynomials with degree smaller than the one of $p$.   },
    MSC2010 = {12E05 11T55 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Constant Polynomials},
    SECTION3 = {Monic Polynomials},
    SECTION4 = {The Canonical Homomorphism from $R$ into $R[X]$},
    SECTION5 = {Units and Irreducible Polynomials},
    SECTION6 = {The Field $F[X]\slash\!\!<\!p\!>$},
    SECTION7 = {Polynomial GCDs},
    EXTERNALREFS = {Mizar-State-2015; HH90; WE2009; },
    INTERNALREFS = {ALGSTR_1.ABS;BINOP_1.ABS;CARD_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_7.ABS;GCD_1.ABS;
GROUP_1.ABS;GROUP_6.ABS;HURWITZ.ABS;IDEAL_1.ABS;INT_1.ABS;INT_3.ABS;MOD_4.ABS;ORDINAL1.ABS;PARTFUN1.ABS;POLYNOM1.ABS;
POLYNOM3.ABS;POLYNOM5.ABS;RATFUNC1.ABS;REALSET1.ABS;RELAT_1.ABS;RINGCAT1.ABS;RING_1.ABS;RING_2.ABS;RING_3.ABS;
RLVECT_1.ABS;SUBSET_1.ABS;VECTSP10.ABS;VECTSP_1.ABS;VECTSP_2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {polynomial; polynomial ring; polynomial GCD; },
    SUBMITTED = {June 30, 2016}}
        
    
@ARTICLE{ANPROJ_8.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 4}},
    PAGES = {239--251},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0020},
    VERSION = {8.1.05 5.39.1282},
    TITLE = {{H}omography in {$\mathbb{RP}^2$}},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {The real projective plane has been formalized in Isabelle/HOL by Timothy Makarios \cite{Makarios} and in Coq by Nicolas Magaud, Julien Narboux and Pascal Schreck \cite{magaud2008formalizing}. \par Some definitions on the real projective spaces were introduced early in the Mizar Mathematical Library by Wojciech Leonczuk \cite{ANPROJ_1.ABS}, Krzysztof Prazmowski \cite{ANPROJ_2.ABS} and by Wojciech Skaba \cite{COLLSP.ABS}. \par In this article, we check with the Mizar system \cite{Mizar-State-2015}, some properties on the determinants and the Grassmann-Pl{\"u}cker relation in rank 3 \cite{apel2010cancellation}, \cite{apel2014phd}, \cite{fuchs2010formalization}, \cite{richter1995mechanical}, \cite{richter2011perspectives}. \par Then we show that the projective space induced (in the sense defined in \cite{ANPROJ_1.ABS}) by $\mathbb{R}^3$ is a projective plane (in the sense defined in \cite{ANPROJ_2.ABS}). \par Finally, in the real projective plane, we define the homography induced by a 3-by-3 invertible matrix and we show that the images of 3 collinear points are themselves collinear.   },
    MSC2010 = {51N15 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Grassmann-Pl{\"u}cker Relation},
    SECTION3 = {Some Properties about the Cross Product},
    SECTION4 = {Real Projective Plane and Homography},
    EXTERNALREFS = {Makarios; magaud2008formalizing; Mizar-State-2015; apel2010cancellation; apel2014phd; fuchs2010formalization; richter1995mechanical; richter2011perspectives; },
    INTERNALREFS = {ANPROJ_1.ABS;ANPROJ_2.ABS;CARD_1.ABS;COLLSP.ABS;DOMAIN_1.ABS;DUALSP01.ABS;EUCLID.ABS;EUCLID_5.ABS;
EUCLID_8.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSEQ_4.ABS;FINSET_1.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FVSUM_1.ABS;
GROUP_1.ABS;LAPLACE.ABS;MATRIX13.ABS;MATRIXR1.ABS;MATRIXR2.ABS;MATRIX_1.ABS;MATRIX_3.ABS;MATRIX_6.ABS;MATRTOP1.ABS;
MOD_2.ABS;ORDINAL1.ABS;PRVECT_1.ABS;RELAT_1.ABS;RLVECT_1.ABS;RLVECT_3.ABS;RVSUM_1.ABS;SQUARE_1.ABS;SUBSET_1.ABS;
VECTSP_1.ABS;VECTSP_7.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {projectivity; projective transformation; projective collineation; real projective plane; Grassmann-Pl{\"u}cker relation; },
    SUBMITTED = {October 18, 2016}}
        
    
    @ARTICLE{INTEGR23.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 4}},
    PAGES = {253--259},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0021},
    VERSION = {8.1.05 5.37.1275},
    TITLE = {{T}he Basic Existence Theorem of {R}iemann-{S}tieltjes Integral},
    AUTHOR = {Nakasho, Kazuhisa and Narita, Keiko and Shidama, Yasunari},
    ADDRESS1 = {Akita Prefectural University\\Akita, Japan},
      ADDRESS2 = {Hirosaki-city\\Aomori, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, the basic existence theorem of Riemann-Stielt\-jes integral is formalized. This theorem states that if $f$ is a continuous function and $\rho$ is a function of bounded variation in a closed interval of real line, $f$ is Riemann-Stieltjes integrable with respect to $\rho$. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on \cite{Stroock1999}, \cite{Kestelman1960}, \cite{Gupta1986}, and \cite{Hille1974}.   },
    MSC2010 = {26A42 26A45 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Existence of Riemann-Stieltjes Integral for Continuous Functions},
    EXTERNALREFS = {Stroock1999; Kestelman1960; Gupta1986; Hille1974; },
    INTERNALREFS = {CARD_1.ABS;COMPLEX1.ABS;COMSEQ_2.ABS;FCONT_1.ABS;FCONT_2.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSET_1.ABS;
FUNCT_1.ABS;FUNCT_2.ABS;INTEGR22.ABS;INTEGRA1.ABS;INTEGRA2.ABS;INTEGRA3.ABS;INT_1.ABS;MEASURE5.ABS;MEMBERED.ABS;
NAT_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RAT_1.ABS;RCOMP_1.ABS;RELAT_1.ABS;RELSET_1.ABS;RVSUM_1.ABS;SEQ_2.ABS;SEQ_4.ABS;
SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Riemann-Stieltjes integral; bounded variation; continuous function; },
    SUBMITTED = {October 18, 2016}}
    
    
    @ARTICLE{NEWTON04.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 4}},
    PAGES = {261--273},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0022},
    VERSION = {8.1.05 5.39.1282},
    TITLE = {{O}n Subnomials},
    AUTHOR = {Ziobro, Rafa{\l}},
    ADDRESS1 = {Department of Carbohydrate Technology\\University of Agriculture\\Krakow, Poland},
      SUMMARY = {While discussing the sum of consecutive powers as a result of division of two binomials W.W. Sawyer \cite{sawyer1970} observes \begin{quote} ``It is a curious fact that most algebra textbooks give our ast result twice. It appears in two different chapters and usually there is no mention in either of these that it also occurs in the other. The first chapter, of course, is that on factors. The second is that on geometrical progressions. Geometrical progressions are involved in nearly all financial questions involving compound interest -- mortgages, annuities, etc." \end{quote} It's worth noticing that the first issue involves a simple arithmetical division of 99...9 by 9. While the above notion seems not have changed over the last 50 years, it reflects only a special case of a broader class of problems involving two variables. It seems strange, that while binomial formula is discussed and studied widely \cite{CAO201096}, \cite{KHANDUJA2011300}, little research is done on its counterpart with all coefficients equal to one, which we will call here the subnomial. The study focuses on its basic properties and applies it to some simple problems usually proven by~induction~\cite{gancarzewicz2000}.   },
    MSC2010 = {40-04 03B35},
    EXTERNALREFS = {sawyer1970; CAO201096; KHANDUJA2011300; gancarzewicz2000; },
    INTERNALREFS = {ABIAN.ABS;CARD_1.ABS;CLASSES1.ABS;COMPLEX1.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSEQ_5.ABS;FINSET_1.ABS;
FOMODEL0.ABS;FUNCT_1.ABS;FUNCT_2.ABS;INT_1.ABS;MEMBERED.ABS;NEWTON.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PARTFUN3.ABS;
RAT_1.ABS;RELAT_1.ABS;RFINSEQ.ABS;RFINSEQ2.ABS;RVSUM_1.ABS;SETFAM_1.ABS;SQUARE_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {binomial formula; geometrical progression; polynomials; },
    ACKNOWLEDGEMENT = {Ad Maiorem Dei Gloriam},
    SUBMITTED = {October 18, 2016}}
    
    
    @ARTICLE{LEIBNIZ1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 4}},
    PAGES = {275--280},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0023},
    VERSION = {8.1.05 5.39.1282},
    TITLE = {{L}eibniz Series for $\pi$},
    ANNOTE = {This work has been financed by the resources of the Polish National Science Centre granted by decision no. DEC-2012/07/N/ST6/02147.},
    AUTHOR = {P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
      SUMMARY = {In this article we prove the Leibniz series for $\pi$ which states that $$\frac{\pi}{4}=\sum_{n=0}^\infty \frac{(-1)^n}{2\cdot n+1}.$$ \par The formalization follows K. Knopp \cite{Knopp}, \cite{aar1999} and \cite{debnath2010}. {\it {L}eibniz's {S}eries for {P}i} is item {\tt{\#26}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} .   },
    MSC2010 = {40G99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Euler Transformation},
    SECTION3 = {Main Theorem},
    EXTERNALREFS = {Knopp; aar1999; debnath2010; },
    INTERNALREFS = {ABIAN.ABS;COMPLEX1.ABS;COMSEQ_2.ABS;FCONT_1.ABS;FDIFF_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;INTEGRA5.ABS;
INT_1.ABS;MEASURE5.ABS;MEMBERED.ABS;NAT_1.ABS;NEWTON.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PARTFUN3.ABS;RCOMP_1.ABS;RELAT_1.ABS;
RELSET_1.ABS;RFUNCT_1.ABS;SEQ_2.ABS;SEQ_4.ABS;SERIES_1.ABS;SIN_COS.ABS;SIN_COS9.ABS;SQUARE_1.ABS;SUBSET_1.ABS;
TAYLOR_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {$\pi$ approximation; Leibniz theorem; Leibniz series; },
    SUBMITTED = {October 18, 2016}}
    
    
    @ARTICLE{PL_AXIOM.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 4}},
    PAGES = {281--290},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0024},
    VERSION = {8.1.05 5.39.1282},
    TITLE = {{T}he Axiomatization of Propositional Logic},
    ANNOTE = {This work was supported by the University of Bialystok grants: BST447 {\it Formalization of temporal logics in a proof-assistant. Application to System Verification}, and BST225 {\it Database of mathematical texts checked by computer}.},
    AUTHOR = {Giero, Mariusz},
    ADDRESS1 = {Faculty of Economics and Informatics\\ University of Bia{\l}ystok\\ Kalvariju 135, LT-08221 Vilnius\\ Lithuania},
      SUMMARY = {This article introduces propositional logic as a formal system (\cite{AniWas}, \cite{wp1992}, \cite{wp1994}). The formulae of the language are as follows $\phi::=\,\,\perp|\, p\,|\,\phi\rightarrow\phi$. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes \begin{itemize} \item $\alpha\Rightarrow(\beta\Rightarrow\alpha)$, \item $(\alpha\Rightarrow(\beta\Rightarrow\gamma))\Rightarrow((\alpha\Rightarrow\beta)\Rightarrow (\alpha\Rightarrow\gamma))$, \item $(\neg\beta\Rightarrow\neg\alpha)\Rightarrow ((\neg\beta\Rightarrow\alpha)\Rightarrow\beta)$. \end{itemize} Modus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved. The proof of the completeness theorem is carried out by a counter-model existence method. In order to prove the completeness theorem, Lindenbaum's Lemma is proved. Some most widely used tautologies are presented.   },
    MSC2010 = {03B05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {The Syntax},
    SECTION3 = {The Semantics},
    SECTION4 = {The Axioms. Derivability.},
    SECTION5 = {Soundness Theorem. Deduction Theorem.},
    SECTION6 = {Strong Completeness Theorem},
    EXTERNALREFS = {AniWas; wp1992; wp1994; },
    INTERNALREFS = {DOMAIN_1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;HILBERT1.ABS;INTPRO_1.ABS;MARGREL1.ABS;
NAT_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RELAT_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {completeness; formal system; Lindenbaum's lemma; },
    SUBMITTED = {October 18, 2016}}
    
    
    @ARTICLE{ALGNUM_1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 4}},
    PAGES = {291--299},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0025},
    VERSION = {8.1.05 5.39.1282},
    TITLE = {{A}lgebraic Numbers},
    AUTHOR = {Watase, Yasushige},
    ADDRESS1 = {Suginami-ku Matsunoki 6\\3-21 Tokyo, Japan},
      SUMMARY = {This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of ``integral". Definitions for an integral closure, an algebraic integer and a transcendental numbers \cite{Zariski1975}, \cite{atiyah1969introduction}, \cite{nagata1985} and \cite{matsumura1989} are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field $\mathbb{Q}$ induced by substitution of an algebraic number to the polynomial ring of $\mathbb{Q}[x]$ turns to be a field.   },
    MSC2010 = {11R04 13B21 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Extended Evaluation Function},
    SECTION3 = {Integral Element and Algebraic Numbers},
    SECTION4 = {Properties of Polynomial Ring over Principal Ideal Domain},
    EXTERNALREFS = {Zariski1975; atiyah1969introduction; nagata1985; matsumura1989; },
    INTERNALREFS = {ALGSEQ_1.ABS;ALGSTR_1.ABS;BINOP_1.ABS;C0SP1.ABS;CARD_1.ABS;COMPLFLD.ABS;DOMAIN_1.ABS;EC_PF_1.ABS;
FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GAUSSINT.ABS;GCD_1.ABS;GROUP_1.ABS;GROUP_6.ABS;HURWITZ.ABS;IDEAL_1.ABS;
INT_1.ABS;INT_3.ABS;MCART_1.ABS;MEMBERED.ABS;NAT_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PBOOLE.ABS;POLYNOM1.ABS;POLYNOM3.ABS;
POLYNOM4.ABS;POLYNOM5.ABS;RATFUNC1.ABS;RAT_1.ABS;REALSET1.ABS;RELAT_1.ABS;RELSET_1.ABS;RINGCAT1.ABS;RING_1.ABS;
RING_2.ABS;RING_4.ABS;RLVECT_1.ABS;SUBSET_1.ABS;VECTSP_1.ABS;VECTSP_2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {algebraic number; integral dependency; },
    SUBMITTED = {December 15, 2016}}
    
    
    @ARTICLE{NIVEN.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {24},
    NUMBER = {{\bf 4}},
    PAGES = {301--308},
    YEAR = {2016},
    DOI = {10.1515/forma-2016-0026},
    VERSION = {8.1.05 5.39.1282},
    TITLE = {{N}iven's Theorem},
    ANNOTE = {The work on the formalization presented in this article was completed thanks to the Mizar Mathematical Library maintenance and refactoring carried out at the Computer Center of the University of Bia{\l}ystok.},
    AUTHOR = {Korni{\l}owicz, Artur and Naumowicz, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {This article formalizes the proof of Niven's theorem \cite{Niven1956} which states that if $x/\pi$ and $sin(x)$ are both rational, then the sine takes values $0$, $\pm 1/2$, and $\pm 1$. The main part of the formalization follows the informal proof presented at Pr$\infty$fWiki (\url{https://proofwiki.org/wiki/Niven's_Theorem#Source_of_Name}). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients \cite{king2006, SLang}.   },
    MSC2010 = {97G60 12D10 03B35},
    EXTERNALREFS = {Niven1956; king2006, SLang; },
    INTERNALREFS = {ALGSEQ_1.ABS;CARD_1.ABS;COMPLEX1.ABS;ENUMSET1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;
FUNCT_7.ABS;GROUP_1.ABS;HURWITZ.ABS;INT_1.ABS;INT_2.ABS;NAT_1.ABS;NEWTON.ABS;NORMSP_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;
POLYNOM1.ABS;POLYNOM3.ABS;POLYNOM4.ABS;POLYNOM5.ABS;RATFUNC1.ABS;RAT_1.ABS;RELAT_1.ABS;RELSET_1.ABS;RLVECT_1.ABS;RVSUM_1.ABS;
SIN_COS.ABS;SUBSET_1.ABS;VECTSP_1.ABS;VFUNCT_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Niven's theorem; rational root theorem; integral root theorem; },
    SUBMITTED = {December 15, 2016}}


@ARTICLE{MEASUR11.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 1}},
    PAGES = {1--29},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0001},
    VERSION = {8.1.05 5.40.1286},
    TITLE = {{F}ubini's Theorem on Measure},
    AUTHOR = {Endou, Noboru},
    ADDRESS1 = {National Institute of Technology, Gifu College\\2236-2 Kamimakuwa, Motosu, Gifu, Japan},
      SUMMARY = {The purpose of this article is to show Fubini's theorem on measure \cite{Halmos74}, \cite{Bauer:2002}, \cite{Bogachev2007measure}, \cite{FOLLAND}, \cite{Rao2004}. Some theorems have the possibility of slight generalization, but we have priority to avoid the complexity of the description. First of all, for the product measure constructed in \cite{MEASUR10.ABS}, we show some theorems. Then we introduce the section which plays an important role in Fubini's theorem, and prove the relevant proposition. Finally we show Fubini's theorem on measure.   },
    MSC2010 = {28A35 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Product Measure and Product $\sigma$-measure},
    SECTION3 = {Sections},
    SECTION4 = {Measurable Sections},
    SECTION5 = {Finite Sequence of Functions},
    SECTION6 = {Some Properties of Integral},
    SECTION7 = {$\sigma$-finite Measure},
    SECTION8 = {Fubini's Theorem on Measure},
    EXTERNALREFS = {Halmos74; Bauer:2002; Bogachev2007measure; FOLLAND; Rao2004; },
    INTERNALREFS = {BINOP_1.ABS;CARD_1.ABS;CARD_3.ABS;DBLSEQ_3.ABS;EXTREAL1.ABS;FINSEQOP.ABS;FINSEQ_1.ABS;FINSET_1.ABS;
FINSUB_1.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_3.ABS;INT_1.ABS;KURATO_0.ABS;MCART_1.ABS;MEASUR10.ABS;
MEASURE1.ABS;MEASURE3.ABS;MEASURE4.ABS;MEASURE8.ABS;MEASURE9.ABS;MESFUNC1.ABS;MESFUNC5.ABS;MESFUNC9.ABS;NAT_1.ABS;
ORDINAL1.ABS;PARTFUN1.ABS;PROB_1.ABS;PROB_2.ABS;PROB_3.ABS;RAT_1.ABS;RELAT_1.ABS;RELSET_1.ABS;SETFAM_1.ABS;
SRINGS_3.ABS;SUBSET_1.ABS;SUPINF_2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Fubini's theorem; product measure; },
    SUBMITTED = {February 23, 2017}}
    
    
@ARTICLE{POLYDIFF.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 1}},
    PAGES = {31--37},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0002},
    VERSION = {8.1.05 5.40.1286},
    TITLE = {{D}ifferentiability of Polynomials over Reals},
    AUTHOR = {Korni{\l}owicz, Artur},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {In this article, we formalize in the Mizar system \cite{FourDecades} the notion of the derivative of polynomials over the field of real numbers \cite{KURATOWSKI_RRC}. To define it, we use the derivative of functions between reals and reals \cite{FDIFF_1.ABS}.   },
    MSC2010 = {26A24 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Differentiability of Real Functions},
    SECTION3 = {Polynomials},
    SECTION4 = {Differentiability of Polynomials over Reals},
    EXTERNALREFS = {FourDecades; KURATOWSKI_RRC; },
    INTERNALREFS = {ALGSEQ_1.ABS;ALGSTR_1.ABS;FCONT_1.ABS;FDIFF_1.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_7.ABS;
GROUP_1.ABS;INT_1.ABS;NAT_1.ABS;NEWTON.ABS;NORMSP_1.ABS;ORDINAL1.ABS;PARTFUN1.ABS;POLYNOM1.ABS;POLYNOM3.ABS;POLYNOM4.ABS;
POLYNOM5.ABS;RATFUNC1.ABS;RELAT_1.ABS;RLVECT_1.ABS;SUBSET_1.ABS;TAYLOR_1.ABS;VECTSP_1.ABS;VFUNCT_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {differentiation of real polynomials; derivative of real polynomials; },
    SUBMITTED = {February 23, 2017}}
    
    
@ARTICLE{LIOUVIL1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 1}},
    PAGES = {39--48},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0003},
    VERSION = {8.1.05 5.40.1286},
    TITLE = {{I}ntroduction to {L}iouville Numbers},
    AUTHOR = {Grabowski, Adam and Korni{\l}owicz, Artur},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 \cite{Liouville1844} as an example of an object which can be approximated ``quite closely" by a sequence of rational numbers. A real number $x$ is a Liouville number iff for every positive integer $n$, there exist integers $p$ and $q$ such that $q> 1$ and $$0< \left|x - \frac{p}{q}\right|<\frac{1}{q^n}.$$ It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum $$\sum\limits_{k=1}^{\infty}\frac{a_k}{b^{k!}}$$ \noindent for a finite sequence $\{a_k\}_{k\in {\mathbb N}}$ and $b \in {\mathbb N}$. Based on this definition, we also introduced the so-called Liouville number as $$L = \sum\limits_{k=1}^{\infty} 10^{-k!} = 0.110001000000000000000001 \dots,$$ substituting in the definition of $L(a_k,b)$ the constant sequence of 1's and $b = 10$. Another important examples of transcendental numbers are $e$ and $\pi$ \cite{Bingham:2011}, \cite{Eberl-AFP:2015}, \cite{Bernard:2016}. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number \cite{Conway:1996}, \cite{Apostol:1997}. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abad's list of ``Top 100 Theorems". We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar \cite{FourDecades} proof, we follow closely \url{https://en.wikipedia.org/wiki/Liouville_number}. The aim is to show that all Liouville numbers are transcendental.   },
    MSC2010 = {11J81 11K60 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Sequences},
    SECTION3 = {Transformations between Real Functions and Finite Sequences},
    SECTION4 = {Sequences not Vanishing at Infinity},
    SECTION5 = {Liouville Numbers},
    SECTION6 = {Liouville Constant},
    EXTERNALREFS = {Liouville1844; Bingham:2011; Eberl-AFP:2015; Bernard:2016; Conway:1996; Apostol:1997; FourDecades; },
    INTERNALREFS = {ASYMPT_0.ABS;ASYMPT_3.ABS;CARD_1.ABS;COMPLEX1.ABS;COMSEQ_2.ABS;EC_PF_2.ABS;FINSEQ_1.ABS;FINSET_1.ABS;
FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUNCT_4.ABS;INT_1.ABS;MEMBERED.ABS;NAT_1.ABS;NEWTON.ABS;ORDINAL1.ABS;PARTFUN1.ABS;
PARTFUN3.ABS;POWER.ABS;PREPOWER.ABS;RAT_1.ABS;RELAT_1.ABS;RSSPACE.ABS;RVSUM_1.ABS;SEQ_1.ABS;SEQ_2.ABS;SERIES_1.ABS;
SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Liouville number; Diophantine approximation; transcendental number; Liouville constant; },
    SUBMITTED = {February 23, 2017}}


@ARTICLE{LIOUVIL2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 1}},
    PAGES = {49--54},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0004},
    VERSION = {8.1.05 5.40.1286},
    TITLE = {{A}ll {L}iouville Numbers are Transcendental},
    AUTHOR = {Korni{\l}owicz, Artur and Naumowicz, Adam and Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS3 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {In this Mizar article, we complete the formalization of one of the items from Abad and Abad's challenge list of ``Top 100 Theorems" about Liouville numbers and the existence of transcendental numbers. It is item {\tt{\#18}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/}. Liouville numbers were introduced by Joseph Liouville in 1844 \cite{Liouville1844} as an example of an object which can be approximated ``quite closely" by a sequence of rational numbers. A real number $x$ is a Liouville number iff for every positive integer $n$, there exist integers $p$ and $q$ such that $q > 1$ and $$0< \left|x - \frac{p}{q}\right|<\frac{1}{q^n}.$$ It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in \cite{Conway:1996}, \cite{Apostol:1997}, and \cite{LIOUVIL1.ABS}. Liouvile constant, which is defined formally in \cite{LIOUVIL1.ABS}, is the first explicit transcendental (not algebraic) number, another notable examples are $e$ and $\pi$ \cite{Bingham:2011}, \cite{Eberl-AFP:2015}, and \cite{Bernard:2016}. Algebraic numbers were formalized with the help of the Mizar system \cite{FourDecades} very recently, by Yasushige Watase in \cite{ALGNUM_1.ABS} and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more set-theoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouville's theorem on Diophantine approximation.   },
    MSC2010 = {11J81 11K60 03B35},
    EXTERNALREFS = {Liouville1844; Conway:1996; Apostol:1997; Bingham:2011; Eberl-AFP:2015; Bernard:2016; FourDecades; },
    INTERNALREFS = {ALGNUM_1.ABS;ALGSEQ_1.ABS;ALGSTR_1.ABS;C0SP1.ABS;COMPLEX1.ABS;COMPLFLD.ABS;FINSEQ_1.ABS;FINSET_1.ABS;
FUNCT_1.ABS;FUNCT_2.ABS;GAUSSINT.ABS;GROUP_1.ABS;INT_1.ABS;INT_3.ABS;LIOUVIL1.ABS;MEASURE5.ABS;MEMBERED.ABS;NEWTON.ABS;
ORDINAL1.ABS;POLYNOM1.ABS;POLYNOM3.ABS;POLYNOM5.ABS;RATFUNC1.ABS;RAT_1.ABS;RCOMP_1.ABS;RELAT_1.ABS;RLVECT_1.ABS;
RVSUM_1.ABS;SUBSET_1.ABS;UPROOTS.ABS;VECTSP_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Liouville numbers; Diophantine approximation; transcendental numbers; Liouville's constant; },
    SUBMITTED = {February 23, 2017}}    

    
@ARTICLE{ANPROJ_9.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 1}},
    PAGES = {55--62},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0005},
    VERSION = {8.1.05 5.40.1286},
    TITLE = {{G}roup of Homography in Real Projective Plane},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {Using the Mizar system \cite{Mizar-State-2015}, we formalized that homographies of the projective real plane (as defined in \cite{ANPROJ_8.ABS}), form a group. \par Then, we prove that, using the notations of Borsuk and Szmielew in \cite{BORSUK:1} \begin{quote} ``Consider in space $\mathbb{RP}^2$ points $P_1,P_2,P_3,P_4$ of which three points are not collinear and points $Q_1,Q_2,Q_3,Q_4$ each three points of which are also not collinear. There exists one homography $h$ of space $\mathbb{RP}^2$ such that $h(P_i)=Q_i$ for $i = 1,2,3,4$." \end{quote} (Existence Statement 52 and Existence Statement 53) \cite{BORSUK:1}. Or, using notations of Richter \cite{richter2011perspectives} \begin{quote} ``Let $[a]$, $[b]$, $[c]$, $[d]$ in $\mathbb{RP}^2$ be four points of which no three are collinear and let $[a']$,$[b']$,$[c']$,$[d']$ in $\mathbb{RP}^2$ be another four points of which no three are collinear, then there exists a $3\times 3$ matrix $M$ such that $[Ma] = [a']$, $[Mb] = [b']$, $[Mc]$ = $[c']$, and $[Md] = [d']$" \end{quote} Makarios has formalized the same results in Isabelle/Isar (the collineations form a group, lemma statement52-existence and lemma statement 53-existence) and published it in Archive of Formal Proofs\footnote{\url{http://isa-afp.org/entries/Tarskis_Geometry.shtml}} \cite{makarios}, \cite{Tarskis_Geometry-AFP}.   },
    MSC2010 = {51N15 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Group of Homography},
    SECTION3 = {Main Results},
    EXTERNALREFS = {Mizar-State-2015; BORSUK:1; richter2011perspectives; makarios; Tarskis_Geometry-AFP; },
    INTERNALREFS = {ANPROJ_1.ABS;ANPROJ_8.ABS;BINOP_1.ABS;CARD_1.ABS;COLLSP.ABS;EUCLID.ABS;EUCLID_5.ABS;EUCLID_8.ABS;
FINSEQ_1.ABS;FINSEQ_2.ABS;FINSEQ_4.ABS;FINSET_1.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GROUP_1.ABS;MATRIX13.ABS;
MATRIX_1.ABS;MATRIX_3.ABS;MATRIX_6.ABS;MONOID_0.ABS;ORDINAL1.ABS;RELAT_1.ABS;RLVECT_1.ABS;SUBSET_1.ABS;VECTSP_1.ABS;
ZFMISC_1.ABS;},
    KEYWORDS = {projectivity; projective transformation; real projective plane; group of homography; },
    SUBMITTED = {March 17, 2017}}
    
    
@ARTICLE{REALALG1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 1}},
    PAGES = {63--72},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0006},
    VERSION = {8.1.05 5.40.1286},
    TITLE = {{O}rdered Rings and Fields},
    AUTHOR = {Schwarzweller, Christoph},
    ADDRESS1 = {Institute of Informatics\\University of Gda{\'n}sk\\Poland},
      SUMMARY = {We introduce ordered rings and fields following Artin-Schreier's approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways \cite{Pre84,KS89,Jac64,Rad91}. This is the continuation of the development of algebraic hierarchy in Mizar \cite{FourDecades,GrabKornSchwarz:2016}.   },
    MSC2010 = {12J15 03B35},
    SECTION1 = {On Order Relations},
    SECTION2 = {On Minimal Non Zero Indices of Polynomials},
    SECTION3 = {Preliminaries},
    SECTION4 = {Squares and Sums of Squares},
    SECTION5 = {Positive Cones and Orderings},
    SECTION6 = {Orderings vs. Order Relations},
    SECTION7 = {Some Ordered (and Non-ordered) Rings},
    SECTION8 = {Ordered Polynomial Rings},
    EXTERNALREFS = {Pre84; KS89; Jac64; Rad91; FourDecades; GrabKornSchwarz:2016; },
    INTERNALREFS = {ALGSTR_1.ABS;ARROW.ABS;C0SP1.ABS;CARD_1.ABS;COMPLFLD.ABS;DOMAIN_1.ABS;EC_PF_1.ABS;FINSEQ_1.ABS;
FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GAUSSINT.ABS;GROUP_1.ABS;GROUP_2.ABS;IDEAL_1.ABS;INT_1.ABS;INT_3.ABS;MEMBERED.ABS;
NAT_1.ABS;ORDINAL1.ABS;O_RING_1.ABS;PARTFUN1.ABS;POLYNOM1.ABS;POLYNOM3.ABS;RATFUNC1.ABS;RAT_1.ABS;RELAT_1.ABS;
RELAT_2.ABS;RELSET_1.ABS;RING_2.ABS;RING_3.ABS;RLVECT_1.ABS;SUBSET_1.ABS;UPROOTS.ABS;VECTSP_1.ABS;VECTSP_2.ABS;
VFUNCT_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {commutative algebra; ordered fields; positive cones; },
    SUBMITTED = {March 17, 2017}}
    
    
@ARTICLE{ZMODLAT2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 1}},
    PAGES = {73--86},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0007},
    VERSION = {8.1.05 5.40.1286},
    TITLE = {{E}mbedded Lattice and Properties of {G}ram Matrix},
    ANNOTE = {This work was supported by JSPS KAKENHI Grant Number JP15K00183.},
    AUTHOR = {Futa, Yuichi and Shidama, Yasunari},
    ADDRESS1 = {Tokyo University of Technology\\Tokyo, Japan},
      ADDRESS2 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, we formalize in Mizar \cite{FourDecades} the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of $\mathbb Z$-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm \cite{LLL} and cryptographic systems with lattice \cite{LATTICE2002}.   },
    MSC2010 = {15A09 15A63 03B35},
    SECTION1 = {Inner Product of Embedded Module},
    SECTION2 = {Embedding of Lattice},
    SECTION3 = {Properties of Gram Matrix},
    EXTERNALREFS = {FourDecades; LLL; LATTICE2002; },
    INTERNALREFS = {BINOP_1.ABS;CARD_1.ABS;DOMAIN_1.ABS;EQREL_1.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSET_1.ABS;FUNCOP_1.ABS;
FUNCT_1.ABS;FUNCT_2.ABS;GAUSSINT.ABS;GROUP_1.ABS;INT_1.ABS;INT_3.ABS;LAPLACE.ABS;MATRIX13.ABS;MATRIX_3.ABS;MATRIX_6.ABS;
ORDINAL1.ABS;PARTFUN1.ABS;PRVECT_1.ABS;RANKNULL.ABS;RAT_1.ABS;REALSET1.ABS;RELAT_1.ABS;RELSET_1.ABS;RLVECT_1.ABS;
SUBSET_1.ABS;VECTSP_1.ABS;VECTSP_4.ABS;VECTSP_6.ABS;VECTSP_7.ABS;ZFMISC_1.ABS;ZMATRLIN.ABS;ZMODLAT1.ABS;ZMODUL01.ABS;
ZMODUL03.ABS;ZMODUL04.ABS;ZMODUL06.ABS;ZMODUL08.ABS;},
    KEYWORDS = {$\mathbb{Z}$-lattice; Gram matrix; rational $\mathbb{Z}$-lattice; },
    SUBMITTED = {March 17, 2017}}


@ARTICLE{POLYVIE1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 2}},
    PAGES = {87--92},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0008},
    VERSION = {8.1.06 5.43.1297},
    TITLE = {{V}ieta's Formula about the Sum of Roots of Polynomials},
    AUTHOR = {Korni{\l}owicz, Artur and P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    SUMMARY = {In the article we formalized in the Mizar system \cite{Mizar-State-2015} the Vieta formula about the sum of roots of a~polynomial $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ defined over an~algebraically closed field. The formula says that $x_1 + x_2 + \cdots + x_{n-1} + x_n = -\frac{a_{n-1}}{a_n}$, where $x_1, x_2, \dots, x_n$ are (not necessarily distinct) roots of the polynomial \cite{Vinberg}. In the article the sum is denoted by \texttt{SumRoots}.   },
    MSC2010 = {12E05 03B35},
    EXTERNALREFS = {Mizar-State-2015; Vinberg; },
    INTERNALREFS = {ALGSEQ_1.ABS;ALGSTR_1.ABS;BINOM.ABS;CARD_1.ABS;DOMAIN_1.ABS;FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;
FVSUM_1.ABS;GROUP_1.ABS;INT_1.ABS;NAT_1.ABS;NIVEN.ABS;ORDINAL1.ABS;PARTFUN1.ABS;POLYNOM1.ABS;POLYNOM3.ABS;POLYNOM4.ABS;
POLYNOM5.ABS;RELAT_1.ABS;RELSET_1.ABS;RLAFFIN3.ABS;RLVECT_1.ABS;SUBSET_1.ABS;UPROOTS.ABS;VECTSP_1.ABS;VECTSP_2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {roots of polynomials; Vieta's formula; },
    SUBMITTED = {May 25, 2017}}
    
    
    @ARTICLE{FUZNORM1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 2}},
    PAGES = {93--100},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0009},
    VERSION = {8.1.06 5.43.1297},
    TITLE = {{B}asic Formal Properties of Triangular Norms and Conorms},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    SUMMARY = {In the article we present in the Mizar system \cite{Mizar-State-2015}, \cite{FourDecades} the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets \cite{Zadeh:1965}. The name \emph{triangular} emphasizes the fact that in the framework of probabilistic metric spaces they generalize triangle inequality \cite{DuboisPrade:1980}.\par After defining corresponding Mizar mode using four attributes, we introduced the following t-norms: \begin{itemize} \item minimum t-norm \verb!minnorm! (Def.~6), \item product t-norm \verb!prodnorm! (Def.~8), \item \L ukasiewicz t-norm \verb!Lukasiewicz_norm! (Def.~10), \item drastic t-norm \verb!drastic_norm! (Def.~11), \item nilpotent minimum \verb!nilmin_norm! (Def.~12), \item Hamacher product \verb!Hamacher_norm! (Def.~13), \end{itemize} and corresponding t-conorms: \begin{itemize} \item maximum t-conorm \verb!maxnorm! (Def.~7), \item probabilistic sum \verb!probsum_conorm! (Def.~9), \item bounded sum \verb!BoundedSum_conorm! (Def.~19), \item drastic t-conorm \verb!drastic_conorm! (Def.~14), \item nilpotent maximum \verb!nilmax_conorm! (Def.~18), \item Hamacher t-conorm \verb!Hamacher_conorm! (Def.~17). \end{itemize} Their basic properties and duality are shown; we also proved the predicate of the ordering of norms \cite{Klement:2000}, \cite{Hajek:1998}. It was proven formally that drastic-norm is the pointwise smallest t-norm and minnorm is the pointwise largest t-norm (maxnorm is the pointwise smallest t-conorm and drastic-conorm is the pointwise largest t-conorm). \par This work is a continuation of the development of fuzzy sets in Mizar \cite{GrabowskiFuzzy:2013} started in \cite{FUZZY_1.ABS} and \cite{FUZNUM_1.ABS}; it could be used to give a variety of more general operations on fuzzy sets. Our formalization is much closer to the set theory used within the Mizar Mathematical Library than the development of rough sets \cite{GrabowskiAssisted:2005}, the approach which was chosen allows however for merging both theories \cite{GrabowskiFI:2014}, \cite{GrabowskiMitsuishi:2015}.   },
    MSC2010 = {03E72 94D05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Basic Example of a Triangular Norm and Conorm: min and max},
    SECTION3 = {Further Examples of Triangular Norms},
    SECTION4 = {Basic Triangular Conorms},
    SECTION5 = {Translating between Triangular Norms and Conorms},
    SECTION6 = {The Ordering of Triangular Norms (and Conorms)},
    SECTION7 = {Triangular Conorms Generated from t-Norms},
    EXTERNALREFS = {Mizar-State-2015; FourDecades; Zadeh:1965; DuboisPrade:1980; Klement:2000; Hajek:1998; 
    GrabowskiFuzzy:2013; GrabowskiAssisted:2005; GrabowskiFI:2014; GrabowskiMitsuishi:2015; },
    INTERNALREFS = {BINOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;FUZZY_1.ABS; FUZNUM_1.ABS; MEMBERED.ABS;PARTFUN1.ABS;RCOMP_1.ABS;
REAL_1.ABS;RELAT_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {fuzzy set; triangular norm; triangular conorm; fuzzy logic; },
    SUBMITTED = {June 27, 2017}}
    
    
    @ARTICLE{FINANCE4.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 2}},
    PAGES = {101--105},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0010},
    VERSION = {8.1.06 5.43.1297},
    TITLE = {{I}ntroduction to Stopping Time in Stochastic Finance Theory},
    AUTHOR = {Jaeger, Peter},
    ADDRESS1 = {Siegmund-Schacky-Str. 18a\\80993 Munich, Germany},
    SUMMARY = { We start with the definition of stopping time according to \cite{follmerschied:2004}, p.283. We prove, that different definitions for stopping time can coincide. We give examples of stopping time using constant-functions or functions defined with the operator max or min (defined in \cite{klenke:2006}, pp.37--38). Finally we give an example with some given filtration. Stopping time is very important for stochastic finance. A stopping time is the moment, where a certain event occurs (\cite{kremer:2006}, p.372) and can be used together with stochastic processes (\cite{follmerschied:2004}, p.283). Look at the following example: we install a function ST: \{1,2,3,4\} \to \{0,1,2\} \ $\cup$ \{+$\infty$\}, we define:\par a. ST(1)=1, ST(2)=1, ST(3)=2, ST(4)=2.\par b. The set \{0,1,2\} consists of time points: 0=now,1=tomorrow,2=the day after tomorrow.\par We can prove:\par c. \{w, where w is Element of $\Omega$: ST.w=0\}=$\emptyset$ \& \{w, where w is Element of $\Omega$: ST.w=1\}=\{1,2\} \& \{w, where w is Element of $\Omega$: ST.w=2\}=\{3,4\} and\par ST is a stopping time.\par We use a function Filt as Filtration of \{0,1,2\}, $\Sigma$ where Filt(0)=$\Omega_{now}$, Filt(1)=$\Omega_{fut1}$ and Filt(2)=$\Omega_{fut2}$. From a.,b. and c. we know that:\par d. \{w, where w is Element of $\Omega$: ST.w=0\} in $\Omega_{now}$ and\par \{w, where w is Element of $\Omega$: ST.w=1\} in $\Omega_{fut1}$ and\par \{w, where w is Element of $\Omega$: ST.w=2\} in $\Omega_{fut2}$.\par The sets in d. are events, which occur at the time points 0(=now), 1(=tomorrow) or 2(=the day after tomorrow), see also \cite{kremer:2006}, p.371. Suppose we have ST(1)=$+\infty$, then this means that for 1 the corresponding event never occurs.\par As an interpretation for our installed functions consider the given adapted stochastic process in the article \cite{FINANCE3.ABS}.\par ST(1)=1 means, that the given element 1 in \{1,2,3,4\} is stopped in 1 (=tomorrow). That tells us, that we have to look at the value $f_{2}(1)$ which is equal to 80. The same argumentation can be applied for the element 2 in \{1,2,3,4\}.\par ST(3)=2 means, that the given element 3 in \{1,2,3,4\} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value $f_{3}(3)$ which is equal to 100.\par ST(4)=2 means, that the given element 4 in \{1,2,3,4\} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value $f_{3}(4)$ which is equal to 120.\par In the real world, these functions can be used for questions like: when does the share price exceed a certain limit? (see \cite{kremer:2006}, p.372).   },
    MSC2010 = {60G40 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Definition of Stopping Time},
    SECTION3 = {Examples of Stopping Times},
    EXTERNALREFS = {follmerschied:2004; klenke:2006; kremer:2006; },
    INTERNALREFS = {ENUMSET1.ABS;FINANCE3.ABS;FINSUB_1.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;INT_1.ABS;MATRIX_7.ABS;
MEMBERED.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PROB_1.ABS;RAT_1.ABS;RELAT_1.ABS;SETFAM_1.ABS;SUBSET_1.ABS;SUPINF_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {stopping time; stochastic process; },
    SUBMITTED = {June 27, 2017}}
    
    
    @ARTICLE{PASCAL.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 2}},
    PAGES = {107--119},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0011},
    VERSION = {8.1.06 5.43.1297},
    TITLE = {{P}ascal's Theorem in Real Projective Plane},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
    SUMMARY = {In this article we check, with the Mizar system \cite{Mizar-State-2015}, Pascal's theorem in the real projective plane (in projective geometry Pascal's theorem is also known as the Hexagrammum Mysticum Theorem)\footnote{\url{https://en.wikipedia.org/wiki/Pascal's\_theorem}}. Pappus' theorem is a special case of a degenerate conic of two lines.\par For proving Pascal's theorem, we use the techniques developed in the section ``Projective Proofs of Pappus' Theorem" in the chapter ``Pappus' Theorem: Nine proofs and three variations" \cite{Richter-Gebert2011}. We also follow some ideas from Harrison's work. With HOL Light, he has the proof of Pascal's theorem\footnote{\url{https://github.com/jrh13/hol-light/tree/master/100/pascal.ml}}. For a lemma, we use { \verb PROVER9 }\footnote{\url{https://www.cs.unm.edu/~mccune/prover9/ }} and { \verb OTT2MIZ } by Josef Urban\footnote{\url{https://github.com/JUrban/ott2miz}} \cite{rudnicki2011escape, grabowski2006solving, grabowskiJAR40}. We note, that we don't use Skolem/Herbrand functions (see ``Skolemization'' in \cite{alama2012escape}).   },
    MSC2010 = {51E15 51N15 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Conic in Real Projective Plane},
    SECTION3 = {Pascal's Theorem},
    EXTERNALREFS = {Mizar-State-2015; Richter-Gebert2011; rudnicki2011escape; grabowski2006solving; grabowskiJAR40; alama2012escape; },
    INTERNALREFS = {ANPROJ_1.ABS;ANPROJ_2.ABS;ANPROJ_8.ABS;CARD_1.ABS;COLLSP.ABS;DOMAIN_1.ABS;EUCLID.ABS;EUCLID_5.ABS;
FINSEQ_1.ABS;FINSEQ_2.ABS;FINSEQ_4.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GROUP_1.ABS;LAPLACE.ABS;MATRIXR1.ABS;
MATRIXR2.ABS;MATRIX_3.ABS;MATRIX_6.ABS;MATRPROB.ABS;MONOID_0.ABS;ORDINAL1.ABS;PRE_TOPC.ABS;RELAT_1.ABS;RLTOPSP1.ABS;
RLVECT_1.ABS;SUBSET_1.ABS;VECTSP_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Pascal's theorem; real projective plane; Grassman-Pl{\"u}cker relation; },
    SUBMITTED = {June 27, 2017}}
    
    
    @ARTICLE{ORDERS_5.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 2}},
    PAGES = {121--139},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0012},
    VERSION = {8.1.06 5.43.1297},
    TITLE = {{A}bout Quotient Orders and Ordering Sequences},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
    NOTE1 = {mailto: {\tt skoch02@students.uni-mainz.de}},
    SUMMARY = { In preparation for the formalization in Mizar \cite{Mizar-State-2015} of lotteries as given in \cite{MSZ}, this article closes some gaps in the Mizar Mathematical Library (MML) regarding relational structures. The quotient order is introduced by the equivalence relation identifying two elements $x, y$ of a preorder as equivalent if $x \leqslant y$ and $y \leqslant x$. This concept is known (see e.g. chapter 5 of \cite{OSI}) and was first introduced into the MML in \cite{DICKSON.ABS} and that work is incorporated here. Furthermore given a set $A$, partition $D$ of $A$ and a finite-support function $f:A\rightarrow\mathbb{R}$, a function $\Sigma_f : D \rightarrow\mathbb{R}, \Sigma_f (X) = \sum_{x\in X} f(x)$ can be defined as some kind of natural ``restriction" from $f$ to $D$. The first main result of this article can then be formulated as: \[\sum_{x\in A} f(x) = \sum_{X\in D}\Sigma_f (X) \left(= \sum_{X\in D}\sum_{x\in X} f(x)\right)\] After that (weakly) ascending/descending finite sequences (based on \cite{FINSEQ_1.ABS}) are introduced, in analogous notation to their infinite counterparts introduced in \cite{WELLFND1.ABS} and \cite{DICKSON.ABS}.\par The second main result is that any finite subset of any transitive connected relational structure can be sorted as a ascending or descending finite sequence, thus generalizing the results from \cite{RFINSEQ2.ABS}, where finite sequence of real numbers were sorted.\par The third main result of the article is that any weakly ascending/weakly descending finite sequence on elements of a preorder induces a weakly ascending/weakly descending finite sequence on the projection of these elements into the quotient order. Furthermore, weakly ascending finite sequences can be interpreted as directed walks in a directed graph, when the set of edges is described by ordered pairs of vertices, which is quite common (see e.g. \cite{IAlg}).\par Additionally, some auxiliary theorems are provided, e.g. two schemes to find the smallest or the largest element in a finite subset of a connected transitive relational structure with a given property and a lemma I found rather useful: Given two finite one-to-one sequences $s, t$ on a set $X$, such that $\rng t \subseteq \rng s$, and a function $f:X\rightarrow\mathbb{R}$ such that $f$ is zero for every $x\in\rng s\setminus\rng t$, we have $\sum f\circ s = \sum f\circ t$.   },
    MSC2010 = {06A05 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Orders},
    SECTION3 = {Quotient Order},
    SECTION4 = {Ordering Finite Sequences},
    EXTERNALREFS = {Mizar-State-2015; MSZ; OSI; IAlg; },
    INTERNALREFS = {CARD_1.ABS;
DICKSON.ABS;EQREL_1.ABS;FINSEQOP.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSEQ_5.ABS;FINSET_1.ABS;FUNCOP_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;
MEMBERED.ABS;NECKLACE.ABS;ORDERS_1.ABS;ORDERS_2.ABS;ORDINAL1.ABS;PARTFUN1.ABS;PARTFUN3.ABS;RELAT_1.ABS;RELAT_2.ABS;RELSET_1.ABS;RFINSEQ2.ABS;
RVSUM_1.ABS;SETFAM_1.ABS;SUBSET_1.ABS;WELLFND1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {quotient order; ordered finite sequences; },
    ACKNOWLEDGEMENT = {I thank Dr. Adam Grabowski for his encouragement and the team behind {\tt mus@mizar.uwb.edu.pl} for their quick and helpful replies to my beginner's questions.},
    SUBMITTED = {June 27, 2017}}
    
    
    @ARTICLE{BASEL_1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 2}},
    PAGES = {141--147},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0013},
    VERSION = {8.1.06 5.43.1297},
    TITLE = {{B}asel Problem -- Preliminaries},
    AUTHOR = {Korni{\l}owicz, Artur and P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    SUMMARY = {In the article we formalize in the Mizar system \cite{Mizar-State-2015} preliminary facts needed to prove the Basel problem \cite{Cauchy1821,PFTB}. Facts that are independent from the notion of structure are included here.   },
    MSC2010 = {11M06 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Limits of Sequences $\frac{an+b}{cn+d}$},
    SECTION3 = {Trigonometry},
    SECTION4 = {Some Special Functions and Sequences},
    EXTERNALREFS = {Mizar-State-2015; Cauchy1821; PFTB; },
    INTERNALREFS = {CARD_1.ABS;COMSEQ_2.ABS;COMSEQ_3.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;
INT_1.ABS;MEMBERED.ABS;ORDINAL1.ABS;PARTFUN1.ABS;POLYEQ_1.ABS;RCOMP_1.ABS;RELAT_1.ABS;RELSET_1.ABS;RVSUM_1.ABS;SEQ_2.ABS;
SERIES_1.ABS;SETFAM_1.ABS;SIN_COS.ABS;SIN_COS4.ABS;SQUARE_1.ABS;SUBSET_1.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Basel problem; },
    SUBMITTED = {June 27, 2017}}
    
    
    @ARTICLE{BASEL_2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 2}},
    PAGES = {149--155},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0014},
    VERSION = {8.1.06 5.43.1297},
    TITLE = {{B}asel Problem},
    ANNOTE = {This work has been financed by the resources of the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.},
    AUTHOR = {P\k{a}k, Karol and Korni{\l}owicz, Artur},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    SUMMARY = {A~rigorous elementary proof of the Basel problem \cite{Cauchy1821,PFTB} $$\Sigma_{n=1}^\infty\frac{1}{n^2} = \frac{\pi^2}{6}$$ is formalized in the Mizar system \cite{Mizar-State-2015}. This theorem is item {\tt{\#14}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} .   },
    MSC2010 = {11M06 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Imaginary Complex Numbers},
    SECTION3 = {Main Facts},
    EXTERNALREFS = {Cauchy1821; PFTB; Mizar-State-2015; },
    INTERNALREFS = {ABIAN.ABS;ALGSEQ_1.ABS;ALGSTR_1.ABS;BASEL_1.ABS;BINOM.ABS;BINOP_1.ABS;CARD_1.ABS;COMPLEX1.ABS;COMPLFLD.ABS;
FINSEQ_1.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;GROUP_1.ABS;HAHNBAN1.ABS;HURWITZ2.ABS;INT_1.ABS;NAT_1.ABS;NEWTON.ABS;
NIVEN.ABS;NORMSP_1.ABS;ORDINAL1.ABS;POLYNOM1.ABS;POLYNOM3.ABS;POLYNOM4.ABS;POLYNOM5.ABS;RELAT_1.ABS;RLVECT_1.ABS;RVSUM_1.ABS;
SERIES_1.ABS;SIN_COS.ABS;SQUARE_1.ABS;SUBSET_1.ABS;UPROOTS.ABS;VECTSP_1.ABS;VECTSP_2.ABS;ZFMISC_1.ABS;},
    KEYWORDS = {Basel problem; },
    SUBMITTED = {June 27, 2017}}
    
    
    @ARTICLE{ZMODLAT3.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 2}},
    PAGES = {157--169},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0015},
    VERSION = {8.1.06 5.43.1297},
    TITLE = {{D}ual Lattice of {$\mathbb{Z}$}-module Lattice},
    ANNOTE = {This work was supported by JSPS KAKENHI grant number JP15K00183.},
    AUTHOR = {Futa, Yuichi and Shidama, Yasunari},
    ADDRESS1 = {Tokyo University of Technology\\Tokyo, Japan},
    ADDRESS2 = {Shinshu University\\Nagano, Japan},
    SUMMARY = {In this article, we formalize in Mizar \cite{Mizar-State-2015} the definition of dual lattice and their properties. We formally prove that a set of all dual vectors in a rational lattice has the construction of a lattice. We show that a dual basis can be calculated by elements of an inverse of the Gram Matrix. We also formalize a summation of inner products and their properties. Lattice of $\mathbb Z$-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lov\'asz) base reduction algorithm and cryptographic systems with lattice \cite{LATTICE2002}, \cite{LANDC} and \cite{LLL}.   },
    MSC2010 = {15A03 15A09 03B35},
    SECTION1 = {Summation of Inner Products},
    SECTION2 = {Dual Lattice},
    EXTERNALREFS = {Mizar-State-2015; LATTICE2002; LANDC; LLL; },
    INTERNALREFS = {BINOP_1.ABS;CARD_1.ABS;DOMAIN_1.ABS;FINSEQ_1.ABS;FINSEQ_2.ABS;FINSET_1.ABS;FUNCT_1.ABS;FUNCT_2.ABS;
GAUSSINT.ABS;INT_1.ABS;INT_3.ABS;MATRIX13.ABS;MATRIX_6.ABS;MCART_1.ABS;MEMBERED.ABS;ORDINAL1.ABS;PARTFUN1.ABS;RAT_1.ABS;
REALSET1.ABS;RELAT_1.ABS;RELSET_1.ABS;RLVECT_1.ABS;SUBSET_1.ABS;VECTSP_1.ABS;VECTSP_4.ABS;VECTSP_6.ABS;VECTSP_7.ABS;
ZFMISC_1.ABS;ZMATRLIN.ABS;ZMODLAT1.ABS;ZMODLAT2.ABS;ZMODUL01.ABS;ZMODUL03.ABS;ZMODUL04.ABS;ZMODUL06.ABS;ZMODUL08.ABS;},
    KEYWORDS = {$\mathbb{Z}$-lattice; dual lattice of $\mathbb{Z}$-lattice; dual basis of $\mathbb{Z}$-lattice; },
    SUBMITTED = {June 27, 2017}}

@ARTICLE{VECTSP12.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 3}},
    PAGES = {171--178},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0016},
    VERSION = {8.1.06 5.44.1305},
    TITLE = {{I}somorphism Theorem on Vector Spaces over a Ring},
    ANNOTE = {This work was supported by JSPS KAKENHI grant number JP15K00183.},
    AUTHOR = {Futa, Yuichi and Shidama, Yasunari},
    ADDRESS1 = {Tokyo University of Technology\\Tokyo, Japan},
      ADDRESS2 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, we formalize in the Mizar system \cite{Mizar-State-2015,FourDecades} some properties of vector spaces over a ring. We formally prove the first isomorphism theorem of vector spaces over a ring. We also formalize the product space of vector spaces. $\mathbb Z$-modules are useful for lattice problems such as LLL (Lenstra, Lenstra and Lov\'asz) \cite{LLL} base reduction algorithm and cryptographic systems \cite{LATTICE2002, LANDC}.   },
    MSC2010 = {15A03 15A04 03B35},
    SECTION1 = {Bijective Linear Transformation},
    SECTION2 = {Properties of Linear Combinations of Modules over a Ring},
    SECTION3 = {First Isomophism Theorem},
    SECTION4 = {The Product Space of Vector Spaces},
    SECTION5 = {Cartesian Product of Vector Spaces},
    EXTERNALREFS = {Mizar-State-2015; FourDecades; LLL; LATTICE2002; LANDC; },
    INTERNALREFS = {ZMODUL04.ABS; NDIFF_5.ABS; },
    KEYWORDS = {isomorphism theorem; vector space; },
    SUBMITTED = {August 30, 2017}}
    
    
    @ARTICLE{DUALSP05.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 3}},
    PAGES = {179--184},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0017},
    VERSION = {8.1.06 5.44.1305},
    TITLE = {{F}. {R}iesz Theorem},
    AUTHOR = {Narita, Keiko and Nakasho, Kazuhisa and Shidama, Yasunari},
    ADDRESS1 = {Hirosaki-city\\Aomori, Japan},
      ADDRESS2 = {Akita Prefectural University\\Akita, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, we formalize in the Mizar system \cite{Mizar-State-2015,FourDecades} the F. Riesz theorem. In the first section, we defined Mizar functor \verb!ClstoCmp!, compact topological spaces as closed interval subset of real numbers. Then using the former definition and referring to the article \cite{C0SP1.ABS} and the article \cite{C0SP2.ABS}, we defined the normed spaces of continuous functions on closed interval subset of real numbers, and defined the normed spaces of bounded functions on closed interval subset of real numbers. We also proved some related properties. \par In Sec.2, we proved some lemmas for the proof of F. Riesz theorem. In Sec.3, we proved F. Riesz theorem, about the dual space of the space of continuous functions on closed interval subset of real numbers, finally. We applied Hahn-Banach theorem (36) in \cite{DUALSP01.ABS}, to the proof of the last theorem. For the description of theorems of this section, we also referred to the article \cite{INTEGR22.ABS} and the article \cite{INTEGR23.ABS}. These formalizations are based on \cite{Brezis2011}, \cite{Dax2002}, \cite{rudin1991functional}, and \cite{yoshida:1980}.   },
    MSC2010 = {46E15 46B10 03B35},
    SECTION1 = {The Normed Space of Continuous Functions on Closed Interval},
    SECTION2 = {Preliminaries},
    SECTION3 = {F. Riesz Theorem},
    EXTERNALREFS = {Mizar-State-2015; FourDecades; Brezis2011; Dax2002; rudin1991functional; yoshida:1980; },
    INTERNALREFS = {C0SP1.ABS; C0SP2.ABS; DUALSP01.ABS; INTEGR22.ABS; INTEGR23.ABS; },
    KEYWORDS = {F. Riesz theorem; dual spaces; continuous functions; },
    SUBMITTED = {August 30, 2017}}
    
    
    @ARTICLE{RING_5.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 3}},
    PAGES = {185--195},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0018},
    VERSION = {8.1.06 5.44.1305},
    TITLE = {{O}n Roots of Polynomials and Algebraically Closed Fields},
    AUTHOR = {Schwarzweller, Christoph},
    ADDRESS1 = {Institute of Informatics\\University of Gda{\'n}sk\\Poland},
      SUMMARY = {In this article we further extend the algebraic theory of polynomial rings in Mizar \cite{Mizar-State-2015,FourDecades,GrabKornSchwarz:2016}. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed \cite{Jacobson2009,Rad91alg1}. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial's degree \cite{HH90,HL99}.   },
    MSC2010 = {13A05 13B25 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Some More Properties of Polynomials},
    SECTION3 = {On Roots of Polynomials},
    SECTION4 = {More about Bags},
    SECTION5 = {On Multiple Roots of Polynomials},
    SECTION6 = {The Polynomial $X^n + 1$},
    SECTION7 = {The Polynomials $(x-a_1) * (x-a_2) * \ldots * (x-a_n)$},
    SECTION8 = {Main Theorems},
    EXTERNALREFS = {Mizar-State-2015; FourDecades; GrabKornSchwarz:2016; Jacobson2009; Rad91alg1; HH90; HL99; },
    INTERNALREFS = {HURWITZ.ABS; RING_4.ABS; },
    KEYWORDS = {commutative algebra; polynomials; algebraic closed fields; },
    SUBMITTED = {August 30, 2017}}
    
    
    @ARTICLE{PELLS_EQ.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 3}},
    PAGES = {197--204},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0019},
    VERSION = {8.1.06 5.44.1305},
    TITLE = {{P}ell's Equation},
    ANNOTE = {This work has been financed by the resources of the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.},
    AUTHOR = {Acewicz, Marcin and P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {In this article we formalize several basic theorems that correspond to Pell's equation. We focus on two aspects: that the Pell's equation $x^2-Dy^2 = 1$ has infinitely many solutions in positive integers for a given $D$ not being a perfect square, and that based on the least fundamental solution of the equation when we can simply calculate algebraically each remaining solution. \par ``Solutions to Pell's Equation'' are listed as item {\tt{\#39}} from the ``Formalizing 100 Theorems'' list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} ~.   },
    MSC2010 = {11D45 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Solutions to Pell's Equation -- Construction},
    SECTION3 = {Pell's Equation},
    SECTION4 = {Solutions to Pell's Equation -- Shape},
    EXTERNALREFS = {Lenstra; Weil:1983; Lagrange; Matiyasevich; Mizar-State-2015; HOL; metamath; KrumbiegelAmthor;
                    SIERPINSKI:1; },
    INTERNALREFS = {POWER.ABS; },
    KEYWORDS = {Pell's equation; Diophantine equation; Hilbert's 10th problem; },
    SUBMITTED = {August 30, 2017}}
    
    
    @ARTICLE{NOMIN_1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 3}},
    PAGES = {205--216},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0020},
    VERSION = {8.1.06 5.44.1305},
    TITLE = {{S}imple-Named Complex-Valued Nominative Data -- Definition and Basic Operations},
    AUTHOR = {Ivanov, Ievgen and Nikitchenko, Mykola and Kryvolap, Andrii and Korni{\l}owicz, Artur},
    ADDRESS1 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      ADDRESS2 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      ADDRESS3 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      ADDRESS4 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {In this paper we give a~formal definition of the notion of nominative data with simple names and complex values \cite{Nikitch98,Nikitch2001,Skobelev2014} and formal definitions of the basic operations on such data, including naming, denaming and overlapping, following the work \cite{Skobelev2014}. \par The notion of nominative data plays an important role in the composition-nominative approach to program formalization \cite{Nikitch98,Nikitch2001} which is a~development of composition programming \cite{Redko1979}. Both approaches are compared in \cite{Nikitch2009}. \par The composition-nominative approach considers mathematical models of computer software and data on various levels of abstraction and generality and provides mathematical tools for reasoning about their properties. In particular, nominative data are mathematical models of data which are stored and processed in computer systems. The composition-nominative approach considers different types \cite{Nikitch2009,Skobelev2014} of nominative data, but all of them are based on the name-value relation. One powerful type of nominative data, which is suitable for representing many kinds of data commonly used in programming like lists, multidimensional arrays, trees, tables, etc. is the type of nominative data with simple (abstract) names and complex (structured) values. The set of nominative data of given type together with a~number of basic operations on them like naming, denaming and overlapping \cite{Skobelev2014} form an algebra which is called {\em data algebra}. \par In the composition-nominative approach computer programs which process data are modeled as partial functions which map nominative data from the carrier of a~given data algebra (input data) to nominative data (output data). Such functions are also called {\em binominative functions}. Programs which evaluate conditions are modeled as partial predicates on nominative data (nominative predicates). Programming language constructs like sequential execution, branching, cycle, etc. which construct programs from the existing programs are modeled as operations which take binominative functions and predicates and produce binominative functions. Such operations are called {\em compositions}. A~set of binominative functions and a~set of predicates together with appropriate compositions form an algebra which is called {\em program algebra}. This algebra serves as a~semantic model of a~programming language. \par For functions over nominative data a~special computability called abstract computability is introduces and complete classes of computable functions are specified \cite{Nikitch2001}. \par For reasoning about properties of programs modeled as binominative functions a~Floyd-Hoare style logic \cite{Floyd1967,Hoare1969} is introduced and applied \cite{Kryvolap2013,NikitchKryvolap2013,DBLP:journals/csjm/IvanovNS16,KornilowiczetalICTERI2017,DBLP:conf/fedcsis/KornilowiczKNI17, DBLP:conf/isat/KornilowiczKNI17}. One advantage of this approach to reasoning about programs is that it naturally handles programs which process complex data structures (which can be quite straightforwardly represented as nominative data). Also, unlike classical Floyd-Hoare logic, the mentioned logic allows reasoning about assertions which include partial pre- and post-conditions \cite{KornilowiczetalICTERI2017}. \par Besides modeling data processed by programs, nominative data can be also applied to modeling data processed by signal processing systems in the context of the mathematical systems theory \cite{Ivanov2014,IvanovNikitchAbr2014,DBLP:journals/fm/IvanovNA15,Ivanov2016,DBLP:journals/corr/Ivanov17}.   },
    MSC2010 = {68Q60 03B70 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Definition of Simple-Named Complex-Valued Nominative Data},
    SECTION3 = {Examples of Simple-Named Complex-Valued Nominative Data},
    SECTION4 = {Operations on Simple-Named Complex-Valued Nominative Data},
    EXTERNALREFS = {Nikitch98; Nikitch2001; Skobelev2014; Redko1979; Nikitch2009; Floyd1967; Hoare1969; Kryvolap2013; 
    NikitchKryvolap2013; DBLP:journals/csjm/IvanovNS16; KornilowiczetalICTERI2017; DBLP:conf/fedcsis/KornilowiczKNI17;  
    Ivanov2014; IvanovNikitchAbr2014; DBLP:journals/fm/IvanovNA15; Ivanov2016; DBLP:journals/corr/Ivanov17; },
    INTERNALREFS = {CGAMES_1.ABS; PETERSON.ABS;},
    KEYWORDS = {program semantics; software verification; nominative data; },
    SUBMITTED = {August 30, 2017}}
    
    
@ARTICLE{COUSIN2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 3}},
    PAGES = {217--225},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0021},
    VERSION = {8.1.06 5.44.1305},
    TITLE = {{G}auge Integral},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {Some authors have formalized the integral in the Mizar Mathematical Library (MML). The first article in a series on the Darboux/Riemann integral was written by Noboru Endou and Artur Korni{\l}owicz: \cite{INTEGRA1.ABS}. The Lebesgue integral was formalized a little later \cite{shidama2007formalization} and recently the integral of Riemann-Stieltjes was introduced in the MML by Keiko Narita, Kazuhisa Nakasho and Yasunari Shidama \cite{INTEGR22.ABS}.\par A presentation of definitions of integrals in other proof assistants or proof checkers (ACL2, COQ, Isabelle/HOL, HOL4, HOL Light, PVS, ProofPower) may be found in \cite{harrison2007formalizing} and \cite{boldo2016formalization}.\par Using the Mizar system \cite{Mizar-State-2015}, we define the Gauge integral (Henstock-Kurzweil) of a real-valued function on a real interval $\lbrack a,b \rbrack$ (see \cite{bartle1996return}, \cite{bartle2001modern}, \cite{yee2000integral}, \cite{peng2008integral}, \cite{mawhin2001eternel}). In the next section we formalize that the Henstock-Kurzweil integral is linear.\par In the last section, we verified that a real-valued bounded integrable (in sense Darboux/Riemann \cite{INTEGRA1.ABS,INTEGRA3.ABS,INTEGRA4.ABS}) function over a interval $\[a,b\]$ is Gauge integrable.\par Note that, in accordance with the possibilities of the MML \cite{grabowski2007revisions}, we reuse a large part of demonstrations already present in another article. Instead of rewriting the proof already contained in \cite{INTEGRA3.ABS} (MML Version: 5.42.1290), we slightly modified this article in order to use directly the expected results.   },
    MSC2010 = {26A39 26A42 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Vector Lattice / Riesz Space},
    SECTION3 = {Some Properties of the ${\raise.4ex\hbox{$\chi$}}$ Function},
    SECTION4 = {Refinement of Tagged Partition},
    SECTION5 = {Definition of the Gauge Integral on a Real Bounded Interval},
    SECTION6 = {The Linearity Property of the Gauge Integral},
    SECTION7 = {Riemann Integrability and Gauge Integrability},
    EXTERNALREFS = {shidama2007formalization; harrison2007formalizing; boldo2016formalization; Mizar-State-2015; 
    bartle1996return; bartle2001modern; yee2000integral; peng2008integral; mawhin2001eternel; grabowski2007revisions; },
    INTERNALREFS = {COUSIN.ABS; INTEGRA1.ABS; INTEGR22.ABS; INTEGRA3.ABS; INTEGRA4.ABS; },
    KEYWORDS = {Gauge integral; Henstock-Kurzweil integral; generalized Riemann integral; },
    SUBMITTED = {September 3, 2017}}


    @ARTICLE{MESFUN11.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 3}},
    PAGES = {227--240},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0022},
    VERSION = {8.1.06 5.44.1305},
    TITLE = {{I}ntegral of Non Positive Functions},
    AUTHOR = {Endou, Noboru},
    ADDRESS1 = {National Institute of Technology, Gifu College\\2236-2 Kamimakuwa, Motosu, Gifu, Japan},
      SUMMARY = {In this article, we formalize in the Mizar system \cite{Mizar-State-2015,FourDecades} the Lebesgue type integral and convergence theorems for non positive functions \cite{Halmos74},\cite{bogachev2007measure}. Many theorems are based on our previous results \cite{MESFUNC5.ABS}, \cite{MESFUNC9.ABS}.   },
    MSC2010 = {28A25 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Integral of Non Positive Measurable Functions},
    SECTION3 = {Convergence Theorems for Non Positive Function's Integration},
    EXTERNALREFS = {Mizar-State-2015; FourDecades; Halmos74; bogachev2007measure; },
    INTERNALREFS = {DBLSEQ_3.ABS; MEASUR11.ABS; MESFUNC5.ABS; MESFUNC9.ABS; },
    KEYWORDS = {integration of non positive function; },
    SUBMITTED = {September 3, 2017}}
    
    
    @ARTICLE{FUZIMPL1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {25},
    NUMBER = {{\bf 3}},
    PAGES = {241--248},
    YEAR = {2017},
    DOI = {10.1515/forma-2017-0023},
    VERSION = {8.1.06 5.44.1305},
    TITLE = {{F}ormal Introduction to Fuzzy Implications},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {In the article we present in the Mizar system the catalogue of nine basic fuzzy implications, used especially in the theory of fuzzy sets. This work is a continuation of the development of fuzzy sets in Mizar; it could be used to give a variety of more general operations, and also it could be a good starting point towards the formalization of fuzzy logic (together with t-norms and t-conorms, formalized previously).   },
    MSC2010 = {03B52 68T37 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Basic Attributes Defining Fuzzy Implications},
    SECTION3 = {Examples Showing Independence of Axioms},
    SECTION4 = {Catalogue of Fuzzy Implications},
    SECTION5 = {Boundary Fuzzy Implications},
    EXTERNALREFS = {Baczynski:2008; FourDecades; Grabowski2018; GrabowskiFuzzy:2013; GrabowskiLTRS; 
    GrabowskiMitsuishi:2015; GrabowskiPerspective:2007; Pawlak1982; Zadeh:1965; },
    INTERNALREFS = {FUZNORM1.ABS; FUZNUM_1.ABS; FUZZY_1.ABS; },
    KEYWORDS = {fuzzy implication; fuzzy set; fuzzy logic; },
    SUBMITTED = {September 3, 2017}}
    
    
    
    @ARTICLE{REALALG2.ABS,
        JOURNAL = {Formalized Mathematics},
    	ISSN = {1426-2630},
    	EISSN = {1898-9934},
          VOLUME = {25},
        NUMBER = {{\bf 4}},
        PAGES = {249--259},
        YEAR = {2017},
        DOI = {10.1515/forma-2017-0024},
        VERSION = {8.1.06 5.45.1311},
        TITLE = {{F}ormally Real Fields},
        AUTHOR = {Schwarzweller, Christoph},
        ADDRESS1 = {Institute of Informatics\\Faculty of Mathematics, Physics and Informatics\\University of Gda{\'n}sk\\Wita Stwosza 57, 80-308 Gda{\'n}sk, Poland},
          SUMMARY = {We extend the algebraic theory of ordered fields \cite{Rad91,Pre84} in Mizar \cite{Mizar-State-2015,FourDecades,GrabKornSchwarz:2016}: we show that every preordering can be extended into an ordering, i.e. that formally real and ordered fields coincide. We further prove some characterizations of formally real fields, in particular the one by Artin and Schreier using sums of squares \cite{Jac64}. In the second part of the article we define absolute values and the square root function \cite{KS89}.   },
        MSC2010 = {12J15 03B35},
        SECTION1 = {Preliminaries},
        SECTION2 = {More on Ring Characteristic},
        SECTION3 = {Maximal Preorderings},
        SECTION4 = {Formally Real Fields},
        SECTION5 = {Order Relations and Strict Order Relations Revisited},
        SECTION6 = {Absolute Values},
        SECTION7 = {Squares and Square Roots},
        EXTERNALREFS = {Rad91; Pre84; Mizar-State-2015; FourDecades; GrabKornSchwarz:2016; Jac64; KS89; },
        INTERNALREFS = {RING_3.ABS; BINOM.ABS; RING_5.ABS; REALALG1.ABS; },
        KEYWORDS = {formally real fields; ordered fields; abstract value; square roots; },
        SUBMITTED = {November 29, 2017}}
        
        
    @ARTICLE{FINANCE5.ABS,
        JOURNAL = {Formalized Mathematics},
    	ISSN = {1426-2630},
    	EISSN = {1898-9934},
          VOLUME = {25},
        NUMBER = {{\bf 4}},
        PAGES = {261--268},
        YEAR = {2017},
        DOI = {10.1515/forma-2017-0025},
        VERSION = {8.1.06 5.45.1311},
        TITLE = {{I}ntroduction to Stopping Time in Stochastic Finance Theory. {P}art {II}},
        AUTHOR = {Jaeger, Peter},
        ADDRESS1 = {Siegmund-Schacky-Str. 18a\\80993 Munich, Germany},
          SUMMARY = {We start proceeding with the stopping time theory in discrete time with the help of the Mizar system \cite{Mizar-State-2015}, \cite{FourDecades}. We prove, that the expression for two stopping times $k_{1}$ and $k_{2}$ not always implies a stopping time $(k_{1}+k_{2})$ (see Theorem 6 in this paper). If you want to get a stopping time, you have to cut the function e.g. $(k_{1}+k_{2}) \cap T $ (see \cite[p.~283 Remark 6.14]{follmerschied:2004}). \par Next we introduce the stopping time in continuous time. We are focused on the intervals $[0,r]$ where $r \in \mathbb{R}$. We prove, that for $I = [0,r]$ or $I=[0,+\infty[$ the set $\{ A \cap I: A \in \mathop{\rm Borel\hbox{-}Sets} \}$ is a $\sigma$-algebra of $I$ (see Definition 6 in this paper, and more general given in \cite[p.12 1.8e]{georgii:2004}). The interval $I$ can be considered as a timeline from now to some point in the future. \par This set is necessary to define our next lemma. We prove the existence of the $\sigma$-algebra of the $\tau$-past, where $\tau$ is a stopping time (see Definition 11 in this paper and \cite[p.187, Definition 9.19]{klenke:2006}). If $\tau_{1}$ and $\tau_{2}$ are stopping times with $\tau_{1}$ is smaller or equal than $\tau_{2}$ we can prove, that the $\sigma$-algebra of the $\tau_{1}$-past is a subset of the $\sigma$-algebra of the $\tau_{2}$-past (see Theorem 9 in this paper and \cite[p.187 Lemma 9.21]{klenke:2006}). \par Suppose, that you want to use Lemma 9.21 with some events, that never occur, see as a comparison the paper \cite{FINANCE4.ABS} and the example for ST(1)=$\{ + \infty \}$ in the Summary. We don't have the element $+\infty$ in our above-mentioned time intervals $[0,r[$ and [0,$+\infty$[. This is only possible if we construct a new $\sigma$-algebra on $\mathbb{R} \cup \{-\infty,+\infty\}$. This construction is similar to the Borel-Sets and we call this $\sigma$-algebra extended Borel sets (see Definition 13 in this paper and \cite[p.~21]{georgii:2004}). It can be proved, that $\{ +\infty \}$ is an Element of extended Borel sets (see Theorem 21 in this paper). Now we use the interval $[0,+\infty]$ as a basis. We construct a $\sigma$-algebra on $[0,+\infty]$ similar to the book (\cite[p.~12 18e]{georgii:2004}), see Definition 18 in this paper, and call it extended Borel subsets. We prove for stopping times with this given $\sigma$-algebra, that for $\tau_{1}$ and $\tau_{2}$ are stopping times with $\tau_{1}$ is smaller or equal than $\tau_{2}$ we have the $\sigma$-algebra of the $\tau_{1}$-past is a subset of the $\sigma$-algebra of the $\tau_{2}$-past, see Theorem 25 in this paper. It is obvious, that $\{ +\infty \} \in$ extended Borel subsets. \par In general, Lemma 9.21 is important for the proof of the Optional Sampling Theorem, see 10.11 Proof of (i) in \cite[p.~203]{klenke:2006}.   },
        MSC2010 = {60G40 03B35},
        SECTION1 = {Preliminaries},
        SECTION2 = {Stopping Time in Discrete Time},
        SECTION3 = {Stopping Time in Continuous Time},
        SECTION4 = {Borel-Sets},
        SECTION5 = {$\sigma$-Algebra of the $\tau$-Past},
        EXTERNALREFS = {Mizar-State-2015; follmerschied:2004; georgii:2004; FourDecades; klenke:2006; },
        INTERNALREFS = {FINANCE4.ABS; PROB_1.ABS; },
        KEYWORDS = {stopping time; stochastic process; },
        SUBMITTED = {November 29, 2017}}
        
        
    @ARTICLE{NDIFF_8.ABS,
        JOURNAL = {Formalized Mathematics},
    	ISSN = {1426-2630},
    	EISSN = {1898-9934},
          VOLUME = {25},
        NUMBER = {{\bf 4}},
        PAGES = {269--281},
        YEAR = {2017},
        DOI = {10.1515/forma-2017-0026},
        VERSION = {8.1.06 5.45.1311},
        TITLE = {{I}mplicit Function Theorem. {P}art {I}},
        ANNOTE = {This study was supported in part by JSPS KAKENHI Grant Number JP17K00182.},
        AUTHOR = {Nakasho, Kazuhisa and Futa, Yuichi and Shidama, Yasunari},
        ADDRESS1 = {Osaka University\\Osaka, Japan},
          ADDRESS2 = {Tokyo University of Technology\\Tokyo, Japan},
          ADDRESS3 = {Shinshu University\\Nagano, Japan},
          SUMMARY = {In this article, we formalize in Mizar \cite{Mizar-State-2015}, \cite{FourDecades} the existence and uniqueness part of the implicit function theorem. In the first section, some composition properties of Lipschitz continuous linear function are discussed. In the second section, a definition of closed ball and theorems of several properties of open and closed sets in Banach space are described. In the last section, we formalized the existence and uniqueness of continuous implicit function in Banach space using Banach fixed point theorem. We referred to \cite{Schwartz1997a}, \cite{Schwartz1997b}, and \cite{driver2003} in this formalization.   },
        MSC2010 = {26B10 53A07 03B35},
        SECTION1 = {Properties of Lipschitz Continuous Linear Function},
        SECTION2 = {Properties of Open and Closed Sets in Banach Space},
        SECTION3 = {Existence and Uniqueness of Continuous Implicit Function},
        EXTERNALREFS = {Mizar-State-2015; driver2003; FourDecades; Schwartz1997a; Schwartz1997b; },
        INTERNALREFS = {LOPBAN_7.ABS; NFCONT_1.ABS; PRVECT_3.ABS; LOPBAN_1.ABS; },
        KEYWORDS = {implicit function theorem; Banach fixed point theorem; Lipschitz continuity; },
        SUBMITTED = {November 29, 2017}}
        
        
    @ARTICLE{DIOPHAN2.ABS,
        JOURNAL = {Formalized Mathematics},
    	ISSN = {1426-2630},
    	EISSN = {1898-9934},
          VOLUME = {25},
        NUMBER = {{\bf 4}},
        PAGES = {283--288},
        YEAR = {2017},
        DOI = {10.1515/forma-2017-0027},
        VERSION = {8.1.06 5.45.1311},
        TITLE = {{I}ntroduction to {D}iophantine Approximation. {P}art {II}},
        AUTHOR = {Watase, Yasushige},
        ADDRESS1 = {Suginami-ku Matsunoki 3-21-6 Tokyo\\Japan},
          SUMMARY = {In the article we present in the Mizar system \cite{Mizar-State-2015}, \cite{FourDecades} the formalized proofs for Hurwitz' theorem \cite[1891]{HURWITZ:1891} and Minkowski's theorem \cite{MINKOWSKI:1907}. Both theorems are well explained as a basic result of the theory of Diophantine approximations appeared in \cite{HardyWright:2008}, \cite{NIVEN:2008}. \par A formal proof of Dirichlet's theorem, namely an inequation $| \theta - y/x | \leq 1/x^2$ has infinitely many integer solutions $(x,y)$ where $\theta$ is an irrational number, was given in \cite{DIOPHAN1.ABS}. A finer approximation is given by Hurwitz' theorem: $| \theta - y/x | \leq 1/\sqrt5 x^2$ .\par Minkowski's theorem concerns an inequation of a product of non-homogeneous binary linear forms such that $| a_1 x + b_1 y + c_1 |\cdot | a_2 x + b_2 y + c_2 | \leq \Delta/4$ where $\Delta = |a_1 b_2 - a_2 b_1| \neq 0$, has at least one integer solution.   },
        MSC2010 = {11J20 11J25 03B35},
        SECTION1 = {Preliminaries},
        SECTION2 = {Hurwitz' Theorem \cite[1891]{HURWITZ:1891}},
        SECTION3 = {Minkowski's Theorem \cite[Zweites Kapitel, \textsection 11, 1907]{MINKOWSKI:1907}},
        EXTERNALREFS = {Mizar-State-2015; FourDecades; HardyWright:2008; HURWITZ:1891; MINKOWSKI:1907; NIVEN:2008; },
        INTERNALREFS = {DIOPHAN1.ABS; AFINSQ_1.ABS; },
        KEYWORDS = {Diophantine approximation; rational approximation; Dirichlet; Hurwitz; Minkowski; },
        SUBMITTED = {November 29, 2017}}
        
        
    @ARTICLE{GTARSKI3.ABS,
        JOURNAL = {Formalized Mathematics},
    	ISSN = {1426-2630},
    	EISSN = {1898-9934},
          VOLUME = {25},
        NUMBER = {{\bf 4}},
        PAGES = {289--313},
        YEAR = {2017},
        DOI = {10.1515/forma-2017-0028},
        VERSION = {8.1.06 5.45.1311},
        TITLE = {{T}arski Geometry Axioms. {P}art {III}},
        AUTHOR = {Coghetto, Roland and Grabowski, Adam},
        ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
          ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
          SUMMARY = {In the article, we continue the formalization of the work devoted to Tarski's geometry -- the book ``Metamathematische Methoden in der Geometrie" by W. Schwabh\"auser, W. Szmielew, and A. Tarski. After we prepared some introductory formal framework in our two previous Mizar articles, we focus on the regular translation of underlying items faithfully following the abovementioned book (our encoding covers first seven chapters). Our development utilizes also other formalization efforts of the same topic, e.g. Isabelle/HOL by Makarios, Metamath or even proof objects obtained directly from Prover9. \par In addition, using the native Mizar constructions (cluster registrations) the propositions (``Satz") are reformulated under weaker conditions, i.e. by using fewer axioms or by proposing an alternative version that uses just another axioms (ex. Satz 2.1 or Satz 2.2).   },
        MSC2010 = {51A05 51M04 03B35},
        SECTION1 = {Congruence Properties},
        SECTION2 = {Betweenness Relation},
        SECTION3 = {Collinearity},
        SECTION4 = {Line Segments},
        SECTION5 = {Lines and Halflines},
        SECTION6 = {Point Reflection},
        SECTION7 = {Note about Simplification of Tarski's Axioms of Geometry by Makarios},
        SECTION8 = {Main Results and Corollaries},
        EXTERNALREFS = {beeson2014otter; Braun:2017; dhurdjevic2015automated; Grabowski:FedCSIS2016; Gupta:1965; 
                        Makarios; Tarskis_Geometry-AFP; makarios:2014; Narboux:2007; Schwabhauser:1983; },
        INTERNALREFS = {GTARSKI1.ABS; GTARSKI2.ABS; },
        KEYWORDS = {Tarski's geometry axioms; foundations of geometry; Euclidean plane; },
        SUBMITTED = {November 29, 2017}}
        
        
    @ARTICLE{HILB10_1.ABS,
        JOURNAL = {Formalized Mathematics},
    	ISSN = {1426-2630},
    	EISSN = {1898-9934},
          VOLUME = {25},
        NUMBER = {{\bf 4}},
        PAGES = {315--322},
        YEAR = {2017},
        DOI = {10.1515/forma-2017-0029},
        VERSION = {8.1.06 5.45.1311},
        TITLE = {{T}he {M}atiyasevich Theorem. {P}reliminaries},
        ANNOTE = {This work has been financed by the resources of the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.},
        AUTHOR = {P\k{a}k, Karol},
        ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
          SUMMARY = {In this article, we prove selected properties of Pell's equation that are essential to finally prove the Diophantine property of two equations. These equations are explored in the proof of Matiyasevich's negative solution of Hilbert's tenth problem.   },
        MSC2010 = {11D45 03B35},
        SECTION1 = {Preliminaries},
        SECTION2 = {Solutions of Pell's Equation -- Special Case},
        SECTION3 = {Solutions of Pell's Equation -- Inequalities},
        SECTION4 = {Solutions of Pell's Equation -- Equality},
        SECTION5 = {Solutions of Pell's Equation -- Congruences},
        SECTION6 = {Solutions of Pell's Equation -- Divisibility},
        SECTION7 = {Special Case of Pell's Equation is Diophantine},
        SECTION8 = {Exponential Function is Diophantine},
        EXTERNALREFS = {AdamowiczZbierski; Hilbert10; },
        INTERNALREFS = {PELLS_EQ.ABS; INT_4.ABS; NEWTON02.ABS; EULER_1.ABS; NAT_2.ABS; },
        KEYWORDS = {Pell's equation; Diophantine equation; Hilbert's 10th problem; },
        SUBMITTED = {November 29, 2017}}
 
   
@ARTICLE{FINANCE6.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},                                    
    NUMBER = {{\bf 1}},
    PAGES = {1--9},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0001},
    VERSION = {8.1.07 5.47.1318},
    TITLE = {{I}ntroduction to Stochastic Finance: Random Variables and Arbitrage Theory},
    AUTHOR = {Jaeger, Peter},
    ADDRESS1 = {Siegmund-Schacky-Str. 18a\\80993 Munich, Germany},
    SUMMARY = {Using the Mizar system \cite{Mizar-State-2015}, \cite{FourDecades}, we start to show, that the Call-Option, the Put-Option and the Straddle (more generally defined as in the literature) are random variables (\cite{follmerschied:2004}, p.~15), see (Def.~\hyperlink{def:1}{1}) and (Def.~\hyperlink{def:2}{2}). Next we construct and prove the simple random variables (\cite{heinz:2002}, p.~14) in (Def.~\hyperlink{def:8}{8}). \par In the third section, we introduce the definition of arbitrage opportunity, see (Def.~\hyperlink{def:12}{12}). Next we show, that this definition can be characterized in a different way (Lemma 1.3. in \cite{follmerschied:2004}, p.~5), see (\hyperlink{th:17}{17}). In our formalization for Lemma 1.3 we make the assumption that $\varphi$ is a sequence of real numbers (there are only finitely many valued of interest, the values of $\varphi$ in $R^d$). For the definition of almost sure with probability 1 see p.~6 in \cite{heinz:2002}. Last we introduce the risk-neutral probability (Definition 1.4, p.~6 in \cite{follmerschied:2004}), here see (Def.~\hyperlink{def:16}{16}). \par We give an example in real world: Suppose you have some assets like bonds (riskless assets). Then we can fix our price for these bonds with $x$ for today and $x\cdot(1+r)$ for tomorrow, $r$ is the interest rate. So we simply assume, that in every possible market evolution of tomorrow we have a determinated value. Then every probability measure of $\Omega_{fut1}$ is a risk-neutral measure, see (\hyperlink{th:21}{21}). This example shows the existence of some risk-neutral measure. If you find more than one of them, you can determine -- with an additional condition to the probability measures -- whether a market model is arbitrage free or not (see Theorem 1.6. in \cite{follmerschied:2004}, p.~6.) \par A short graph for (\hyperlink{th:21}{21}): \par Suppose we have a portfolio with many (in this example infinitely many) assets. For asset $d$ we have the price $\pi(d)$ for today, and the price $\pi(d)\cdot(1+r)$ for tomorrow with some interest rate $r>0$. \par Let G be a sequence of random variables on $\Omega_{fut1}$, Borel sets. So you have many functions $f_{k}:\{1,2,3,4\} \rightarrow R$ with $G(k)=f_{k}$ and $f_{k}$ is a random variable of $\Omega_{fut1}$, Borel sets. For every $f_{k}$ we have $f_{k}(w)=\pi(k)\cdot(1+r)$ for $w \in \{1,2,3,4\}$. \\ \\ \begin{center} $ {\begin{array}{cp{0.2cm}c} \strut\mkern80mu Today & & Tomorrow\strut\mkern80mu\\ \\ \multicolumn{1}{r}{\rm ~only~one~scenario} & & \multicolumn{1}{l}{\left\{ \begin{array}{l} w_{21}=\{1,2\},\\ w_{22}=\{3,4\},\\ \end{array}\right \phantom{\}}.} \\ \\ \multicolumn{1}{r}{{\rm ~for~all~} d \in \mathbb{N} {\rm ~holds}~ \pi(d)} & & \left\{ \begin{array}{l} f_{d}(w)=G(d)(w)=\pi(d)\cdot(1+r),\\ w \in w_{21} {\rm ~or~} w \in w_{22},\\ r>0 {\rm ~is~the~interest~rate.} \end{array}\right \phantom{\}}. \\ \end{array}} $ \end{center} Here, every probability measure of $\Omega_{fut1}$ is a risk-neutral measure.   },
    MSC2010 = {28A05 03B35},
    SECTION1 = {Put-Option, Call-Option and Straddle are Random Variables},
    SECTION2 = {Simple Random Variables},
    SECTION3 = {Arbitrage Theory: Definition and Alternative Representation},
    SECTION4 = {Risk-Neutral Probability Measure},
    EXTERNALREFS = {Mizar-State-2015; FourDecades; follmerschied:2004; heinz:2002; },
    INTERNALREFS = {MESFUNC2.ABS; FINANCE1.ABS; FINANCE3.ABS; },
    KEYWORDS = {random variable; arbitrage theory; risk-neutral measure; },
    SUBMITTED = {March 27, 2018}}

      
@ARTICLE{PARTPR_1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 1}},
    PAGES = {11--20},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0002},
    VERSION = {8.1.07 5.47.1318},
    TITLE = {{K}leene Algebra of Partial Predicates},
    AUTHOR = {Korni{\l}owicz, Artur and Ivanov, Ievgen and Nikitchenko, Mykola},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    ADDRESS2 = {Taras Shevchenko National University\\Kyiv, Ukraine},
    ADDRESS3 = {Taras Shevchenko National University\\Kyiv, Ukraine},
    SUMMARY = {We show that the set of all partial predicates over a set $D$ together with the disjunction, conjunction, and negation operations, defined in accordance with the truth tables of S.C. Kleene's strong logic of indeterminacy \cite{Kleene1952}, forms a~Kleene algebra. A~Kleene algebra is a~De Morgan algebra \cite{Brignole1964} (also called quasi-Boolean algebra) which satisfies the condition $x \wedge \neg x \le y \vee \neg y$ (sometimes called the normality axiom). We use the formalization of De Morgan algebras from \cite{ROBBINS1.ABS}. \par The term ``Kleene algebra'' was introduced by A.~Monteiro and D.~Brignole in \cite{Brignole1964}. A~similar notion of a~``normal i-lattice'' had been previously studied by J.A. Kalman \cite{Kalman1958}. More details about the origin of this notion and its relation to other notions can be found in \cite{Monteiro1996,Cignoli1975,Balbes,Blyth1994}. It should be noted that there is a~different widely known class of algebras, also called Kleene algebras \cite{Kozen1990,Conway1971}, which generalize the algebra of regular expressions, however, the term ``Kleene algebra'' used in this paper does not refer to them. \par Algebras of partial predicates naturally arise in computability theory in the study on partial recursive predicates. They were studied in connection with non-classical logics \cite{Kleene1952,Cleave1991,Korner1966,RasiowaNonClassical,NikitchShkilniak2013,NikShk2017:QE-calculus}. A~partial predicate also corresponds to the notion of a~partial set \cite{Negri1998} on a~given domain, which represents a (partial) property which for any given element of this domain may hold, not hold, or neither hold nor not hold. The field of all partial sets on a~given domain is an algebra with generalized operations of union, intersection, complement, and three constants (0, 1, $n$ which is the fixed point of complement) which can be generalized to an equational class of algebras called DMF-algebras (De Morgan algebras with a~single fixed point of involution) \cite{Negri1996}. In \cite{Negri2013} partial sets and DMF-algebras were considered as a~basis for unification of set-theoretic and linguistic approaches to probability. \par Partial predicates over classes of mathematical models of data were used for formalizing semantics of computer programs in the composition-nominative approach to program formalization \cite{Nikitch98,NikitchShkilniak2008,Skobelev2014,DBLP:journals/csjm/IvanovNS16}, for formalizing extensions of the Floyd-Hoare logic \cite{Floyd1967,Hoare1969} which allow reasoning about properties of programs in the case of partial pre- and postconditions \cite{Kryvolap2013,KornilowiczetalICTERI2017,DBLP:conf/fedcsis/KornilowiczKNI17,Kornilowicz2018}, for formalizing dynamical models with partial behaviors in the context of the mathematical systems theory \cite{Ivanov2014,Ivanov2014a,PETERSON.ABS,Ivanov2016,DBLP:journals/corr/Ivanov17}.   },
    MSC2010 = {03B70 03G25 03B35},
    SECTION1 = {Partial Predicates},
    SECTION2 = {Algebra of Partial Connectives with (strong) Kleene Logical Connectives},
    EXTERNALREFS = {Kleene1952; Brignole1964; Kalman1958; Monteiro1996; Cignoli1975; Balbes1975; Blyth1994; Kozen1990; Conway1971; 
Cleave1991; Korner1966; Rasiowa1974; NikitchShkilniak2013; NikShk2017:QE-calculus; Negri1998; Negri1996; Negri2013; Nikitch98; 
NikitchShkilniak2008; Skobelev2014; DBLP:journals/csjm/IvanovNS16; Floyd1967; Hoare1969; Kryvolap2013; KornilowiczetalICTERI2017; 
DBLP:conf/fedcsis/KornilowiczKNI17; Kornilowicz2018; Ivanov2014; Ivanov2014a; 
Ivanov2016; DBLP:journals/corr/Ivanov17; },
    INTERNALREFS = {ROBBINS1.ABS; PETERSON.ABS; },
    KEYWORDS = {partial predicate; Kleene algebra; },
    SUBMITTED = {March 27, 2018}}

@ARTICLE{BKMODEL1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 1}},
    PAGES = {21--32},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0003},
    VERSION = {8.1.07 5.47.1318},
    TITLE = {{K}lein-{B}eltrami Model. {P}art {I}},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
    SUMMARY = {Tim Makarios (with Isabelle/HOL\footnote{\url{https://www.isa-afp.org/entries/Tarskis_Geometry.html}}) and John Harrison (with HOL-Light\footnote{\url{https://github.com/jrh13/hol-light/blob/master/100/independence.ml}}) shown that ``the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski's axioms except his Euclidean axiom" \cite{beltrami1868saggio}, \cite{beltrami1869essai}, \cite{BORSUK_1.ABS}, \cite{BS55}. \par With the Mizar system \cite{Mizar-State-2015}, \cite{FourDecades} we use some ideas are taken from Tim Makarios' MSc thesis \cite{makarios} for the formalization of some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. This work can be also treated as further development of Tarski's geometry in the formal setting \cite{GrabowskiFed2016}. Note that the model presented here, may also be called ``Beltrami-Klein Model", ``Klein disk model", and the ``Cayley-Klein model" \cite{a2014klein}.   },
    MSC2010 = {51A05 51M10 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Absolute},
    EXTERNALREFS = {beltrami1868saggio; beltrami1869essai; BS55; GrabowskiFed2016; Mizar-State-2015; FourDecades; makarios; a2014klein; },
    INTERNALREFS = {EUCLID_5.ABS; TOPREAL9.ABS; TOPREALB.ABS; EUCLID_4.ABS; EUCLID_8.ABS; BORSUK_1.ABS; MATRIX_6.ABS; },
    KEYWORDS = {Tarski's geometry axioms; foundations of geometry; Klein-Beltrami model; },
    SUBMITTED = {March 27, 2018}}


@ARTICLE{BKMODEL2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 1}},
    PAGES = {33--48},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0004},
    VERSION = {8.1.07 5.47.1318},
    TITLE = {{K}lein-{B}eltrami Model. {P}art {II}},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
    SUMMARY = {Tim Makarios (with Isabelle/HOL\footnote{\url{https://www.isa-afp.org/entries/Tarskis_Geometry.html}}) and John Harrison (with HOL-Light\footnote{\url{https://github.com/jrh13/hol-light/blob/master/100/independence.ml}}) have shown that ``the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski's axioms except his Euclidean axiom" \cite{beltrami1868saggio,beltrami1869essai,BORSUK_1.ABS,BS55}. \par With the Mizar system \cite{Mizar-State-2015}, \cite{FourDecades} we use some ideas are taken from Tim Makarios' MSc thesis \cite{makarios} for formalized some definitions (like the tangent) and lemmas necessary for the verification of the independence of the parallel postulate. This work can be also treated as a further development of Tarski's geometry in the formal setting \cite{GrabowskiFed2016}.   },
    MSC2010 = {51A05 51M10 03B35},
    SECTION1 = {Beltrami-Cayley-Klein Disk Model},
    SECTION2 = {Tangent},
    SECTION3 = {Subgroup of $K$-Isometry},
    SECTION4 = {Main Lemmas},
    EXTERNALREFS = {beltrami1868saggio; beltrami1869essai; BS55; Mizar-State-2015; GrabowskiFed2016; FourDecades; makarios; },
    INTERNALREFS = {ANPROJ_8.ABS; ANPROJ_9.ABS; BKMODEL1.ABS; PASCAL.ABS; EUCLID_5.ABS; JGRAPH_1.ABS; LAPLACE.ABS; BORSUK_1.ABS; },
    KEYWORDS = {Tarski's geometry axioms; foundations of geometry; Klein-Beltrami model; },
    SUBMITTED = {March 27, 2018}}


@ARTICLE{MESFUN12.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 1}},
    PAGES = {49--67},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0005},
    VERSION = {8.1.07 5.47.1318},
    TITLE = {{F}ubini's Theorem for Non-Negative or Non-Positive Functions},
    AUTHOR = {Endou, Noboru},
    ADDRESS1 = {National Institute of Technology, Gifu College\\2236-2 Kamimakuwa, Motosu, Gifu, Japan},
    SUMMARY = {The goal of this article is to show Fubini's theorem for non-negative or non-positive measurable functions \cite{Halmos74}, \cite{Bauer:2002}, \cite{Bogachev2007measure}, using the Mizar system \cite{Mizar-State-2015}, \cite{FourDecades}. We formalized Fubini's theorem in our previous article \cite{MEASUR11.ABS}, but in that case we showed the Fubini's theorem for measurable sets and it was not enough as the integral does not appear explicitly. \par On the other hand, the theorems obtained in this paper are more general and it can be easily extended to a general integrable function. Furthermore, it also can be easy to extend to functional space $L^p$ \cite{LPSPACE1.ABS}. It should be mentioned also that H{\"o}lzl and Heller \cite{hoelzl2011measuretheory} have introduced the Lebesgue integration theory in Isabelle/HOL and have proved Fubini's theorem there.   },
    MSC2010 = {28A35 03B35},
    SECTION1 = {Extended Real-Valued Characteristic Function},
    SECTION2 = {Some Properties of Integration},
    SECTION3 = {Sections of Partial Function},
    SECTION4 = {Fubini's Theorem for Non-negative or Non-positive Functions},
    EXTERNALREFS = {Halmos74; Bauer:2002; Bogachev2007measure; Mizar-State-2015; FourDecades; hoelzl2011measuretheory; },
    INTERNALREFS = {MEASUR10.ABS; MEASUR11.ABS; MESFUNC5.ABS; EXTREAL1.ABS; MESFUNC2.ABS; LPSPACE1.ABS; },
    KEYWORDS = {Fubini's theorem; extended real-valued non-negative (or non-positive) measurable function; },
    SUBMITTED = {March 27, 2018}}


@ARTICLE{MOEBIUS3.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 1}},
    PAGES = {69--79},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0006},
    VERSION = {8.1.07 5.47.1318},
    TITLE = {{S}equences of Prime Reciprocals. {P}reliminaries},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    SUMMARY = {In the article we formalize some properties needed to prove that sequences of prime reciprocals are divergent. The aim is to show that the series exhibits log-log growth. We introduce some auxiliary notions as harmonic numbers, telescoping series, and prove some standard properties of logarithms and exponents absent in the Mizar Mathematical Library. At the end we proceed with square-free and square-containing parts of a natural number and reciprocals of corresponding products.   },
    MSC2010 = {11A51 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Harmonic Numbers},
    SECTION3 = {On Exponents and Logarithms},
    SECTION4 = {Some Special Sequences},
    SECTION5 = {Square-free and Square-containing Parts of a Natural Number},
    SECTION6 = {Generating Bags from Subsets of Prime Numbers},
    SECTION7 = {On Reciprocals of Products of Prime Numbers},
    EXTERNALREFS = {BancerekJAR:2018; Euler:1737; GrabowskiDuplication; grabowski2007revisions; },
    INTERNALREFS = {INTEGRA5.ABS; INTEGRA6.ABS; BASEL_1.ABS; NAT_3.ABS; NAT_2.ABS; TAYLOR_2.ABS; BASEL_2.ABS; },
    KEYWORDS = {prime factorization; sequence of prime reciprocals; harmonic number; },
    SUBMITTED = {March 27, 2018}}


@ARTICLE{HILB10_2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 1}},
    PAGES = {81--90},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0007},
    VERSION = {8.1.07 5.47.1318},
    TITLE = {{D}iophantine sets. {P}reliminaries},
    ANNOTE = {This work has been financed by the resources of the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.},
    AUTHOR = {P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
    SUMMARY = {In this article, we define Diophantine sets using the Mizar formalism. We focus on selected properties of multivariate polynomials, i.e., functions of several variables to show finally that the class of Diophantine sets is closed with respect to the operations of union and intersection. \par This article is the next in a series \cite{PELLS_EQ.ABS}, \cite{HILB10_1.ABS} aiming to formalize the proof of Matiyasevich's negative solution of Hilbert's tenth problem.   },
    MSC2010 = {11D45 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Polynomial Extended by 0},
    SECTION3 = {Polynomial Permuted by Permutation},
    SECTION4 = {Main Lemmas},
    SECTION5 = {Diophantine Sets},
    EXTERNALREFS = {AdamowiczZbierski; FourDecades; LNT:Smorynski; },
    INTERNALREFS = {PELLS_EQ.ABS; FINSEQ_5.ABS; HILB10_1.ABS; POLYNOM1.ABS; AFINSQ_1.ABS; },
    KEYWORDS = {Hilbert's 10th problem; Pell's equation; multivariate polynomials; },
    SUBMITTED = {March 27, 2018}}


@ARTICLE{NEWTON05.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 2}},
    PAGES = {91--100},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0008},
    VERSION = {8.1.08 5.52.1328},
    TITLE = {{P}arity as a Property of Integers},
    AUTHOR = {Ziobro, Rafa{\l}},
    ADDRESS1 = {Department of Carbohydrate Technology\\University of Agriculture\\Krakow, Poland},
      ACKNOWLEDGEMENT = {Ad Maiorem Dei Gloriam},
    SUMMARY = {Even and odd numbers appear early in history of mathematics \cite{Zazkis1998}, as they serve to describe the property of objects easily noticeable by human eye \cite{sawyer2003vision}. Although the use of parity allowed to discover irrational numbers \cite{russo2013forgotten}, there is a common opinion that this property is ``not rich enough to become the main content focus of any particular research" \cite{Zazkis1998}. \par On the other hand, due to the use of decimal system, divisibility by 2 is often regarded as the property of the last digit of a number (similarly to divisibility by 5, but not to divisibility by any other primes), which probably restricts its use for any advanced purposes. \par The article aims to extend the definition of parity towards its notion in binary representation of integers, thus making an alternative to the articles grouped in \cite{BINARITH.ABS}, \cite{NUMERAL1.ABS}, and \cite{RADIX_1.ABS} branches, formalized in Mizar \cite{Mizar-State-2015}, \cite{BancerekJAR:2018}.   },
    MSC2010 = {11A51 03B35 68T99},
    EXTERNALREFS = {Zazkis1998; sawyer2003vision; russo2013forgotten; Mizar-State-2015; BancerekJAR:2018; },
    INTERNALREFS = {RADIX_1.ABS; NUMERAL1.ABS; BINARITH.ABS; INT_6.ABS; NEWTON02.ABS; },
    KEYWORDS = {divisibility; binary representation; },
    SUBMITTED = {June 29, 2018}}
    

@ARTICLE{GLIB_006.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 2}},
    PAGES = {101--124},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0009},
    VERSION = {8.1.08 5.52.1328},
    TITLE = {{A}bout Supergraphs. {P}art {I}},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
      NOTE1 = {mailto: {\tt skoch02@students.uni-mainz.de}},
      SUMMARY = {Drawing a finite graph is usually done by a finite sequence of the following three operations. \begin{enumerate} \item Draw a vertex of the graph. \item Draw an edge between two vertices of the graph. \item Draw an edge starting from a vertex of the graph and immediately draw a vertex at the other end of it. \end{enumerate} By this procedure any finite graph can be constructed. This property of graphs is so obvious that the author of this article has yet to find a reference where it is mentioned explicitly. In introductionary books (like \cite{SIMPLE}, \cite{MULTI}, \cite{German}) as well as in advanced ones (like \cite{SELECTED}), after the initial definition of graphs the examples are usually given by graphical representations rather than explicit set theoretic descriptions, assuming a mutual understanding how the representation is to be translated into a description fitting the definition. However, Mizar \cite{Mizar-State-2015}, \cite{BancerekJAR:2018} does not possess this innate ability of humans to translate pictures into graphs. Therefore, if one wants to create graphs in Mizar without directly providing a set theoretic description (i.e. using the {\texttt{createGraph}} functor), a rigorous approach to the constructing operations is required.\par In this paper supergraphs are defined as an inverse mode to subgraphs as given in \cite{GLIB_000.ABS}. The three graph construction operations are defined as modes extending {\texttt{Supergraph}} similar to how the various remove operations were introduced as submodes of {\texttt{Subgraph}} in \cite{GLIB_000.ABS}. Many theorems are proven that describe how graph properties are transferred to special supergraphs. In particular, to prove that disconnected graphs cannot become connected by adding an edge between two vertices that lie in the same component, the theory of replacing a part of a walk with another walk is introduced in the preliminaries.   },
    MSC2010 = {05C76 03B35 68T99},
    SECTION1 = {General Preliminaries},
    SECTION2 = {Graph Preliminaries},
    SECTION3 = {Supergraphs},
    EXTERNALREFS = {SIMPLE; BancerekJAR:2018; German; SELECTED; Mizar-State-2015; MULTI; },
    INTERNALREFS = {POLYFORM.ABS; GLIB_001.ABS; GLIB_002.ABS; GLIB_000.ABS; },
    KEYWORDS = {supergraph; graph operations; },
    SUBMITTED = {June 29, 2018}}
    

@ARTICLE{GLIB_007.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 2}},
    PAGES = {125--140},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0010},
    VERSION = {8.1.08 5.52.1328},
    TITLE = {{A}bout Supergraphs. {P}art {II}},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
      NOTE1 = {mailto: {\tt skoch02@students.uni-mainz.de}},
      SUMMARY = {In the previous article \cite{GLIB_006.ABS} supergraphs and several specializations to formalize the process of drawing graphs were introduced. In this paper another such operation is formalized in Mizar \cite{Mizar-State-2015}, \cite{BancerekJAR:2018}: drawing a vertex and then immediately drawing edges connecting this vertex with a subset of the other vertices of the graph. In case the new vertex is joined with all vertices of a given graph $G$, this is known as the join of $G$ and the trivial loopless graph $K_1$. While the join of two graphs is known and found in standard literature (like \cite{SIMPLE}, \cite{MULTI}, \cite{German} and \cite{SELECTED}), the operation discribed in this article is not.\par Alongside the new operation a mode to reverse the directions of a subset of the edges of a graph is introduced. When all edge directions of a graph are reversed, this is commonly known as the converse of a (directed) graph.   },
    MSC2010 = {05C76 03B35 68T99},
    SECTION1 = {Reversing Edge Directions},
    SECTION2 = {Adding a Vertex and Several Edges to a Graph},
    EXTERNALREFS = {Mizar-State-2015; BancerekJAR:2018; SIMPLE; MULTI; German; SELECTED; },
    INTERNALREFS = {GLIB_006.ABS; GLIB_001.ABS; ABIAN.ABS; },
    KEYWORDS = {supergraph; graph operations; },
    SUBMITTED = {June 29, 2018}}
    

@ARTICLE{PARTPR_2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 2}},
    PAGES = {141--147},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0011},
    VERSION = {8.1.08 5.53.1335},
    TITLE = {{O}n Algebras of Algorithms and Specifications over Uninterpreted Data},
    AUTHOR = {Ivanov, Ievgen and Korni{\l}owicz, Artur and Nikitchenko, Mykola},
    ADDRESS1 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS3 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      SUMMARY = {This paper continues formalization in Mizar \cite{FourDecades,BancerekJAR:2018} of basic notions of the composition-nominative approach to program semantics \cite{Nikitch98} which was started in \cite{NOMIN_1.ABS,PARTPR_1.ABS}.\par The composition-nominative approach studies mathematical models of computer programs and data on various levels of abstraction and generality and provides tools for reasoning about their properties. Besides formalization of semantics of programs, certain elements of the composition-nominative approach were applied to abstract systems in a~mathematical systems theory \cite{Ivanov2014,Ivanov2014a,DBLP:journals/fm/IvanovNA15,Ivanov2016,DBLP:journals/corr/Ivanov17}.\par In the paper we introduce a~definition of the notion of a~binominative function over a~set $D$ understood as a~partial function which maps elements of $D$ to $D$. The sets of binominative functions and nominative predicates \cite{PARTPR_1.ABS} over given sets form the carrier of the generalized Glushkov algorithmic algebra \cite{Skobelev2014}. This algebra can be used to formalize algorithms which operate on various data structures (such as multidimensional arrays, lists, etc.) and reason about their properties.\par We formalize the operations of this algebra (also called compositions) which are valid over uninterpretated data and which include among others the sequential composition, the prediction composition, the branching composition, the monotone Floyd-Hoare composition, and the cycle composition. The details on formalization of nominative data and the operations of the algorithmic algebra over them are described in \cite{DBLP:conf/fedcsis/KornilowiczKNI17,Kornilowicz2018,Moldova2018}.   },
    MSC2010 = {68Q60 68T37 03B70 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Operations in Algebras of Algorithms and Specifications over Uninterpreted Data},
    EXTERNALREFS = {BancerekJAR:2018; FourDecades; DBLP:journals/corr/Ivanov17; Ivanov2014; Ivanov2016; Ivanov2014a; 
                    DBLP:journals/fm/IvanovNA15; Moldova2018; DBLP:conf/fedcsis/KornilowiczKNI17; Kornilowicz2018; 
                    Nikitch98; Skobelev2014; },
    INTERNALREFS = {NOMIN_1.ABS; PARTPR_1.ABS; },
    KEYWORDS = {Glushkov algorithmic algebra; uninterpreted data; },
    SUBMITTED = {June 29, 2018}}
    

@ARTICLE{NOMIN_2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 2}},
    PAGES = {149--158},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0012},
    VERSION = {8.1.08 5.52.1328},
    TITLE = {{O}n an Algorithmic Algebra over Simple-Named Complex-Valued Nominative Data},
    AUTHOR = {Ivanov, Ievgen and Korni{\l}owicz, Artur and Nikitchenko, Mykola},
    ADDRESS1 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS3 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      SUMMARY = {This paper continues formalization in the Mizar system \cite{FourDecades,BancerekJAR:2018} of basic notions of the composition-nominative approach to program semantics \cite{Nikitch98} which was started in \cite{NOMIN_1.ABS,PARTPR_1.ABS,PARTPR_2.ABS}. \par The composition-nominative approach studies mathematical models of computer programs and data on various levels of abstraction and generality and provides tools for reasoning about their properties. In particular, data in computer systems are modeled as nominative data \cite{Skobelev2014}. Besides formalization of semantics of programs, certain elements of the composition-nominative approach were applied to abstract systems in a~mathematical systems theory \cite{Ivanov2014,Ivanov2014a,DBLP:journals/fm/IvanovNA15,Ivanov2016,DBLP:journals/corr/Ivanov17}. \par In the paper we give a~formal definition of the notions of a~binominative function over given sets of names and values (i.e. a~partial function which maps simple-named complex-valued nominative data to such data) and a~nominative predicate (a~partial predicate on simple-named complex-valued nominative data). The sets of such binominative functions and nominative predicates form the carrier of the generalized Glushkov algorithmic algebra for simple-named complex-valued nominative data \cite{Skobelev2014}. This algebra can be used to formalize algorithms which operate on various data structures (such as multidimensional arrays, lists, etc.) and reason about their properties. \par In particular, we formalize the operations of this algebra which require a~specification of a~data domain and which include the existential quantifier, the assignment composition, the composition of superposition into a~predicate, the composition of superposition into a~binominative function, the name checking predicate. The details on formalization of nominative data and the operations of the algorithmic algebra over them are described in \cite{DBLP:conf/fedcsis/KornilowiczKNI17,Kornilowicz2018,Moldova2018}.   },
    MSC2010 = {68Q60 68T37 03B70 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {On an Algorithmic algebra over Simple-Named Complex-Valued Nominative Data},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; Nikitch98; Skobelev2014; Ivanov2014; Ivanov2014a; 
                   DBLP:journals/fm/IvanovNA15; Ivanov2016; DBLP:journals/corr/Ivanov17; 
                   DBLP:conf/fedcsis/KornilowiczKNI17; Kornilowicz2018; Moldova2018; },
    INTERNALREFS = {NOMIN_1.ABS; PARTPR_2.ABS; PARTPR_1.ABS; },
    KEYWORDS = {Glushkov algorithmic algebra; nominative data; },
    SUBMITTED = {June 29, 2018}}
    

@ARTICLE{NOMIN_3.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 2}},
    PAGES = {159--164},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0013},
    VERSION = {8.1.08 5.53.1335},
    TITLE = {{A}n Inference System of an Extension of {F}loyd-{H}oare Logic for Partial Predicates},
    AUTHOR = {Ivanov, Ievgen and Korni{\l}owicz, Artur and Nikitchenko, Mykola},
    ADDRESS1 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS3 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      SUMMARY = {In the paper we give a~formalization in the Mizar system \cite{FourDecades,BancerekJAR:2018} of the rules of an~inference system for an extended Floyd-Hoare logic with partial pre- and post-conditions which was proposed in \cite{KornilowiczetalICTERI2017,Kryvolap2013}. The rules are formalized on the semantic level. The details of the approach used to implement this formalization are described in \cite{Moldova2018}. \par We formalize the notion of a~semantic Floyd-Hoare triple (for an extended Floyd-Hoare logic with partial pre- and post-conditions) \cite{Moldova2018} which is a~triple of a~pre-condition represented by a~partial predicate, a~program, represented by a~partial function which maps data to data, and a~post-condition, represented by a~partial predicate, which informally means that if the pre-condition on a~program's input data is defined and true, and the program's output after a~run on this data is defined (a~program terminates successfully), and the post-condition is defined on the program's output, then the post-condition is true. \par We formalize and prove the soundness of the rules of the inference system \cite{Kryvolap2013,KornilowiczetalICTERI2017} for such semantic Floyd-Hoare triples. For reasoning about sequential composition of programs and while loops we use the rules proposed in \cite{SeqRule2018}. \par The formalized rules can be used for reasoning about sequential programs, and in particular, for sequential programs on nominative data \cite{NOMIN_1.ABS}. Application of these rules often requires reasoning about partial predicates representing pre- and post-conditions which can be done using the formalized results on the Kleene algebra of partial predicates given in \cite{PARTPR_1.ABS}.   },
    MSC2010 = {68Q60 68T37 03B70 03B35},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; KornilowiczetalICTERI2017; Kryvolap2013; Moldova2018; SeqRule2018; },
    INTERNALREFS = {NOMIN_1.ABS; NOMIN_2.ABS; PARTPR_1.ABS; },
    KEYWORDS = {Floyd-Hoare logic; Floyd-Hoare triple; inference rule; program verification; },
    SUBMITTED = {June 29, 2018}}
    

@ARTICLE{NOMIN_4.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 2}},
    PAGES = {165--173},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0014},
    VERSION = {8.1.08 5.52.1328},
    TITLE = {{P}artial Correctness of {GCD} Algorithm},
    AUTHOR = {Ivanov, Ievgen and Korni{\l}owicz, Artur and Nikitchenko, Mykola},
    ADDRESS1 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS3 = {Taras Shevchenko National University\\Kyiv, Ukraine},
      SUMMARY = {In this paper we present a~formalization in the Mizar system \cite{FourDecades,BancerekJAR:2018} of the correctness of the subtraction-based version of Euclid's algorithm computing the greatest common divisor of natural numbers. The algorithm is written in terms of simple-named complex-valued nominative data \cite{Skobelev2014,NOMIN_1.ABS}.\par The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data \cite{NOMIN_3.ABS}. Proofs of the correctness are based on an~inference system for an~extended Floyd-Hoare logic with partial pre- and post-conditions \cite{KornilowiczetalICTERI2017,Kryvolap2013,Moldova2018,SeqRule2018}.   },
    MSC2010 = {68Q60 68T37 03B70 03B35},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; Skobelev2014; KornilowiczetalICTERI2017; Kryvolap2013; 
                    Moldova2018; SeqRule2018; },
    INTERNALREFS = {NOMIN_1.ABS; NOMIN_2.ABS; NOMIN_3.ABS; PARTPR_1.ABS; },
    KEYWORDS = {greatest common divisor; nominative data; program verification; },
    SUBMITTED = {June 29, 2018}}
 
 
 @ARTICLE{HILB10_3.ABS,
     JOURNAL = {Formalized Mathematics},
 	ISSN = {1426-2630},
 	EISSN = {1898-9934},
       VOLUME = {26},
     NUMBER = {{\bf 2}},
     PAGES = {175--181},
     YEAR = {2018},
     DOI = {10.2478/forma-2018-0015},
     VERSION = {8.1.08 5.52.1328},
     TITLE = {{B}asic {D}iophantine Relations},
     ANNOTE = {This work has been financed by the resources of the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.},
     AUTHOR = {Acewicz, Marcin and P\k{a}k, Karol},
     ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
       ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
       SUMMARY = {The main purpose of formalization is to prove that two equations $\texttt{y}_a(z)=y$, $y=x^z$ are Diophantine. These equations are explored in the proof of Matiyasevich's negative solution of Hilbert's tenth problem. \par In our previous work \cite{HILB10_1.ABS}, we showed that from the Diophantine standpoint these equations can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities. In this formalization, we express these relations in terms of Diophantine set introduced in \cite{HILB10_2.ABS}. We prove that these relations are Diophantine and then we prove several second-order theorems that provide the ability to combine Diophantine relation using conjunctions and alternatives as well as to substitute the right-hand side of a given Diophantine equality as an argument in a given Diophantine relation. Finally, we investigate the possibilities of our approach to prove that the two equations, being the main purpose of this formalization, are Diophantine. \par The formalization by means of Mizar system \cite{BancerekJAR:2018}, \cite{Mizar-State-2015} follows Z.~Adamowicz, P.~Zbierski \cite{AdamowiczZbierski} as well as M.~Davis~\cite{Hilbert10}.   },
     MSC2010 = {11D45 03B35 68T99},
     SECTION1 = {Preliminaries},
     SECTION2 = {Basic Diophantine Relations},
     SECTION3 = {Main Lemmas},
     EXTERNALREFS = {AdamowiczZbierski; Hilbert10; Mizar-State-2015; BancerekJAR:2018; },
     INTERNALREFS = {AFINSQ_1.ABS; AFINSQ_2.ABS; HILB10_1.ABS; HILB10_2.ABS; },
     KEYWORDS = {Hilbert's 10th problem; Diophantine relations; },
     SUBMITTED = {June 29, 2018}}
  
  
@ARTICLE{ROUGHS_5.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 2}},
    PAGES = {183--191},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0016},
    VERSION = {8.1.08 5.52.1328},
    TITLE = {{F}ormalizing Two Generalized Approximation Operators},
    AUTHOR = {Grabowski, Adam and Sielwiesiuk, Micha{\l}},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {Rough sets, developed by Pawlak \cite{Pawlak1982}, are important tool to describe situation of incomplete or partially unknown information. In this article we give the formal characterization of two closely related rough approximations, along the lines proposed in a paper by Gomoli{\'n}ska \cite{Gomolinska2002}. We continue the formalization of rough sets in Mizar \cite{Mizar-State-2015} started in \cite{ROUGHS_1.ABS}.   },
    MSC2010 = {03E99 03B35 68T99},
    SECTION1 = {Preliminaries: Map-Reflexivity},
    SECTION2 = {Properties of Flipping Operator $f^d$},
    SECTION3 = {Uncertainty Mappings $I$ and $\tau$},
    SECTION4 = {Generalized Approximation Mappings},
    SECTION5 = {The Ordering of Approximation Mappings},
    SECTION6 = {Acting on the Empty Set and the Universe},
    SECTION7 = {Standard Properties of Approximations},
    SECTION8 = {Monotonicity of Approximations},
    SECTION9 = {Distributivity wrt. Set-Theoretic Operations},
    EXTERNALREFS = {Mizar-State-2015; Gomolinska2002; GrabowskiFI:2013; GrabowskiLTRS; GrabowskiJ10; 
                    GrabowskiPerspective:2007; GrabowskiDuplication; SchwarzRCA:2004; Jarvinen:2007; 
                    Pawlak1982; yao96; Zhu:2007; },
    INTERNALREFS = {LATTAD_1.ABS; ROUGHS_1.ABS; ROUGHS_2.ABS; ROUGHS_3.ABS; PREFER_1.ABS; },
    KEYWORDS = {rough approximation; rough set; generalized approximation operator ; },
    SUBMITTED = {June 29, 2018}}


@ARTICLE{ROBBINS5.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 2}},
    PAGES = {193--198},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0017},
    VERSION = {8.1.08 5.52.1328},
    TITLE = {{O}n Two Alternative Axiomatizations of Lattices by {M}c{K}enzie and {S}holander},
    AUTHOR = {Grabowski, Adam and Sawicki, Damian},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {The main result of the article is to prove formally that two sets of axioms, proposed by McKenzie and Sholander, axiomatize lattices and distributive lattices, respectively. In our Mizar article we used proof objects generated by Prover9. We continue the work started in \cite{ROBBINS1.ABS}, \cite{ROBBINS2.ABS}, and \cite{SHEFFER1.ABS} of developing lattice theory as initialized in \cite{LATTICES.ABS} as a formal counterpart of \cite{Gratzer}. Complete formal proofs can be found in the Mizar source code of this article available in the Mizar Mathematical Library (MML).   },
    MSC2010 = {03B35 68T99 06B05 06D05},
    SECTION1 = {Sholander Axiom for Distributive Lattices},
    SECTION2 = {Four Axioms for Lattices Proposed by McKenzie},
    EXTERNALREFS = {BancerekJAR:2018; BIRKHOFF:1; DahnRob; Davey:2002; GrabowskiJAR40; GrabowskiLTRS; GrabMo2004; 
                    FourDecades; EqualityFedCSIS; Gratzer; Gratzer2011; McCune:2005; prover9-mace4; Padma1996; 
                    McKenzie1970; Padma2008; rudnicki2011escape; Sholander1951; },
    INTERNALREFS = {ROBBINS1.ABS; SHEFFER1.ABS; ROBBINS2.ABS; LATTICES.ABS; },
    KEYWORDS = {lattice; distributive lattice; lattice axioms; },
    SUBMITTED = {June 29, 2018}}


@ARTICLE{FINSEQ_9.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 3}},
    PAGES = {199--208},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0018},
    VERSION = {8.1.08 5.53.1335},
    TITLE = {{A}rithmetic Operations on Short Finite Sequences},
    AUTHOR = {Ziobro, Rafa{\l}},
    ADDRESS1 = {Department of Carbohydrate Technology\\University of Agriculture\\Krakow, Poland},
      ACKNOWLEDGEMENT = {Ad Maiorem Dei Gloriam},
    SUMMARY = {In contrast to other proving systems Mizar Mathematical Library, considered as one of the largest formal mathematical libraries \cite{Elizarov2017}, is maintained as a single base of theorems, which allows the users to benefit from earlier formalized items \cite{BancerekJAR:2018}, \cite{Mizar-State-2015}. This eventually leads to a development of certain branches of articles using common notation and ideas. Such formalism for finite sequences has been developed since 1989 \cite{FINSEQ_1.ABS} and further developed despite of the controversy over indexing which excludes zero \cite{Rudnicki2003}, also for some advanced and new mathematics \cite{Naumowicz2009}. \par The article aims to add some new machinery for dealing with finite sequences, especially those of short length.   },
    MSC2010 = {11B99 03B35 68T99},
    SECTION1 = {Preliminaries},
    SECTION2 = {The Length of Finite Sequences},
    SECTION3 = {On Positive and Negative Yielding Functions},
    SECTION4 = {Basic Operations on Short Finsequences},
    EXTERNALREFS = {Elizarov2017; BancerekJAR:2018; Mizar-State-2015; Rudnicki2003; Naumowicz2009; },
    INTERNALREFS = {FINSEQ_1.ABS; },
    KEYWORDS = {finite sequences; functions; relations; },
    SUBMITTED = {September 29, 2018}}


@ARTICLE{TOPS_5.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 3}},
    PAGES = {209--222},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0019},
    VERSION = {8.1.08 5.53.1335},
    TITLE = {{S}ome Remarks about Product Spaces},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
      NOTE1 = {The author is enrolled in the Johannes Gutenberg University in Mayence, Germany, mailto: {\tt skoch02@students.uni-mainz.de}},
      SUMMARY = {This article covers some technical aspects about the product topology which are usually not given much of a thought in mathematics and standard literature like \cite{MUNKRES} and \cite{kelley}, not even by Bourbaki in \cite{BOURBAKI}. \par Let $\{\mathcal T_i\}_{i\in I}$ be a family of topological spaces. The prebasis of the product space $\mathcal T=\prod_{i\in I} \mathcal T_i$ is defined in \cite{WAYBEL18.ABS} as the set of all $\pi^{-1}_i(V)$ with $i\in I$ and $V$ open in $\mathcal T_i$. Here it is shown that the basis generated by this prebasis consists exactly of the sets $\prod_{i\in I} V_i$ with $V_i$ open in $ \mathcal T_i$ and for all but finitely many $i\in I$ holds $V_i=\mathcal T_i$. Given $I=\{a\}$ we have $ \mathcal T\cong \mathcal T_a$, given $I=\{a,b\}$ with $a\neq b$ we have $\mathcal T \cong \mathcal T_a \times\mathcal T_b$. Given another family of topological spaces $\{\mathcal S_i\}_{i\in I}$ such that $\mathcal S_i\cong\mathcal T_i$ for all $i\in I$, we have $\mathcal S=\prod_{i\in I} \mathcal S_i\cong \mathcal T$. If instead $S_i$ is a subspace of $T_i$ for each $i\in I$, then $\mathcal S$ is a subspace of $\mathcal T$. \par These results are obvious for mathematicians, but formally proven here by means of the Mizar system \cite{BancerekJAR:2018}, \cite{Mizar-State-2015}.   },
    MSC2010 = {54B10 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Remarks about Product Spaces},
    EXTERNALREFS = {MUNKRES; kelley; BOURBAKI; BancerekJAR:2018; Mizar-State-2015; },
    INTERNALREFS = {YELLOW17; FUNCT_7; PENCIL_3; WAYBEL18.ABS; },
    KEYWORDS = {topology; product spaces; },
    SUBMITTED = {September 29, 2018}}


@ARTICLE{BINARI_6.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 3}},
    PAGES = {223--229},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0020},
    VERSION = {8.1.08 5.53.1335},
    TITLE = {{B}inary Representation of Natural Numbers},
    AUTHOR = {Okazaki, Hiroyuki},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
      NOTE1 = {This study was supported in part by JSPS KAKENHI Grant Numbers JP17K00182. The author would also like to express gratitude to Prof. Yasunari Shidama for his support and encouragement.},
      SUMMARY = {Binary representation of integers \cite{Leibniz1879}, \cite{Knuth1997} and arithmetic operations on them have already been introduced in Mizar Mathematical Library \cite{BINARITH.ABS,BINARI_2.ABS,BINARI_3.ABS,BINARI_4.ABS}. However, these articles formalize the notion of integers as mapped into a certain length tuple of boolean values. \par In this article we formalize, by means of Mizar system \cite{BancerekJAR:2018}, \cite{Mizar-State-2015}, the binary representation of natural numbers which maps ${\mathbb N}$ into bitstreams.   },
    MSC2010 = {68W01 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Homomorphism from the Natural Numbers to the Bitstreams},
    SECTION3 = {Homomorphism from the Bitstreams to the Natural Numbers},
    EXTERNALREFS = {Leibniz1879; Knuth1997; BancerekJAR:2018; Mizar-State-2015; },
    INTERNALREFS = {BINARITH.ABS; BINARI_2.ABS; BINARI_3.ABS; BINARI_4.ABS; },
    KEYWORDS = {algorithms; },
    SUBMITTED = {September 29, 2018}}


@ARTICLE{LOPBAN_8.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 3}},
    PAGES = {231--237},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0021},
    VERSION = {8.1.08 5.53.1335},
    TITLE = {{C}ontinuity of Bounded Linear Operators on Normed Linear Spaces},
    ANNOTE = {This study was supported in part by JSPS KAKENHI Grant Number JP17K00182.},
    AUTHOR = {Nakasho, Kazuhisa and Futa, Yuichi and Shidama, Yasunari},
    ADDRESS1 = {Yamaguchi University\\Yamaguchi, Japan},
      ADDRESS2 = {Tokyo University of Technology\\Tokyo, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, using the Mizar system \cite{Mizar-State-2015}, \cite{BancerekJAR:2018}, we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized. \par In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to \cite{miyadera:1972}, \cite{yoshida:1980}, and \cite{Dunford:1958} in this formalization.   },
    MSC2010 = {46-00 47A07 47A30 68T99 03B35},
    SECTION1 = {Uniform Continuity and Lipschitz Continuity of Bounded Linear Operators},
    SECTION2 = {Basic Properties of Currying},
    SECTION3 = {Equivalence of Some Definitions of Continuity of Bilinear Operators},
    EXTERNALREFS = {Mizar-State-2015; BancerekJAR:2018; miyadera:1972; yoshida:1980; Dunford:1958; },
    INTERNALREFS = {NORMSP_3.ABS; NFCONT_1.ABS; NFCONT_2.ABS; NDIFF_8; PRVECT_3.ABS; INTEGR20; LOPBAN_1.ABS; LOPBAN_3.ABS; },
    KEYWORDS = {Lipschitz continuity; uniform continuity; bounded linear operators; bilinear operators; },
    SUBMITTED = {September 29, 2018}}
                       

@ARTICLE{MUSIC_S1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 4}},
    PAGES = {239--269},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0022},
    VERSION = {8.1.08 5.53.1335},
    TITLE = {{P}ythagorean Tuning: Pentatonic and Heptatonic Scale},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      ACKNOWLEDGEMENT = {I would like to thank H\'el\`ene Cambier (Professor of music history at the Music Academy La Louvi\`ere) and Marie Barbier (musical instrument: Recorder) for valuable suggestions.},
    SUMMARY = {In this article, using the Mizar system \cite{Mizar-State-2015}, \cite{BancerekJAR:2018}, we define a structure \cite{Bancerek:2003}, \cite{GrabKornSchwarz:2016} in order to build a Pythagorean pentatonic scale and a Pythagorean heptatonic scale\footnote{\url{https://en.wikipedia.org/wiki/Pythagorean_tuning}} ~ \cite{baskevitch2008}, \cite{parzysz1984musique}.   },
    MSC2010 = {00A65 97M80 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Real Frequency},
    SECTION3 = {Rational Frequency},
    SECTION4 = {Musical Structure and Some Axioms},
    SECTION5 = {Harmonic},
    SECTION6 = {Spiral of Fifths},
    SECTION7 = {Pentatonic Pythagorean Scale},
    SECTION8 = {Heptatonic Pythagorean Scale},
    EXTERNALREFS = {Mizar-State-2015; BancerekJAR:2018; Bancerek:2003; GrabKornSchwarz:2016; baskevitch2008; parzysz1984musique; },
    INTERNALREFS = {NAT_1.ABS; },
    KEYWORDS = {music; Pythagorean tuning; pentatonic scale; heptatonic scale; },
    SUBMITTED = {September 29, 2018}}


@ARTICLE{FUZIMPL2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 4}},
    PAGES = {271--276},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0023},
    VERSION = {8.1.08 5.53.1335},
    TITLE = {{F}undamental Properties of Fuzzy Implications},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {In the article we continue in the Mizar system \cite{FourDecades}, \cite{BancerekJAR:2018} the formalization of fuzzy implications according to the monograph of Baczy\'nski and Jayaram \emph{``Fuzzy Implications"} \cite{Baczynski:2008}. We develop a framework of Mizar attributes allowing us for a smooth proving of basic properties of these fuzzy connectives \cite{Hajek:1998}. We also give a set of theorems about the ordering of nine fundamental implications: {\L}ukasiewicz ($I_{\rm LK}$), G\"odel ($I_{\rm GD}$), Reichenbach ($I_{\rm RC}$), Kleene-Dienes ($I_{\rm KD}$), Goguen ($I_{\rm GG}$), Rescher ($I_{\rm RS}$), Yager ($I_{\rm YG}$), Weber ($I_{\rm WB}$), and Fodor ($I_{\rm FD}$). \par This work is a continuation of the development of fuzzy sets in Mizar \cite{GrabowskiFuzzy:2013}; it could be used to give a variety of more general operations on fuzzy sets \cite{Zadeh:1965}. The formalization follows \cite{FUZZY_1.ABS},~\cite{FUZNORM1.ABS}, and \cite{FUZIMPL1.ABS}.   },
    MSC2010 = {03B52 68T37 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {The Ordering of Fuzzy Implications},
    SECTION3 = {Additional Properties of Fuzzy Implications},
    SECTION4 = {Dependencies between Chosen Properties},
    SECTION5 = {Properties of Nine Classical Fuzzy Implications},
    EXTERNALREFS = {Mizar-State-2015; BancerekJAR:2018; Baczynski:2008; GrabowskiFuzzy:2013; Zadeh:1965; },
    INTERNALREFS = {FUZZY_1.ABS; FUZNORM1.ABS; FUZIMPL1.ABS; TAYLOR_1.ABS; },
    KEYWORDS = {fuzzy implication; fuzzy set; fuzzy logic; },
    SUBMITTED = {September 29, 2018}}


@ARTICLE{TOPZARI1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {26},
    NUMBER = {{\bf 4}},
    PAGES = {277--283},
    YEAR = {2018},
    DOI = {10.2478/forma-2018-0024},
    VERSION = {8.1.08 5.53.1335},
    TITLE = {{Z}ariski Topology},
    AUTHOR = {Watase, Yasushige},
    ADDRESS1 = {Suginami-ku Matsunoki\\3-21-6 Tokyo, Japan},
      SUMMARY = {We formalize in the Mizar system \cite{Mizar-State-2015}, \cite{BancerekJAR:2018} basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring \cite{IITAKA1982}, \cite{IITAKA2013}, then formalize proofs of some related theorems along with the first chapter of \cite{atiyah1969introduction}. \par The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring $A$ is called the prime spectrum of $A$ denoted by $\mathrm{Spectrum} \ A$. A new functor $\mathrm{Spec}$ generates Zariski topology to make $\mathrm{Spectrum} \ A$ a topological space. A different role is given to $\mathrm{Spec}$ as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism $h:A \longrightarrow B$, we defined $(\mathrm{Spec} \ h) : \mathrm{Spec} \ B \longrightarrow \mathrm{Spec} \ A$ by $(\mathrm{Spec} \ h)(\mathfrak{p}) = h^{-1}(\mathfrak{p})$ where $\mathfrak{p} \in \mathrm{Spec} \ B$.   },
    MSC2010 = {14A05 16D25 68T99 03B35},
    SECTION1 = {Preliminaries: Some Properties of Ideals},
    SECTION2 = {Spectrum of Prime Ideals (Spectrum) and Maximal Ideals (m-Spectrum)},
    SECTION3 = {Local and Semi-Local Ring},
    SECTION4 = {Nilradical and Jacobson Radical},
    SECTION5 = {Construction of Zariski Topology of the Prime Spectrum of $A$},
    SECTION6 = {Continous Map of Zariski Topology Associated with a Ring Homomorphism},
    EXTERNALREFS = {Mizar-State-2015; BancerekJAR:2018; IITAKA1982; IITAKA2013; atiyah1969introduction; },
    INTERNALREFS = {BINOM.ABS; IDEAL_1.ABS; },
    KEYWORDS = {prime spectrum; local ring; semi-local ring; nilradical; Jacobson radical; Zariski topology; },
    SUBMITTED = {October 16, 2018}}


@ARTICLE{RVSUM_4.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 1}},
    PAGES = {1--13},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0001},    
    VERSION = {8.1.09 5.54.1341},
    TITLE = {{C}oncatenation of Finite Sequences},
    AUTHOR = {Ziobro, Rafa{\l}},
    ADDRESS1 = {Department of Carbohydrate Technology\\University of Agriculture\\Krakow, Poland},
      ACKNOWLEDGEMENT = {Ad Maiorem Dei Gloriam},
    SUMMARY = {The coexistence of ``classical" {\it finite sequences} \cite{FINSEQ_1.ABS} and their zero-based equivalents {\it finite 0-sequences} \cite{AFINSQ_1.ABS} in Mizar has been regarded as a disadvantage. However the suggested replacement of the former type with the latter \cite{Rudnicki2003} has not yet been implemented, despite of several advantages of this form, such as the identity of length and domain operators \cite{kornilowicz2009define}. On the other hand the number of theorems formalized using {\it finite sequence} notation is much larger then of those based on {\it finite 0-sequences}, so such translation would require quite an effort. \par The paper addresses this problem with another solution, using the Mizar system \cite{FourDecades}, \cite{BancerekJAR:2018}. Instead of removing one notation it is possible to introduce operators which would concatenate sequences of various types, and in this way allow utilization of the whole range of formalized theorems. While the operation could replace existing {\tt FS2XFS}, {\tt XFS2FS} commands (by using empty sequences as initial elements) its universal notation (independent on sequences that are concatenated to the initial object) allows to ``forget" about the type of sequences that are concatenated on further positions, and thus simplify the proofs.   },
    MSC2010 = {11B99 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Complex-Valued Sequences},
    SECTION3 = {On Product and Sum of Complex Sequences},
    SECTION4 = {Finite 0-sequences},
    SECTION5 = {Shifting Sequences},
    SECTION6 = {Converting Complex 0-sequences into Ordinary Ones},
    SECTION7 = {Properties of Concatenation},
    SECTION8 = {Sum of Finite 0-sequences},
    SECTION9 = {Product of Finite 0-sequences},
    EXTERNALREFS = {Rudnicki2003; kornilowicz2009define; FourDecades; BancerekJAR:2018; },
    INTERNALREFS = {AFINSQ_1.ABS; FINSEQ_1.ABS; NEWTON02.ABS; NEWTON04.ABS; },
    KEYWORDS = {finite sequence; finite 0-sequence; concatenation; },
    SUBMITTED = {February 27, 2019}}


@ARTICLE{LOPBAN_9.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 1}},
    PAGES = {15--23},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0002},
    VERSION = {8.1.09 5.54.1341},
    TITLE = {{B}ilinear Operators on Normed Linear Spaces},
    AUTHOR = {Nakasho, Kazuhisa},
    ADDRESS1 = {Yamaguchi University\\Yamaguchi, Japan},
      ACKNOWLEDGEMENT = {I would like to express my gratitude to Professor Yasunari Shidama for his helpful advice.},
    SUMMARY = {The main aim of this article is proving properties of bilinear operators on normed linear spaces formalized by means of Mizar \cite{BancerekJAR:2018}. In the first two chapters, algebraic structures \cite{GrabKornSchwarz:2016} of bilinear operators on linear spaces are discussed. Especially, the space of bounded bilinear operators on normed linear spaces is developed here. In the third chapter, it is remarked that the algebraic structure of bounded bilinear operators to a certain Banach space also constitutes a Banach space. \par In the last chapter, the correspondence between the space of bilinear operators and the space of composition of linear opearators is shown. We referred to \cite{miyadera:1972}, \cite{yoshida:1980}, \cite{Dunford:1958}, \cite{Schwartz1997a} and \cite{Schwartz1997b} in this formalization.   },
    MSC2010 = {46-00 47A07 47A30 68T99 03B35},
    SECTION1 = {Real Vector Space of Bilinear Operators},
    SECTION2 = {Real Normed Linear Space of Bounded Bilinear Operators},
    SECTION3 = {Real Banach Space of Bounded Bilinear Operators},
    SECTION4 = {Isomorphisms between the Space of Bilinear Operators and the Space of Composition of Linear Operators},
    EXTERNALREFS = {BancerekJAR:2018; GrabKornSchwarz:2016; miyadera:1972; yoshida:1980; Dunford:1958; Schwartz1997a; 
                    Schwartz1997b; },
    INTERNALREFS = {PRVECT_3.ABS; LOPBAN_1.ABS; RSSPACE3.ABS; LOPBAN_8.ABS; },
    KEYWORDS = {Lipschitz continuity; bounded linear operator; bilinear operator; algebraic structure; Banach space; },
    SUBMITTED = {February 27, 2019}}
 
 
@ARTICLE{PDIFFEQ1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 1}},
    PAGES = {25--34},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0003},
    VERSION = {8.1.09 5.54.1341},
    TITLE = {{A} Simple Example for Linear Partial Differential Equations and Its Solution Using the Method of Separation of Variables},
    AUTHOR = {Otsuki, Sora and Kawamoto, Pauline N. and Yamazaki, Hiroshi},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
      ADDRESS2 = {Shinshu University\\Nagano, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      ACKNOWLEDGEMENT = {We would like to thank Yasunari Shidama for useful advice on formalizing theorems.},
    SUMMARY = {In this article, we formalized in Mizar \cite{FourDecades}, \cite{BancerekJAR:2018} simple partial differential equations. In the first section, we formalized partial differentiability and partial derivative. The next section contains the method of separation of variables for one-dimensional wave equation. In the last section, we formalized the superposition principle. We referred to \cite{Sneddon1957}, \cite{Fritz1990}, \cite{Nakao1992} and \cite{Yano1982} in this formalization.   },
    MSC2010 = {35A08 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {The Method of Separation of Variables for One-dimensional Wave Equation},
    SECTION3 = {The Superposition Principle},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; Sneddon1957; Fritz1990; Nakao1992; Yano1982; },
    INTERNALREFS = {FINSEQ_7.ABS; SIN_COS.ABS; PDIFF_9.ABS; },
    KEYWORDS = {partial differential equations; separation of variables; superposition principle; },
    SUBMITTED = {February 27, 2019}}


@ARTICLE{LOPBAN10.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 1}},
    PAGES = {35--45},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0004},
    VERSION = {8.1.09 5.54.1341},
    TITLE = {{M}ultilinear Operator and Its Basic Properties},
    AUTHOR = {Nakasho, Kazuhisa},
    ADDRESS1 = {Yamaguchi University\\Yamaguchi, Japan},
      ACKNOWLEDGEMENT = {I would like to express my gratitude to Professor Yasunari Shidama for his helpful advice.},
    SUMMARY = {In the first chapter, the notion of multilinear operator on real linear spaces is discussed. The algebraic structure \cite{GrabKornSchwarz:2016} of multilinear operators is introduced here. In the second chapter, the results of the first chapter are extended to the case of the normed spaces. This chapter shows that bounded multilinear operators on normed linear spaces constitute the algebraic structure. We referred to \cite{miyadera:1972}, \cite{yoshida:1980}, \cite{Schwartz1997a}, \cite{Schwartz1997b} in this formalization.   },
    MSC2010 = {46-00 47A07 47A30 68T99 03B35},
    SECTION1 = {Multilinear Operator on Real Linear Spaces},
    SECTION2 = {Bounded Multilinear Operator on Normed Linear Spaces},
    EXTERNALREFS = {GrabKornSchwarz:2016; miyadera:1972; yoshida:1980; Schwartz1997a; Schwartz1997b; },
    INTERNALREFS = {RVSUM_1.ABS; NAT_4.ABS; },
    KEYWORDS = {Lipschitz continuity; bounded linear operators; bilinear operators; algebraic structure; Banach space; },
    SUBMITTED = {February 27, 2019}}


@ARTICLE{ANPROJ10.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 1}},
    PAGES = {47--60},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0005},
    VERSION = {8.1.09 5.54.1341},
    TITLE = {{C}ross-Ratio in Real Vector Space},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {Using Mizar \cite{Mizar-State-2015}, in the context of a real vector space, we introduce the concept of affine ratio of three aligned points (see \cite{bsl0}). \par It is also equivalent to the notion of ``Mesure alg{\' e}brique"\footnote{\tt{https://fr.wikipedia.org/wiki/Mesure\_alg{\' e}brique}}, to the opposite of the notion of Teilverh\"{a}ltnis\footnote{\tt{https://de.wikipedia.org/wiki/Teilverh{\" a}ltnis}} or to the opposite of the ordered length-ratio \cite{richter2011perspectives}. \par In the second part, we introduce the classic notion of ``cross-ratio" of 4 points aligned in a real vector space. \par Finally, we show that if the real vector space is the real line, the notion corresponds to the classical notion\footnote{\url{https://en.wikipedia.org/wiki/Cross-ratio}} \cite{richter2011perspectives}: \begin{quote} The cross-ratio of a quadruple of distinct points on the real line with coordinates $x_1, x_2,x_3,x_4$ is given by: $$(x_1,x_2;x_3,x_4) = \frac{x_3 - x_1}{x_3 - x_2}. \frac{x_4 - x_2}{x_4 - x_1}$$ \end{quote} \par In the Mizar Mathematical Library, the vector spaces were first defined by Kusak, Leo\'{n}czuk and Muzalewski in the article \cite{VECTSP_1.ABS}, while the actual real vector space was defined by Trybulec \cite{RLVECT_1.ABS} and the complex vector space was defined by Endou \cite{CLVECT_1.ABS}. Nakasho and Shidama have developed a solution to explore the notions introduced by different authors\footnote{\url{http://webmizar.cs.shinshu-u.ac.jp/mmlfe/current}} \cite{nakasho2015documentation}. The definitions can be directly linked in the HTMLized version of the Mizar library\footnote{\scriptsize Example: {\tt RealLinearSpace} \tt{http://mizar.org/version/current/html/rlvect\_1.html\#NM2}}. \par The study of the cross-ratio will continue within the framework of the Klein-Beltrami model \cite{BKMODEL1.ABS}, \cite{BKMODEL2.ABS}. For a generalized cross-ratio, see Papadopoulos \cite{papadopoulos2015projective}.   },
    MSC2010 = {15A03 51A05 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Affine Ratio},
    SECTION3 = {Cross-Ratio},
    SECTION4 = {Cross-Ratio and the Real Line},
    EXTERNALREFS = {Mizar-State-2015; bsl0; richter2011perspectives; richter2011perspectives; 
                    nakasho2015documentation; papadopoulos2015projective; },
    INTERNALREFS = {VECTSP_1.ABS; RLVECT_1.ABS; CLVECT_1.ABS; BKMODEL1.ABS; BKMODEL2.ABS; },
    KEYWORDS = {affine ratio; cross-ratio; real vector space; geometry; },
    SUBMITTED = {February 27, 2019}}


@ARTICLE{LOPBAN11.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 1}},
    PAGES = {61--65},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0006},
    VERSION = {8.1.09 5.54.1341},
    TITLE = {{C}ontinuity of Multilinear Operator on Normed Linear Spaces},
    ANNOTE = {This study was supported in part by JSPS KAKENHI Grant Number JP17K00182.},
    AUTHOR = {Nakasho, Kazuhisa and Shidama, Yasunari},
    ADDRESS1 = {Yamaguchi University\\Yamaguchi, Japan},
      ADDRESS2 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, various definitions of contuity of multilinear operators on normed linear spaces are discussed in the Mizar formalism \cite{FourDecades}, \cite{Mizar-State-2015} and \cite{BancerekJAR:2018}. In the first chapter, several basic theorems are prepared to handle the norm of the multilinear operator, and then it is formalized that the linear space of bounded multilinear operators is a complete Banach space. \par In the last chapter, the continuity of the multilinear operator on finite normed spaces is addressed. Especially, it is formalized that the continuity at the origin can be extended to the continuity at every point in its whole domain. We referred to \cite{miyadera:1972}, \cite{yoshida:1980}, \cite{Schwartz1997a}, \cite{Schwartz1997b} in this formalization.   },
    MSC2010 = {46-00 47A07 47A30 68T99 03B35},
    SECTION1 = {Completeness of the Space of Multilinear Operators},
    SECTION2 = {Equivalence of Continuity Definitions of Multilinear Operators},
    EXTERNALREFS = {FourDecades; Mizar-State-2015; BancerekJAR:2018; miyadera:1972; yoshida:1980; Schwartz1997a; Schwartz1997b; },
    INTERNALREFS = {LOPBAN_1.ABS; NDIFF_8.ABS; NFCONT_1.ABS; PRVECT_2.ABS; },
    KEYWORDS = {Lipschitz continuity; bounded linear operators; multilinear operators; Banach space; },
    SUBMITTED = {February 27, 2019}}


@ARTICLE{MESFUN13.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 1}},
    PAGES = {67--74},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0007},
    VERSION = {8.1.09 5.54.1344},
    TITLE = {{F}ubini's Theorem},
    AUTHOR = {Endou, Noboru},
    ADDRESS1 = {National Institute of Technology, Gifu College\\2236-2 Kamimakuwa, Motosu, Gifu, Japan},
      SUMMARY = {Fubini theorem is an essential tool for the analysis of high-dimensional space \cite{Halmos74}, \cite{Bauer:2002}, \cite{Bogachev2007measure}, a theorem about the multiple integral and iterated integral. The author has been working on formalizing Fubini's theorem over the past few years \cite{MEASUR11.ABS}, \cite{MESFUN12.ABS} in the Mizar system \cite{FourDecades}, \cite{BancerekJAR:2018}. As a result, Fubini's theorem (\hyperlink{th:30}{30}) was proved in complete form by this article.   },
    MSC2010 = {28A35 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Markov's Inequality},
    SECTION3 = {Fubini's Theorem},
    EXTERNALREFS = {Halmos74; Bauer:2002; Bogachev2007measure; FourDecades; BancerekJAR:2018; },
    INTERNALREFS = {MEASUR11.ABS; MESFUN12.ABS; MESFUN11.ABS; },
    KEYWORDS = {Fubini's theorem; product measure; multiple integral; iterated integral; },
    SUBMITTED = {March 11, 2019}}


@ARTICLE{GTARSKI4.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 1}},
    PAGES = {75--85},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0008},
    VERSION = {8.1.09 5.54.1344},
    TITLE = {{T}arski Geometry Axioms. {P}art {IV} -- Right Angle},
    AUTHOR = {Coghetto, Roland and Grabowski, Adam},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {In the article, we continue \cite{GTARSKI3.ABS} the formalization of the work devoted to Tarski's geometry -- the book ``Metamathematische Methoden in der Geometrie" (SST for short) by W. Schwabh\"auser, W. Szmielew, and A. Tarski \cite{Schwabhauser:1983}, \cite{Grabowski:FedCSIS2016}, \cite{GrabowskiCoghetto:2016}. We use the Mizar system to systematically formalize Chapter~8 of the SST book.\par We define the notion of right angle and prove some of its basic properties, a theory of intersecting lines (including orthogonality). Using the notion of perpendicular foot, we prove the existence of the midpoint (Satz 8.22), which will be used in the form of the Mizar functor (as the uniqueness can be easily shown) in Chapter~10. In the last section we give some lemmas proven by means of Otter during \emph{Tarski Formalization Project} by M. Beeson (the so-called Section 8A of SST).   },
    MSC2010 = {51A05 51M04 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Right Angle},
    SECTION3 = {Orthogonality},
    SECTION4 = {Intersection of Lines},
    SECTION5 = {Perpendicular Foot},
    SECTION6 = {Additional Lemmas Needed by Otter: Chapter 8A},
    EXTERNALREFS = {Schwabhauser:1983; Grabowski:FedCSIS2016; GrabowskiCoghetto:2016; Mizar-State-2015; BancerekJAR:2018;
    Gupta:1965; 10.1007/978-3-540-77356-6_9; boutry:hal-01483457; boutry2019parallel; Beeson2019; dhurdjevic2015automated;
    beeson2014otter; },
    INTERNALREFS = {GTARSKI1.ABS; GTARSKI3.ABS; },
    KEYWORDS = {Tarski geometry; foundations of geometry; right angle; },
    SUBMITTED = {March 11, 2019}}


@ARTICLE{NTALGO_2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 1}},
    PAGES = {87--91},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0009},
    VERSION = {8.1.09 5.54.1344},
    TITLE = {{M}aximum Number of Steps Taken by Modular Exponentiation and {E}uclidean Algorithm},
    ANNOTE = {This study was supported in part by JSPS KAKENHI Grant Numbers JP17K00182 and JP15K00183.},
    AUTHOR = {Okazaki, Hiroyuki and Nagao, Koh-ichi and Futa, Yuichi},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
      ADDRESS2 = {Kanto Gakuin University\\Kanagawa, Japan},
      ADDRESS3 = {Tokyo University of Technology\\Tokyo, Japan},
      ACKNOWLEDGEMENT = {The authors would like to express our gratitude to Prof. Yasunari Shidama for his support and encouragement.},
    SUMMARY = {In this article we formalize in Mizar \cite{Mizar-State-2015}, \cite{BancerekJAR:2018} the maximum number of steps taken by some number theoretical algorithms, ``right--to--left binary algorithm" for modular exponentiation and ``Euclidean algorithm" \cite{KNUTH2}. For any natural numbers $a$, $b$, $n$, ``right--to--left binary algorithm" can calculate the natural number, see (Def.~\hyperlink{def:2}{2}), ${\rm Algo}_{\rm BPow}(a,n,m):=a^b \bmod n$ and for any integers $a$, $b$, ``Euclidean algorithm" can calculate the non negative integer ${\rm gcd}(a, b)$. We have not formalized computational complexity of algorithms yet, though we had already formalize the ``Euclidean algorithm" in \cite{NTALGO_1.ABS}. \par For ``right-to-left binary algorithm", we formalize the theorem, which says that the required number of the modular squares and modular products in this algorithms are $1+\lfloor \log_2 n \rfloor$ and for ``Euclidean algorithm", we formalize the Lam{\' e}'s theorem \cite{Lame:1844}, which says the required number of the divisions in this algorithm is at most $5 \log_{10} \min(|a|,|b|)$. Our aim is to support the implementation of number theoretic tools and evaluating computational complexities of algorithms to prove the security of cryptographic systems.   },
    MSC2010 = {68W40 11A05 11A15 03B35},
    SECTION1 = {Right--to--Left Binary Algorithm for Modular Exponentiation},
    SECTION2 = {Lam{\' e}'s Theorem},
    EXTERNALREFS = {Mizar-State-2015; BancerekJAR:2018; KNUTH2; Lame:1844; },
    INTERNALREFS = {NTALGO_1.ABS; NAT_4.ABS; EULER_2.ABS; FIB_NUM2.ABS; },
    KEYWORDS = {algorithms; power residues; Euclidean algorithm; },
    SUBMITTED = {March 11, 2019}}


@ARTICLE{FIELD_1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {93--100},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0010},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{O}n Roots of Polynomials over \fxp},
    AUTHOR = {Schwarzweller, Christoph},
    ADDRESS1 = {Institute of Informatics\\University of Gda{\'n}sk\\Poland},
      SUMMARY = {This is the first part of a four-article series containing a Mizar \cite{GrabKornSchwarz:2016}, \cite{Mizar-State-2015}, \cite{BancerekJAR:2018} formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field $F$ and every polynomial $p \in F[X] \backslash F$ there exists a field extension $E$ of $F$ such that $p$ has a root over $E$. The formalization follows Kronecker's classical proof using $F[X]\slash\!\!<\!\!p\!\!>$ as the desired field extension $E$ \cite{Rad91alg1}, \cite{Jac85}, \cite{HL99}.\par In this first part we show that an irreducible polynomial $p \in F[X] \backslash F$ has a root over $F[X]\slash\!\!<\!\!p\!\!>$. Note, however, that this statement cannot be true in a rigid formal sense: We do not have $F \subseteq F[X]\slash\!\!<\!\!p\!\!>$ as sets, so $F$ is not a subfield of $F[X]\slash\!\!<\!\!p\!\!>$, and hence formally $p$ is not even a polynomial over $F[X]\slash\!\!<\!\!p\!\!>$. Consequently, we translate $p$ along the canonical monomorphism $\phi : F \longrightarrow F[X]\slash\!\!<\!\!p\!\!>$ and show that the translated polynomial $\phi(p)$ has a root over $F[X]\slash\!\!<\!\!p\!\!>$.\par Because $F$ is not a subfield of $F[X]\slash\!\!<\!\!p\!\!>$ we construct in the second part the field $(E\, \backslash \, \phi F) \cup F$ for a given monomorphism $\phi : F \longrightarrow E$ and show that this field both is isomorphic to $F$ and includes $F$ as a subfield. In the literature this part of the proof usually consists of saying that ``one can identify $F$ with its image $\phi F$ in $F[X]\slash\!\!<\!\!p\!\!>$ and therefore consider $F$ as a subfield of $F[X]\slash\!\!<\!\!p\!\!>$". Interestingly, to do so we need to assume that $F \cap E = \emptyset$, in particular Kronecker's construction can be formalized for fields $F$ with $F \cap F[X] = \emptyset$.\par Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields $F$: With the exception of $\mathbb{Z}_2$ we construct for every field $F$ an isomorphic copy $F'$ of $F$ with $F' \cap F'[X] \neq \emptyset$. We also prove that for Mizar's representations of $\mathbb{Z}_n$, $\mathbb{Q}$ and $\mathbb{R}$ we have $\mathbb{Z}_n \cap \mathbb{Z}_n[X] = \emptyset$, $\mathbb{Q} \cap \mathbb{Q}[X] = \emptyset$ and $\mathbb{R} \cap \mathbb{R}[X] = \emptyset$, respectively.\par In the fourth part we finally define field extensions: $E$ is a field extension of $F$ iff $F$ is a subfield of $E$. Note, that in this case we have $F \subseteq E$ as sets, and thus a polynomial $p$ over $F$ is also a polynomial over $E$. We then apply the construction of the second part to $F[X]\slash\!\!<\!\!p\!\!>$ with the canonical monomorphism $\phi : F \longrightarrow F[X]\slash\!\!<\!\!p\!\!>$. Together with the first part this gives - for fields $F$ with $F \cap F[X] = \emptyset$ - a field extension $E$ of $F$ in which $p \in F[X] \backslash F$ has a root.   },
    MSC2010 = {12E05 12F05 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {The Polynomials $a \cdot x^n$},
    SECTION3 = {More on Homomorphisms},
    SECTION4 = {Lifting Homomorphisms from $R$ to $R[X]$},
    SECTION5 = {Kronecker's Construction},
    EXTERNALREFS = {GrabKornSchwarz:2016; Mizar-State-2015; BancerekJAR:2018; Rad91alg1; Jac85; HL99; },
    INTERNALREFS = {POLYNOM5.ABS; BINOM.ABS; POLYNOM4.ABS; RING_2.ABS; },
    KEYWORDS = {roots of polynomials; field extensions; Kronecker's construction; },
    SUBMITTED = {March 28, 2019}}
    

@ARTICLE{LOPBAN12.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {101--106},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0011},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{I}somorphisms from the Space of Multilinear Operators},
    AUTHOR = {Nakasho, Kazuhisa},
    ADDRESS1 = {Yamaguchi University\\Yamaguchi, Japan},
      ACKNOWLEDGEMENT = {I would like to express my gratitude to Professor Yasunari Shidama for his helpful advice.},
    SUMMARY = {In this article, using the Mizar system \cite{FourDecades}, \cite{BancerekJAR:2018}, the isomorphisms from the space of multilinear operators are discussed. In the first chapter, two isomorphisms are formalized. The former isomorphism shows the correspondence between the space of multilinear operators and the space of bilinear operators.\par The latter shows the correspondence between the space of multilinear operators and the space of the composition of linear operators. In the last chapter, the above isomorphisms are extended to isometric mappings between the normed spaces. We referred to \cite{miyadera:1972}, \cite{yoshida:1980}, \cite{Schwartz1997a}, \cite{Dunford:1958}, \cite{Schwartz1997b} in this formalization.   },
    MSC2010 = {46-00 47A07 47A30 68T99 03B35},
    SECTION1 = {Plain Isomorphisms\\ from the Space of Multilinear Operators},
    SECTION2 = {Extensions to Isometric Isomorphism\\ from the Normed Space of Multilinear Operators},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; miyadera:1972; yoshida:1980; Schwartz1997a; Dunford:1958; Schwartz1997b; },
    INTERNALREFS = {FUNCT_7.ABS; NDIFF_7.ABS; LOPBAN_9.ABS; PRVECT_3.ABS; },
    KEYWORDS = {linear operators; bilinear operators; multilinear operators; isomorphism of linear operator spaces; },
    SUBMITTED = {May 27, 2019}}
    

@ARTICLE{LOPBAN13.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {107--115},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0012},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{I}nvertible Operators on {B}anach Spaces},
    AUTHOR = {Nakasho, Kazuhisa},
    ADDRESS1 = {Yamaguchi University\\Yamaguchi, Japan},
      ACKNOWLEDGEMENT = {I would like to express my gratitude to Professor Yasunari Shidama for his helpful advice.},
    SUMMARY = {In this article, using the Mizar system \cite{FourDecades}, \cite{BancerekJAR:2018}, we discuss invertible operators on Banach spaces. In the first chapter, we formalized the theorem that denotes any operators that are close enough to an invertible operator are also invertible by using the property of Neumann series.\par In the second chapter, we formalized the continuity of an isomorphism that maps an invertible operator on Banach spaces to its inverse. These results are used in the proof of the implicit function theorem. We referred to \cite{miyadera:1972}, \cite{yoshida:1980}, \cite{Schwartz1997a}, \cite{Schwartz1997b} in this formalization.   },
    MSC2010 = {47A05 47J07 68T99 03B35},
    SECTION1 = {Neumann Series and Invertible Operator},
    SECTION2 = {Isomorphic Mapping to Inverse Operators},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; miyadera:1972; yoshida:1980; Schwartz1997a; Schwartz1997b; },
    INTERNALREFS = {LOPBAN_2.ABS; NDIFF_8.ABS; PRVECT_3.ABS; LOPBAN_1.ABS; },
    KEYWORDS = {Banach space; invertible operator; Neumann series; isomorphism of linear operator spaces; },
    SUBMITTED = {May 27, 2019}}
    

@ARTICLE{NDIFF_9.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {117--131},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0013},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{I}mplicit Function Theorem. {P}art {II}},
    ANNOTE = {This study was supported in part by JSPS KAKENHI Grant Number JP17K00182.},
    AUTHOR = {Nakasho, Kazuhisa and Shidama, Yasunari},
    ADDRESS1 = {Yamaguchi University\\Yamaguchi, Japan},
      ADDRESS2 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, we formalize differentiability of implicit function theorem in the Mizar system \cite{FourDecades}, \cite{BancerekJAR:2018}. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here.\par In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in \cite{NDIFF_8.ABS} is cited. We referred to \cite{Schwartz1997a}, \cite{Schwartz1997b}, and \cite{driver2003} in the formalization.   },
    MSC2010 = {26B10 47A05 47J07 53A07 03B35},
    SECTION1 = {Properties of Lipschitz Continuous Linear Operators},
    SECTION2 = {Differentiability of Implicit Function},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; Schwartz1997a; Schwartz1997b; driver2003; },
    INTERNALREFS = {NDIFF_8.ABS; LOPBAN_7.ABS; NDIFF_1.ABS; NFCONT_1.ABS; PRVECT_3.ABS; LOPBAN13.ABS; },
    KEYWORDS = {implicit function theorem; Lipschitz continuity; differentiability; implicit function; },
    SUBMITTED = {May 27, 2019}}
    

@ARTICLE{FIELD_2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {133--137},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0014},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{O}n Monomorphisms and Subfields},
    AUTHOR = {Schwarzweller, Christoph},
    ADDRESS1 = {Institute of Informatics\\University of Gda{\'n}sk\\Poland},
      SUMMARY = {This is the second part of a four-article series containing a Mizar \cite{FourDecades}, \cite{BancerekJAR:2018} formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field $F$ and every polynomial $p \in F[X] \backslash F$ there exists a field extension $E$ of $F$ such that $p$ has a root over $E$. The formalization follows Kronecker's classical proof using $F[X]\slash\!\!<\!\!p\!\!>$ as the desired field extension $E$ \cite{Rad91alg1}, \cite{Jac85}, \cite{HL99}.\par In the first part we show that an irreducible polynomial $p \in F[X] \backslash F$ has a root over $F[X]\slash\!\!<\!\!p\!\!>$. Note, however, that this statement cannot be true in a rigid formal sense: We do not have $F \subseteq F[X]\slash\!\!<\!\!p\!\!>$ as sets, so $F$ is not a subfield of $F[X]\slash\!\!<\!\!p\!\!>$, and hence formally $p$ is not even a polynomial over $F[X]\slash\!\!<\!\!p\!\!>$. Consequently, we translate $p$ along the canonical monomorphism $\phi : F \longrightarrow F[X]\slash\!\!<\!\!p\!\!>$ and show that the translated polynomial $\phi(p)$ has a root over $F[X]\slash\!\!<\!\!p\!\!>$. \par Because $F$ is not a subfield of $F[X]\slash\!\!<\!\!p\!\!>$ we construct in this second part the field $(E\, \backslash \, \phi F) \cup F$ for a given monomorphism $\phi : F \longrightarrow E$ and show that this field both is isomorphic to $F$ and includes $F$ as a subfield. In the literature this part of the proof usually consists of saying that ``one can identify $F$ with its image $\phi F$ in $F[X]\slash\!\!<\!\!p\!\!>$ and therefore consider $F$ as a subfield of $F[X]\slash\!\!<\!\!p\!\!>$". Interestingly, to do so we need to assume that $F \cap E = \emptyset$, in particular Kronecker's construction can be formalized for fields $F$ with $F \cap F[X] = \emptyset$. \par Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields $F$: With the exception of $\mathbb{Z}_2$ we construct for every field $F$ an isomorphic copy $F'$ of $F$ with $F' \cap F'[X] \neq \emptyset$. We also prove that for Mizar's representations of $\mathbb{Z}_n$, $\mathbb{Q}$ and $\mathbb{R}$ we have $\mathbb{Z}_n \cap \mathbb{Z}_n[X] = \emptyset$, $\mathbb{Q} \cap \mathbb{Q}[X] = \emptyset$ and $\mathbb{R} \cap \mathbb{R}[X] = \emptyset$, respectively. \par In the fourth part we finally define field extensions: $E$ is a field extension of $F$ iff $F$ is a subfield of $E$. Note, that in this case we have $F \subseteq E$ as sets, and thus a polynomial $p$ over $F$ is also a polynomial over $E$. We then apply the construction of the second part to $F[X]\slash\!\!<\!\!p\!\!>$ with the canonical monomorphism $\phi : F \longrightarrow F[X]\slash\!\!<\!\!p\!\!>$. Together with the first part this gives - for fields $F$ with $F \cap F[X] = \emptyset$ - a field extension $E$ of $F$ in which $p \in F[X] \backslash F$ has a root.   },
    MSC2010 = {12E05 12F05 68T99 03B35},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; Rad91alg1; Jac85; HL99; },
    INTERNALREFS = {},
    KEYWORDS = {roots of polynomials; field extensions; Kronecker's construction; },
    SUBMITTED = {May 27, 2019}}
    

@ARTICLE{ORDINAL7.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {139--152},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0015},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{N}atural Addition of Ordinals},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
      NOTE1 = {The author is enrolled in the Johannes Gutenberg University in Mayence, Germany, mailto: {\tt skoch02@students.uni-mainz.de}},
      SUMMARY = {In \cite{ORDINAL5.ABS} the existence of the Cantor normal form of ordinals was proven in the Mizar system \cite{FourDecades}. In this article its uniqueness is proven and then used to formalize the natural sum of ordinals.   },
    MSC2010 = {03E10 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {About the Cantor Normal Form},
    SECTION3 = {Natural Addition of Ordinals},
    EXTERNALREFS = {FourDecades; Abian; Bachmann; Cantor; Deiser; sierpinski1965; },
    INTERNALREFS = {ORDINAL5.ABS; POLYNOM1.ABS; AFINSQ_1.ABS; },
    KEYWORDS = {ordinal numbers; Cantor normal form; Hessenberg sum; natural sum; },
    SUBMITTED = {May 27, 2019}}
    

@ARTICLE{GLIB_008.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {153--179},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0016},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{A}bout Supergraphs. {P}art {III}},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
      NOTE1 = {The author is enrolled in the Johannes Gutenberg University in Mayence, Germany, mailto: {\tt skoch02@students.uni-mainz.de}},
      SUMMARY = {The previous articles \cite{GLIB_006.ABS} and \cite{GLIB_007.ABS} introduced formalizations of the step-by-step operations we use to construct finite graphs by hand. That implicitly showed that any finite graph can be constructed from the trivial edgeless graph $K_1$ by applying a finite sequence of these basic operations. In this article that claim is proven explicitly with Mizar \cite{FourDecades}.   },
    MSC2010 = {05C76 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Edgeless and Non Edgeless Graphs},
    SECTION3 = {Finite Graph Sequences},
    SECTION4 = {Construction of Finite Graphs},
    EXTERNALREFS = {FourDecades; SIMPLE; MULTI; German; SELECTED; },
    INTERNALREFS = {GLIB_006.ABS; GLIB_007.ABS; GLIB_000.ABS; FINSEQ_1.ABS; },
    KEYWORDS = {supergraph; graph operations; construction of finite graphs; },
    SUBMITTED = {May 27, 2019}}
    

@ARTICLE{NOMIN_5.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {181--187},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0017},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{P}artial Correctness of a Factorial Algorithm},
    AUTHOR = {Jaszczak, Adrian and Korni{\l}owicz, Artur},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = { In this paper we present a~formalization in the Mizar system \cite{FourDecades},\cite{BancerekJAR:2018} of the partial correctness of the algorithm:\par {\begin{center} \begin{tabular}{l}\tt i := val.1\\\tt j := val.2\\\tt n := val.3\\\tt s := val.4\\\tt while (i <> n)\\\tt $\strut\mkern25mu$i := i + j\\\tt$\strut\mkern25mu$s := s * i\\\tt return s\end{tabular}\end{center}} \par \noindent computing the factorial of given natural number \texttt{n}, where variables {\tt i, n, s} are located as values of a~{\tt V-valued Function}, {\tt loc}, as: \verb!loc/.1 = i!, \verb!loc/.3 = n! and \verb!loc/.4 = s!, and the constant {\tt 1} is located in the location \verb!loc/.2 = j! (set {\tt V} represents simple names of considered nominative data \cite{Skobelev2014}).\par This work continues a~formal verification of algorithms written in terms of simple-named complex-valued nominative data \cite{NOMIN_1.ABS},\cite{NOMIN_2.ABS},\cite{PARTPR_1.ABS},\cite{PARTPR_2.ABS},\cite{DBLP:conf/fedcsis/KornilowiczKNI17},\cite{DBLP:conf/isat/KornilowiczKNI17}. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data \cite{NOMIN_3.ABS}. Proofs of the correctness are based on an~inference system for an~extended Floyd-Hoare logic \cite{Floyd1967},\cite{Hoare1969} with partial pre- and post-conditions \cite{KornilowiczetalICTERI2017},\cite{Kryvolap2013},\cite{Moldova2018},\cite{SeqRule2018}.   },
    MSC2010 = {68Q60 68T37 03B70 03B35},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; Skobelev2014; DBLP:conf/fedcsis/KornilowiczKNI17; 
    DBLP:conf/isat/KornilowiczKNI17; Floyd1967; Hoare1969; KornilowiczetalICTERI2017; Kryvolap2013; Moldova2018; 
    SeqRule2018; },
    INTERNALREFS = {NOMIN_1.ABS; NOMIN_2.ABS; NOMIN_3.ABS; PARTPR_1.ABS; PARTPR_2.ABS; },
    KEYWORDS = {factorial; nominative data; program verification; },
    SUBMITTED = {May 27, 2019}}
    

@ARTICLE{NOMIN_6.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {189--195},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0018},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{P}artial Correctness of a Power Algorithm},
    AUTHOR = {Jaszczak, Adrian},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {This work continues a~formal verification of algorithms written in terms of simple-named complex-valued nominative data \cite{NOMIN_1.ABS},\cite{NOMIN_2.ABS},\cite{PARTPR_1.ABS},\cite{PARTPR_2.ABS},\cite{DBLP:conf/fedcsis/KornilowiczKNI17},\cite{DBLP:conf/isat/KornilowiczKNI17}. In this paper we present a~formalization in the Mizar system \cite{FourDecades},\cite{BancerekJAR:2018} of the partial correctness of the algorithm:\par {\begin{center} \begin{tabular}{l}\tt i := val.1\\\tt j := val.2\\\tt b := val.3\\\tt n := val.4\\\tt s := val.5\\\tt while (i <> n)\\\tt $\strut\mkern25mu$i := i + j\\\tt $\strut\mkern25mu$s := s * b\\\tt return s\end{tabular}\end{center}} \par \noindent computing the natural {\tt n} power of given complex number {\tt b}, where variables {\tt i, b, n, s} are located as values of a~{\tt V-valued Function}, {\tt loc}, as: \verb!loc/.1 = i!, \verb!loc/.3 = b!, \verb!loc/.4 = n! and \verb!loc/.5 = s!, and the constant {\tt 1} is located in the location \verb!loc/.2 = j! (set {\tt V} represents simple names of considered nominative data \cite{Skobelev2014}).\par The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data \cite{NOMIN_3.ABS}. Proofs of the correctness are based on an~inference system for an~extended Floyd-Hoare logic \cite{Floyd1967},\cite{Hoare1969} with partial pre- and post-conditions \cite{KornilowiczetalICTERI2017},\cite{Kryvolap2013},\cite{Moldova2018},\cite{SeqRule2018}.   },
    MSC2010 = {68Q60 68T37 03B70 03B35},
    EXTERNALREFS = {DBLP:conf/fedcsis/KornilowiczKNI17; DBLP:conf/isat/KornilowiczKNI17; FourDecades; 
    BancerekJAR:2018; Skobelev2014; Floyd1967; Hoare1969; KornilowiczetalICTERI2017; Kryvolap2013; Moldova2018; SeqRule2018; },
    INTERNALREFS = {NOMIN_1.ABS; NOMIN_2.ABS; NOMIN_3.ABS; NOMIN_4.ABS; PARTPR_1.ABS; PARTPR_2.ABS; },
    KEYWORDS = {power; nominative data; program verification; },
    SUBMITTED = {May 27, 2019}}
    

@ARTICLE{HILB10_4.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {197--208},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0019},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{D}iophantine Sets. {P}art {II}},
    AUTHOR = {P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {The article is the next in a series aiming to formalize the MDPR-theorem using the Mizar proof assistant \cite{Mizar-State-2015}, \cite{FourDecades}, \cite{BancerekJAR:2018}. We analyze four equations from the Diophantine standpoint that are crucial in the bounded quantifier theorem, that is used in one of the approaches to solve the problem. \par Based on our previous work \cite{HILB10_3.ABS}, we prove that the value of a given binomial coefficient and factorial can be determined by its arguments in a Diophantine way. Then we prove that two products \begin{equation}z = \displaystyle\prod_{i=1}^{x}(1+i\cdot y), \mkern50mu z = \displaystyle\prod_{i=1}^{x}(y +1 -j),\end{equation} where $y>x$ are Diophantine. \par The formalization follows \cite{LNT:Smorynski}, Z. Adamowicz, P. Zbierski \cite{AdamowiczZbierski} as well as M.~Da\-vis~\cite{Hilbert10}.   },
    MSC2010 = {11D45 68T99 03B35},
    SECTION1 = {Product of Zero Based Finite Sequences},
    SECTION2 = {Binomial is Diophantine},
    SECTION3 = {Factorial is Diophantine},
    SECTION4 = {Diophanticity of Selected Products},
    SECTION5 = {Selected Subsets of Zero Based Finite Sequences of $\mathbb{N}$\\as Diophantine Sets},
    EXTERNALREFS = {Mizar-State-2015; FourDecades; BancerekJAR:2018; LNT:Smorynski; AdamowiczZbierski; Hilbert10; },
    INTERNALREFS = {HILB10_3.ABS; NEWTON04.ABS; INT_4.ABS; BASEL_1.ABS; HILB10_2.ABS; AFINSQ_1.ABS; },
    KEYWORDS = {Hilbert's 10th problem; Diophantine relations; },
    SUBMITTED = {May 27, 2019}}


@ARTICLE{HILB10_5.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 2}},
    PAGES = {209--221},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0020},
    VERSION = {8.1.09 5.57.1355},
    TITLE = {{F}ormalization of the {MRDP} Theorem in the {M}izar System},
    ANNOTE = {This work has been financed by the resources of the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.},
    AUTHOR = {P\k{a}k, Karol},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {This article is the final step of our attempts to formalize the negative solution of Hilbert's tenth problem. \par In our approach, we work with the Pell's Equation defined in~\cite{PELLS_EQ.ABS}. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form $x^2-(a^2-1)y^2=1$~\cite{HILB10_1.ABS} and its solutions considered as two sequences $\{x_{i}(a)\}_{i=0}^\infty$, $\{y_{i}(a)\}_{i=0}^\infty$. We showed in~\cite{HILB10_3.ABS} that the $n$-th element of these sequences can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities, or more precisely that the equation $x=y_{i}(a)$ is Diophantine. Following the post-Matiyasevich results we show that the equality determined by the value of the power function $y = x^z$ is Diophantine, and analogously property in cases of the binomial coefficient, factorial and several product~\cite{HILB10_4.ABS}. \par In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that {\it every recursively enumerable set is Diophantine,} where recursively enumerable sets have been defined by the Martin Davis normal form. \par The formalization by means of Mizar system \cite{BancerekJAR:2018}, \cite{FourDecades}, \cite{Mizar-State-2015} follows \cite{LNT:Smorynski}, Z. Adamowicz, P. Zbierski \cite{AdamowiczZbierski} as well as M.~Da\-vis~\cite{Hilbert10}.   },
    MSC2010 = {11D45 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Selected Operations on Polynomials},
    SECTION3 = {Selected Subsets of Zero Based Finite Sequences of $\mathbb{N}$\\ as Diophantine Sets},
    SECTION4 = {Bounded Quantifier Theorem and its Variant},
    SECTION5 = {MRDP Theorem},
    EXTERNALREFS = {BancerekJAR:2018; FourDecades; Mizar-State-2015; LNT:Smorynski; AdamowiczZbierski; Hilbert10; },
    INTERNALREFS = {PELLS_EQ.ABS; HILB10_1.ABS; HILB10_3.ABS; HILB10_4.ABS; },
    KEYWORDS = {Hilbert's 10th problem; Diophantine relations; },
    SUBMITTED = {May 27, 2019}}


@ARTICLE{FIELD_3.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 3}},
    PAGES = {223--228},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0021},
    VERSION = {8.1.09 5.59.1363},
    TITLE = {{O}n the Intersection of Fields \f~with \fx},
    AUTHOR = {Schwarzweller, Christoph},
    ADDRESS1 = {Institute of Informatics\\University of Gda{\'n}sk\\Poland},
      SUMMARY = {This is the third part of a four-article series containing a Mizar \cite{GrabKornSchwarz:2016}, \cite{Mizar-State-2015}, \cite{BancerekJAR:2018} formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field $F$ and every polynomial $p \in F[X] \backslash F$ there exists a field extension $E$ of $F$ such that $p$ has a root over $E$. The formalization follows Kronecker's classical proof using $F[X]\slash\!\!<\!\!p\!\!>$ as the desired field extension $E$ \cite{Rad91alg1}, \cite{Jac85}, \cite{HL99}.\par In the first part we show that an irreducible polynomial $p \in F[X] \backslash F$ has a root over $F[X]\slash\!\!<\!\!p\!\!>$. Note, however, that this statement cannot be true in a rigid formal sense: We do not have $F \subseteq F[X]\slash\!\!<\!\!p\!\!>$ as sets, so $F$ is not a subfield of $F[X]\slash\!\!<\!\!p\!\!>$, and hence formally $p$ is not even a polynomial over $F[X]\slash\!\!<\!\!p\!\!>$. Consequently, we translate $p$ along the canonical monomorphism $\phi : F \longrightarrow F[X]\slash\!\!<\!\!p\!\!>$ and show that the translated polynomial $\phi(p)$ has a root over $F[X]\slash\!\!<\!\!p\!\!>$.\par Because $F$ is not a subfield of $F[X]\slash\!\!<\!\!p\!\!>$ we construct in the second part the field $(E\, \backslash \, \phi F) \cup F$ for a given monomorphism $\phi : F \longrightarrow E$ and show that this field both is isomorphic to $F$ and includes $F$ as a subfield. In the literature this part of the proof usually consists of saying that ``one can identify $F$ with its image $\phi F$ in $F[X]\slash\!\!<\!\!p\!\!>$ and therefore consider $F$ as a subfield of $F [X]\slash\!\!<\!\!p\!\!>$". Interestingly, to do so we need to assume that $F \cap E = \emptyset$, in particular Kronecker's construction can be formalized for fields $F$ with $F \cap F[X] = \emptyset$.\par Surprisingly, as we show in this third part, this condition is not automatically true for arbitrary fields $F$: With the exception of $\mathbb{Z}_2$ we construct for every field $F$ an isomorphic copy $F'$ of $F$ with $F' \cap F'[X] \neq \emptyset$. We also prove that for Mizar's representations of $\mathbb{Z}_n$, $\mathbb{Q}$ and $\mathbb{R}$ we have $\mathbb{Z}_n \cap \mathbb{Z}_n[X] = \emptyset$, $\mathbb{Q} \cap \mathbb{Q}[X] = \emptyset$ and $\mathbb{R} \cap \mathbb{R}[X] = \emptyset$, respectively.\par In the fourth part we finally define field extensions: $E$ is a field extension of $F$ iff $F$ is a subfield of $E$. Note, that in this case we have $F \subseteq E$ as sets, and thus a polynomial $p$ over $F$ is also a polynomial over $E$. We then apply the construction of the second part to $F[X]\slash\!\!<\!\!p\! \!>$ with the canonical monomorphism $\phi : F \longrightarrow F[X]\slash\!\!<\!\!p\!\!>$. Together with the first part this gives -- for fields $F$ with $F \cap F[X] = \emptyset$ -- a field extension $E$ of $F$ in which $p \in F[X] \backslash F$ has a root.   },
    MSC2010 = {12E05 12F05 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Some Negative Results},
    SECTION3 = {An Intuitive ``Solution"},
    SECTION4 = {Some Positive Results},
    EXTERNALREFS = {GrabKornSchwarz:2016; Mizar-State-2015; BancerekJAR:2018; Rad91alg1; Jac85; HL99; },
    INTERNALREFS = {},
    KEYWORDS = {roots of polynomials; field extensions; Kronecker's construction; },
    SUBMITTED = {August 29, 2019}}
    

@ARTICLE{FIELD_4.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 3}},
    PAGES = {229--235},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0022},
    VERSION = {8.1.09 5.59.1363},
    TITLE = {{F}ield Extensions and {K}ronecker's Construction},
    AUTHOR = {Schwarzweller, Christoph},
    ADDRESS1 = {Institute of Informatics\\University of Gda{\'n}sk\\Poland},
      SUMMARY = {This is the fourth part of a four-article series containing a Mizar \cite{GrabKornSchwarz:2016}, \cite{FourDecades}, \cite{BancerekJAR:2018} formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field $F$ and every polynomial $p \in F[X] \backslash F$ there exists a field extension $E$ of $F$ such that $p$ has a root over $E$. The formalization follows Kronecker's classical proof using $F[X]\slash\!\!<\!\!p\!\!>$ as the desired field extension $E$ \cite{Rad91alg1}, \cite{Jac85}, \cite{HL99}.\par In the first part we show that an irreducible polynomial $p \in F[X] \backslash F$ has a root over $F[X]\slash\!\!<\!\!p\!\!>$. Note, however, that this statement cannot be true in a rigid formal sense: We do not have $F \subseteq F[X]\slash\!\!<\!\!p\!\!>$ as sets, so $F$ is not a subfield of $F[X]\slash\!\!<\!\!p\!\!>$, and hence formally $p$ is not even a polynomial over $F[X]\slash\!\!<\!\!p\!\!>$. Consequently, we translate $p$ along the canonical monomorphism $\phi : F \longrightarrow F[X]\slash\!\!<\!\!p\!\!>$ and show that the translated polynomial $\phi(p)$ has a root over $F[X]\slash\!\!<\!\!p\!\!>$.\par Because $F$ is not a subfield of $F[X]\slash\!\!<\!\!p\!\!>$ we construct in the second part the field $(E\, \backslash \, \phi F) \cup F$ for a given monomorphism $\phi : F \longrightarrow E$ and show that this field both is isomorphic to $F$ and includes $F$ as a subfield. In the literature this part of the proof usually consists of saying that ``one can identify $F$ with its image $\phi F$ in $F[X]\slash\!\!<\!\!p\!\!>$ and therefore consider $F$ as a subfield of $F [X]\slash\!\!<\!\!p\!\!>$". Interestingly, to do so we need to assume that $F \cap E = \emptyset$, in particular Kronecker's construction can be formalized for fields $F$ with $F \cap F[X] = \emptyset$.\par Surprisingly, as we show in the third part, this condition is not automatically true for arbitrary fields $F$: With the exception of $\mathbb{Z}_2$ we construct for every field $F$ an isomorphic copy $F'$ of $F$ with $F' \cap F'[X] \neq \emptyset$. We also prove that for Mizar's representations of $\mathbb{Z}_n$, $\mathbb{Q}$ and $\mathbb{R}$ we have $\mathbb{Z}_n \cap \mathbb{Z}_n[X] = \emptyset$, $\mathbb{Q} \cap \mathbb{Q}[X] = \emptyset$ and $\mathbb{R} \cap \mathbb{R}[X] = \emptyset$, respectively.\par In this fourth part we finally define field extensions: $E$ is a field extension of $F$ iff $F$ is a subfield of $E$. Note, that in this case we have $F \subseteq E$ as sets, and thus a polynomial $p$ over $F$ is also a polynomial over $E$. We then apply the construction of the second part to $F[X]\slash\!\!<\!\!p\! \!>$ with the canonical monomorphism $\phi : F \longrightarrow F[X]\slash\!\!<\!\!p\!\!>$. Together with the first part this gives -- for fields $F$ with $F \cap F[X] = \emptyset$ -- a field extension $E$ of $F$ in which $p \in F[X] \backslash F$ has a root.   },
    MSC2010 = {12E05 12F05 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Ring and Field Extensions},
    SECTION3 = {Extensions of Polynomial Rings},
    SECTION4 = {Evaluation of Polynomials in Ring Extensions},
    SECTION5 = {The Degree of Field Extensions},
    SECTION6 = {Kronecker's Construction},
    EXTERNALREFS = {GrabKornSchwarz:2016; FourDecades; BancerekJAR:2018; Rad91alg1; Jac85; HL99; },
    INTERNALREFS = {},
    KEYWORDS = {roots of polynomials; field extensions; Kronecker's construction; },
    SUBMITTED = {August 29, 2019}}
    

@ARTICLE{GLIB_009.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 3}},
    PAGES = {237--259},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0023},
    VERSION = {8.1.09 5.59.1363},
    TITLE = {{U}nderlying Simple Graphs},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
      NOTE1 = {The author is enrolled in the Johannes Gutenberg University in Mayence, Germany, mailto: {\tt skoch02@students.uni-mainz.de}},
      SUMMARY = {In this article the notion of the underlying simple graph of a graph (as defined in \cite{GLIB_000.ABS}) is formalized in the Mizar system \cite{FourDecades}, along with some convenient variants. The property of a graph to be without decorators (as introduced in \cite{GLIB_003.ABS}) is formalized as well to serve as the base of graph enumerations in the future.   },
    MSC2010 = {68T99 03B35 05C76},
    SECTION1 = {Preliminaries},
    SECTION2 = {Plain Graphs},
    SECTION3 = {Graphs with Loops Removed},
    SECTION4 = {Graphs with Parallel Edges Removed},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; SIMPLE; SELECTED; MULTI; GERWAG; },
    INTERNALREFS = {GLIB_000.ABS; GLIB_003.ABS; TREES_1.ABS; GRAPH_1.ABS; SGRAPH1.ABS; SCMYCIEL.ABS; },
    KEYWORDS = {graph operations; underlying simple graph; },
    SUBMITTED = {August 29, 2019}}

    
@ARTICLE{GLIB_010.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 3}},
    PAGES = {261--301},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0024},
    VERSION = {8.1.09 5.59.1363},
    TITLE = {{A}bout Graph Mappings},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
      NOTE1 = {The author is enrolled in the Johannes Gutenberg University in Mayence, Germany, mailto: {\tt skoch02@students.uni-mainz.de}},
      SUMMARY = {In this articles adjacency-preserving mappings from a graph to another are formalized in the Mizar system \cite{FourDecades}, \cite{BancerekJAR:2018}. The generality of the approach seems to be largely unpreceeded in the literature to the best of the author's knowledge. However, the most important property defined in the article is that of two graphs being isomorphic, which has been extensively studied. Another graph decorator is introduced as well.   },
    MSC2010 = {05C60 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Ordering of a Graph},
    SECTION3 = {Graph Mappings},
    SECTION4 = {Walks Induced by Graph Mappings},
    SECTION5 = {Graph Mappings and Graph Modes},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; MULTI; HOMO; COLPROB; AGT2; CAYLEY; EXPATH; SIMPLE; AGT; },
    INTERNALREFS = {FUNCT_1.ABS; FUNCT_2.ABS; TREES_1.ABS; GLIB_007.ABS; GLIB_000.ABS; GLIB_006.ABS; },
    KEYWORDS = {graph homomorphism; graph isomorphism; },
    SUBMITTED = {August 29, 2019}}

    
@ARTICLE{GLIB_011.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 3}},
    PAGES = {303--313},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0025},
    VERSION = {8.1.09 5.59.1363},
    TITLE = {{A}bout Vertex Mappings},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
      NOTE1 = {The author is enrolled in the Johannes Gutenberg University in Mayence, Germany, mailto: {\tt skoch02@students.uni-mainz.de}},
      SUMMARY = {In \cite{GLIB_010.ABS} partial graph mappings were formalized in the Mizar system \cite{FourDecades}. Such mappings map some vertices and edges of a graph to another while preserving adjacency. While this general approach is appropriate for the general form of (multidi)graphs as introduced in \cite{GLIB_000.ABS}, a more specialized version for graphs without parallel edges seems convenient. As such, partial vertex mappings preserving adjacency between the mapped verticed are formalized here.   },
    MSC2010 = {05C60 68T99 03B35},
    SECTION1 = {Vertex Mappings},
    SECTION2 = {The Relation Between Graph Mappings and Vertex Mappings},
    EXTERNALREFS = {FourDecades; BancerekJAR:2018; AGT; DISC; HOMO; SIMPLE; },
    INTERNALREFS = {GLIB_000.ABS; GLIB_010.ABS; },
    KEYWORDS = {graph homomorphism; graph isomorphism; },
    SUBMITTED = {August 29, 2019}}


@ARTICLE{EC_PF_3.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 3}},
    PAGES = {315--320},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0026},
    VERSION = {8.1.09 5.59.1363},
    TITLE = {{O}perations of Points on Elliptic Curve in Affine Coordinates},
    ANNOTE = {This work was supported by JSPS KAKENHI Grant Numbers JP15K00183 and JP17K00182.},
    AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
    ADDRESS1 = {Tokyo University of Technology\\Tokyo, Japan},
      ADDRESS2 = {Shinshu University\\Nagano, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {In this article, we formalize in Mizar \cite{Mizar-State-2015}, \cite{BancerekJAR:2018} a binary operation of points on an elliptic curve over $\bf{GF}(p)$ in affine coordinates. We show that the operation is unital, complementable and commutative. Elliptic curve cryptography \cite{BSS99}, whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.   },
    MSC2010 = {14H52 14K05 68T99 03B35},
    SECTION1 = {Set of Points on Elliptic Curve in Affine Coordinates},
    SECTION2 = {Commutative Property of Operations of Points on Elliptic Curve},
    EXTERNALREFS = {Mizar-State-2015; BancerekJAR:2018; BSS99; },
    INTERNALREFS = {EC_PF_1.ABS; EC_PF_2.ABS; RECDEF_2.ABS; },
    KEYWORDS = {elliptic curve; commutative operation; },
    SUBMITTED = {August 29, 2019}}


@ARTICLE{AIMLOOP.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 4}},
    PAGES = {321--335},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0027},
    VERSION = {8.1.09 5.59.1363},
    TITLE = {{AIM} Loops and the {AIM} Conjecture},
    ANNOTE = {This work has been supported by the European Research Council (ERC) Consolidator grant nr. 649043 \textit{AI4REASON} and the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473. },
    AUTHOR = {Brown, Chad E. and P\k{a}k, Karol},
    ADDRESS1 = {\u{C}esk\'{y} Institut Informatiky Robotiky a Kybernetiky\\ Zikova 4, 166 36 Praha 6,\\ Czech Republic},
      ADDRESS2 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {In this article, we prove, using the Mizar \cite{BancerekJAR:2018} formalism, a number of properties that correspond to the AIM Conjecture. In the first section, we define division operations on loops, inner mappings $T$, $L$ and $R$, commutators and associators and basic attributes of interest. We also consider subloops and homomorphisms. Particular subloops are the nucleus and center of a loop and kernels of homomorphisms. Then in Section~2, we define a set \mbox{Mlt $Q$} of multiplicative mappings of $Q$ and cosets (mostly following Albert 1943 for cosets~\cite{Albert43}). Next, in Section~3 we define the notion of a normal subloop and construct quotients by normal subloops. In the last section we define the set \mbox{InnAut} of inner mappings of $Q$, define the notion of an AIM loop and relate this to the conditions on $T$, $L$, and $R$ defined by satisfies \mbox{TT}, etc. We prove in Theorem~(67) that the nucleus of an AIM loop is normal and finally in Theorem~(68) that the AIM Conjecture follows from knowing every AIM loop satisfies \mbox{aa1}, \mbox{aa2}, \mbox{aa3}, \mbox{Ka}, \mbox{aK1}, \mbox{aK2} and \mbox{aK3}. \par The formalization follows M.K. Kinyon, R. Veroff, P. Vojtechovsky \cite{KinyonVV13} (in \cite{2013mccune}) as well as Veroff's Prover9 files.   },
    MSC2010 = {20N05 68T99 03B35},
    SECTION1 = {Loops -- Introduction},
    SECTION2 = {Multiplicative Mappings and Cosets},
    SECTION3 = {Normal Subloop},
    SECTION4 = {AIM Conjecture},
    EXTERNALREFS = {BancerekJAR:2018; Albert43; KinyonVV13; 2013mccune; },
    INTERNALREFS = {RING_3.ABS; },
    KEYWORDS = {loops; Abelian inner mapping; groups; },
    SUBMITTED = {August 29, 2019}}


@ARTICLE{ROUGHIF1.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {27},
    NUMBER = {{\bf 4}},
    PAGES = {337--345},
    YEAR = {2019},
    DOI = {10.2478/forma-2019-0028},
    VERSION = {8.1.09 5.59.1363},
    TITLE = {{F}ormal Development of Rough Inclusion Functions},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {Rough sets, developed by Pawlak \cite{Pawlak1982}, are important tool to describe situation of incomplete or partially unknown information. In this article, continuing the formalization of rough sets \cite{ROUGHS_5.ABS}, we give the formal characterization of three rough inclusion functions (RIFs). We start with the standard one, $\kappa^{\pounds}$, connected with {\L}ukasiewicz \cite{Lukasiewicz:1913}, and extend this research for two additional RIFs: $\kappa_1$, and $\kappa_2$, following a paper by Gomoli{\'n}ska \cite{GomolinskaRIF2008}, \cite{GomolinskaRIF2007}. We also define q-RIFs and weak q-RIFs \cite{Gomolinska2009}. The paper establishes a formal counterpart of \cite{GrabowskiIJCRS2019} and makes a preliminary step towards rough mereology \cite{PolkowskiRM2011}, \cite{PolkowskiS:1996} in Mizar \cite{FourDecades}.   },
    MSC2010 = {03E70 68T99 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Standard Rough Inclusion Function},
    SECTION3 = {Rough Inclusion Functions},
    SECTION4 = {Defining Two New RIFs},
    SECTION5 = {Characteristic Properties of Rough Inclusions},
    SECTION6 = {On the Connections between Postulates},
    SECTION7 = {Formalization of Proposition 2 \cite{GomolinskaRIF2007}},
    SECTION8 = {Formalization of Proposition 4 \cite{GomolinskaRIF2007}},
    SECTION9 = {Concrete Example},
    SECTION10 = {Continuing Formalization of Theorem 4.1 \cite{Gomolinska2002}},
    EXTERNALREFS = {Pawlak1982; PolkowskiRM2011; PolkowskiS:1996; FourDecades; Lukasiewicz:1913; SkowronS96; Zhu:2007; 
                    Gomolinska2002; GomolinskaRIF2008; GomolinskaRIF2007; Gomolinska2009; 
                    GrabowskiFI:2014; GrabowskiDuplication; GrabowskiAssisted:2005; GrabowskiLTRS; GrabowskiIJCRS2019; },
    INTERNALREFS = {ROUGHS_2.ABS; ROUGHS_3.ABS; ROUGHS_5.ABS; },
    KEYWORDS = {rough set; rough inclusion; approximation space; },
    SUBMITTED = {August 29, 2019}}

@ARTICLE{BKMODEL3.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {1--7},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0001},
    VERSION = {8.1.09 5.60.1371},
    TITLE = {{K}lein-{B}eltrami model. {P}art {III}},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {Timothy Makarios (with Isabelle/HOL\footnote{\url{https://www.isa-afp.org/entries/Tarskis_Geometry.html}}) and John Harrison (with HOL-Light\footnote{\url{https://github.com/jrh13/hol-light/blob/master/100/independence.ml}}) shown that ``the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski's axioms except his Euclidean axiom'' \cite{beltrami1868saggio},\cite{beltrami1869essai},\cite{BORSUK:1},\cite{BS55}.\par With the Mizar system \cite{Mizar-State-2015} we use some ideas taken from Tim Makarios's MSc thesis \cite{makarios} to formalize some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. In this article we prove that our constructed model (we prefer ``Beltrami-Klein'' name over ``Klein-Beltrami'', which can be seen in the naming convention for Mizar functors, and even MML identifiers) satisfies the congruence symmetry, the congruence equivalence relation, and the congruence identity axioms formulated by Tarski (and formalized in Mizar as described briefly in \cite{GrabowskiCoghetto:2016}).   },
    MSC2010 = {51A05 51M10 68V20},
    SECTION1 = {Preliminaries},
    SECTION2 = {Planar Lemmas},
    SECTION3 = {The Construction of Beltrami-Klein Model},
    SECTION4 = {Congruence Symmetry},
    SECTION5 = {Congruence Equivalence Relation},
    SECTION6 = {Congruence Identity},
    EXTERNALREFS = {beltrami1868saggio; beltrami1869essai; BORSUK:1; BS55; Mizar-State-2015; makarios; GrabowskiCoghetto:2016; },
    INTERNALREFS = {ANPROJ_9; BKMODEL2; INCPROJ; },
    KEYWORDS = {Tarski's geometry axioms; foundations of geometry; Klein-Beltrami model; },
    SUBMITTED = {December 30, 2019}}

@ARTICLE{BKMODEL4.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {9--21},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0002},
    VERSION = {8.1.09 5.60.1371},
    TITLE = {{K}lein-{B}eltrami model. {P}art {IV}},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
      SUMMARY = {Timothy Makarios (with Isabelle/HOL\footnote{\url{https://www.isa-afp.org/entries/Tarskis_Geometry.html}}) and John Harrison (with HOL-Light\footnote{\url{https://github.com/jrh13/hol-light/blob/master/100/independence.ml}}) shown that ``the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski's axioms except his Euclidean axiom'' \cite{beltrami1868saggio}, \cite{beltrami1869essai}, \cite{BORSUK:1}, \cite{BS55}.\par With the Mizar system \cite{Mizar-State-2015} we use some ideas taken from Tim Makarios's MSc thesis \cite{makarios} to formalize some definitions and lemmas necessary for the verification of the independence of the parallel postulate. In this article, which is the continuation of \cite{BKMODEL3.ABS}, we prove that our constructed model satisfies the axioms of segment construction, the axiom of betweenness identity, and the axiom of Pasch due to Tarski, as formalized in \cite{GTARSKI1.ABS} and related Mizar articles.   },
    MSC2010 = {51A05 51M10 68V20},
    SECTION1 = {Preliminaries},
    SECTION2 = {The Axiom of Segment Construction},
    SECTION3 = {The Axiom of Betweenness Identity},
    SECTION4 = {The Axiom of Pasch},
    EXTERNALREFS = {beltrami1868saggio; beltrami1869essai; BORSUK:1; BS55; Mizar-State-2015; makarios; },
    INTERNALREFS = {ANPROJ_8.ABS; BKMODEL1.ABS; BKMODEL3.ABS; EUCLID_5.ABS; GTARSKI1.ABS; },
    KEYWORDS = {Tarski's geometry axioms; foundations of geometry; Klein-Beltrami model; },
    SUBMITTED = {December 30, 2019}}

@ARTICLE{GLIBPRE0.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {23--39},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0003},
    VERSION = {8.1.09 5.60.1371},
    TITLE = {{M}iscellaneous Graph Preliminaries},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
      NOTE1 = {The author is enrolled in the Johannes Gutenberg University in Mayence, Germany, mailto: {\tt skoch02@students.uni-mainz.de}},
      SUMMARY = {This article contains many auxiliary theorems which were missing in the Mizar Mathematical Library \cite{BancerekJAR:2018} to the best of the author's knowledge. Most of them regard graph theory as formalized in the \texttt{GLIB} series (cf. \cite{GLIB_000.ABS}) and most of them are preliminaries needed in \cite{GLIB_012.ABS} or other forthcoming articles.   },
    MSC2010 = {05C07 68V20},
    SECTION1 = {Preliminaries not Directly Related to Graphs},
    SECTION2 = {Into GLIB\_000},
    SECTION3 = {Into GLIB\_002},
    SECTION4 = {Into GLIB\_006},
    SECTION5 = {Into GLIB\_007},
    SECTION6 = {Into GLIB\_008},
    SECTION7 = {Into GLIB\_009},
    SECTION8 = {Into GLIB\_010},
    SECTION9 = {Into CHORD},
    SECTION10 = {Into GLIB\_011},
    EXTERNALREFS = {AGT; BancerekJAR:2018; MULTI; SIMPLE; HOMO; Mizar-State-2015; },
    INTERNALREFS = {GLIB_000.ABS; GLIB_010.ABS; GLIB_012.ABS; },
    KEYWORDS = {graph theory; vertex degrees; },
    SUBMITTED = {December 30, 2019}}
       
@ARTICLE{GLIB_012.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {41--63},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0004},
    VERSION = {8.1.09 5.60.1371},
    TITLE = {{A}bout Graph Complements},
    AUTHOR = {Koch, Sebastian},
    ADDRESS1 = {Johannes Gutenberg University\\Mainz, Germany},
      NOTE1 = {The author is enrolled in the Johannes Gutenberg University in Mayence, Germany, mailto: {\tt skoch02@students.uni-mainz.de}},
      SUMMARY = {This article formalizes different variants of the complement graph in the Mizar system \cite{FourDecades}, based on the formalization of graphs in \cite{GLIB_000.ABS}.   },
    MSC2010 = {05C76 68V20},
    SECTION1 = {Loopfull Graphs},
    SECTION2 = {Adding Loops to a Graph},
    SECTION3 = {Directed Graph Complement with Loops},
    SECTION4 = {Undirected Graph Complement with Loops},
    SECTION5 = {Directed Graph Complement without Loops},
    SECTION6 = {Undirected Graph Complement without Loops},
    SECTION7 = {Self-complementary Graphs},
    EXTERNALREFS = {FourDecades; GERWAG; GERDIE; MULTI; },
    INTERNALREFS = {GLIB_000.ABS; GLIB_006.ABS; GLIB_008.ABS; NECKLACE.ABS; },
    KEYWORDS = {graph complement; loop; },
    SUBMITTED = {December 30, 2019}}
  
@ARTICLE{WALLACE.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {65--77},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0005},
    VERSION = {8.1.09 5.60.1371},
    TITLE = {{S}tability of the 7-3 Compressor Circuit for {W}allace Tree. {P}art {I}},
    ANNOTE = {This work has been partially supported by the JSPS KAKENHI Grant Number 19K11821, Japan.},
    AUTHOR = {Wasaki, Katsumi},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
      SUMMARY = {To evaluate our formal verification method on a real-size calculation circuit, in this article, we continue to formalize the concept of the 7-3 Compressor (STC) Circuit \cite{Mehta:1991} for Wallace Tree \cite{Wallace:1964}, to define the structures of calculation units for a very fast multiplication algorithm for VLSI implementation \cite{Vuillemin:1983}. We define the circuit structure of the tree constructions of the Generalized Full Adder Circuits (GFAs). We then successfully prove its circuit stability of the calculation outputs after four and six steps. The motivation for this research is to establish a technique based on formalized mathematics and its applications for calculation circuits with high reliability, and to implement the applications of the reliable logic synthesizer and hardware compiler \cite{Iwasaki:2008}.   },
    MSC2010 = {68M07 68W35 68V20},
    SECTION1 = {Properties of `Intermediate' \\ STC Circuit Structure (LAYER-I)},
    SECTION2 = {Properties of `Intermediate' \\ STC Circuit Structure (LAYER-II)},
    SECTION3 = {Properties of STC Circuit Structure (LAYER-III)},
    EXTERNALREFS = {Mehta:1991; Wallace:1964; Vuillemin:1983; Iwasaki:2008; Mizar-State-2015; BancerekJAR:2018; FourDecades; },
    INTERNALREFS = {CIRCOMB.ABS; CIRCUIT1.ABS; CIRCUIT2.ABS; FACIRC_1.ABS; FTACELL1.ABS; GFACIRC1.ABS; TWOSCOMP.ABS; },
    KEYWORDS = {arithmetic processor; high order compressor; high-speed multiplier; Wallace tree; logic circuit stability; },
    SUBMITTED = {December 30, 2019}}


@ARTICLE{RINGFRAC.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {79--87},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0006},
    VERSION = {8.1.09 5.60.1371},
    TITLE = {{R}ings of {F}ractions and {L}ocalization},
    AUTHOR = {Watase, Yasushige},
    ADDRESS1 = {Suginami-ku Matsunoki\\3-21-6 Tokyo, Japan},
      SUMMARY = {This article formalized rings of fractions in the Mizar system \cite{Mizar-State-2015}, \cite{BancerekJAR:2018}. A~construction of the ring of fractions from an integral domain, namely a quotient field was formalized in \cite{QUOFIELD.ABS}. \par This article generalizes a construction of fractions to a ring which is commutative and has zero divisor by means of a multiplicatively closed set, say $S$, by known manner. Constructed ring of fraction is denoted by $S$\textasciitilde$R$ instead of $S^{-1}R$ appeared in \cite{atiyah1969introduction}, \cite{matsumura1989}. As an important example we formalize a ring of fractions by a particular multiplicatively closed set, namely $R\smallsetminus\mathfrak{p}$, where $\mathfrak{p}$ is a prime ideal of $R$. The resulted local ring is denoted by ${R_{\mathfrak{p}}}$. In our Mizar article it is coded by $R$\textasciitilde ${\mathfrak{p}}$ as a synonym. \par This article contains also the formal proof of a universal property of a ring of fractions, the total-quotient ring, a proof of the equivalence between the total-quotient ring and the quotient field of an integral domain.   },
    MSC2010 = {13B30 16S85 68V20},
    SECTION1 = {Preliminaries: \\ Units, Zero Divisors and Multiplicatively-closed Set},
    SECTION2 = {Equivalence Relation of Fractions},
    SECTION3 = {Construction of Ring of Fractions},
    SECTION4 = {Localization in Terms of Prime Ideals},
    SECTION5 = {Universal Property of Ring of Fractions},
    SECTION6 = {The Total-Quotient Ring \\ and the Quotient Field of Integral Domain},
    EXTERNALREFS = {Mizar-State-2015; BancerekJAR:2018; atiyah1969introduction; matsumura1989; },
    INTERNALREFS = {IDEAL_1.ABS; RING_2.ABS; TOPZARI1.ABS; QUOFIELD.ABS; },
    KEYWORDS = {rings of fractions; localization; total-quotient ring; quotient field; },
    SUBMITTED = {January 13, 2020}}


@ARTICLE{PRSUBSET.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {89--92},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0007},
    VERSION = {8.1.09 5.60.1371},
    TITLE = {{D}ynamic Programming for the Subset Sum Problem},
    ANNOTE = {This work was supported by JSPS KAKENHI Grant Numbers JP16K00033, JP17K00013 and JP17K00183.},
    AUTHOR = {Fujiwara, Hiroshi and Watari, Hokuto and Yamamoto, Hiroaki},
    ADDRESS1 = {Shinshu University\\Nagano, Japan},
      ADDRESS2 = {Nagano Electronics Industrial Co., Ltd.\\Chikuma, Japan},
      ADDRESS3 = {Shinshu University\\Nagano, Japan},
      ACKNOWLEDGEMENT = {We are very grateful to Prof. Yasunari Shidama for his encouraging support. We thank Prof. Pauline N. Kawamoto, Dr. Hiroyuki Okazaki, and Dr. Hiroshi Yamazaki for their helpful discussions.},
    SUMMARY = {The subset sum problem is a basic problem in the field of theoretical computer science, especially in the complexity theory \cite{Complexity:1972}. The input is a sequence of positive integers and a target positive integer. The task is to determine if there exists a subsequence of the input sequence with sum equal to the target integer. It is known that the problem is NP-hard~\cite{Karp1972} and can be solved by dynamic programming in pseudo-polynomial time~\cite{Garey:1979}. In this article we formalize the recurrence relation of the dynamic programming.   },
    MSC2010 = {90C39 68Q25 68V20},
    SECTION1 = {Preliminaries},
    SECTION2 = {Summing Up Finite Sequences},
    SECTION3 = {Recurrence Relation of Dynamic Programming \\ for the Subset Sum Problem},
    EXTERNALREFS = {Complexity:1972; Karp1972; Garey:1979; },
    INTERNALREFS = {FINSEQ_3.ABS; },
    KEYWORDS = {dynamic programming; subset sum problem; complexity theory; },
    SUBMITTED = {January 13, 2020}}


@ARTICLE{MEASUR12.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {93--104},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0008},
    VERSION = {8.1.09 5.60.1371},
    TITLE = {{R}econstruction of the One-Dimensional {L}ebesgue Measure},
    AUTHOR = {Endou, Noboru},
    ADDRESS1 = {National Institute of Technology, Gifu College\\2236-2 Kamimakuwa, Motosu, Gifu, Japan},
      SUMMARY = {In the Mizar system (\cite{Mizar-State-2015}, \cite{BancerekJAR:2018}), J\'ozef Bia{\l}as has already given the one-dimensional Lebesgue measure \cite{MEASURE7.ABS}. However, the measure introduced by Bia{\l}as limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure to a semialgebra of intervals. Furthermore, by repeating the extension of the previous measure, we reconstructed the one-dimensional Lebesgue measure \cite{FOLLAND}, \cite{Bauer:2002}.   },
    MSC2010 = {28A12 28A75 68V20},
    SECTION1 = {Properties of Intervals},
    SECTION2 = {Tools for Extended Real Sequences},
    SECTION3 = {Open Covering of Intervals},
    SECTION4 = {Measure of Intervals by OS-Meas},
    SECTION5 = {Construction of the One-Dimensional Lebesgue Measure},
    EXTERNALREFS = {Mizar-State-2015; BancerekJAR:2018; FOLLAND; Bauer:2002; },
    INTERNALREFS = {MEASURE7.ABS; MEASURE8.ABS; MESFUNC5.ABS; RINFSUP2.ABS; },
    KEYWORDS = {Lebesgue measure; algebra of intervals; },
    SUBMITTED = {January 13, 2020}}


@ARTICLE{ROUGHIF2.ABS,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {105--113},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0009},
    VERSION = {8.1.09 5.60.1371},
    TITLE = {{D}eveloping Complementary Rough Inclusion Functions},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {We continue the formal development of rough inclusion functions (RIFs), continuing the research on the formalization of rough sets \cite{Pawlak1982} -- a well-known tool of modelling of incomplete or partially unknown information. In this article we give the formal characterization of complementary RIFs, following a paper by Gomoli{\'n}ska \cite{GomolinskaRIF2008}. We expand this framework introducing Jaccard index, Steinhaus generate metric, and Marczewski-Steinhaus metric space \cite{deza2009encyclopedia}. This is the continuation of \cite{ROUGHIF1.ABS}; additionally we implement also parts of \cite{Gomolinska2009}, \cite{GomolinskaRIF2007}, and the details of this work can be found in \cite{GrabowskiIJCRS2019}.   },
    MSC2010 = {03E70 68V20 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Complementary Rough Inclusion Functions},
    SECTION3 = {Introducing co-RIFs},
    SECTION4 = {Proposition 6 from \cite{GomolinskaRIF2008}},
    SECTION5 = {Jaccard Index Measuring Similarity of Sets},
    SECTION6 = {Marczewski-Steinhaus Metric},
    SECTION7 = {Steinhaus Generate Metric},
    SECTION8 = {Marczewski-Steinhaus Metric is Generated by Symmetric Difference Metric},
    SECTION9 = {Steinhaus Metric Spaces},
    EXTERNALREFS = {deza2009encyclopedia; Gomolinska2009; GomolinskaRIF2007; GomolinskaRIF2008; GrabowskiAssisted:2005; GrabowskiFI:2014; GrabowskiIJCRS2019; GrabowskiLTRS; GrabowskiDuplication; Lukasiewicz:1913; Pawlak1982; SkowronS96; Zhu:2007; },
    INTERNALREFS = {ROUGHIF1.ABS; ROUGHS_2.ABS; ROUGHS_3.ABS; ROUGHS_5.ABS; },
    KEYWORDS = {rough set; rough inclusion function; Steinhaus generate metric; },
    SUBMITTED = {February 26, 2020}}


@ARTICLE{NUMBER01.MIZ,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {115--120},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0010},
    VERSION = {8.1.09 5.60.1374},
    TITLE = {{E}lementary Number Theory Problems. {P}art~{I}},
    AUTHOR = {Naumowicz, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {In this paper we demonstrate the feasibility of formalizing {\em recreational mathematics} in Mizar (\cite{Mizar-State-2015}, \cite{BancerekJAR:2018}) drawing examples from W. Sierpinski's book ``250 Problems in Elementary Number Theory'' \cite{Sierpinski:1970}. The current work contains proofs of initial ten problems from the chapter devoted to the divisibility of numbers. Included are problems on several levels of difficulty.   },
    MSC2010 = {11A99 68V20 03B35},
    SECTION1 = {Problem 1},
    SECTION2 = {Problem 2},
    SECTION3 = {Problem 3},
    SECTION4 = {Problem 4},
    SECTION5 = {Problem 5},
    SECTION6 = {Problem 6 (due to Kraitchik)},
    SECTION7 = {Problem 7},
    SECTION8 = {Problem 8},
    SECTION9 = {Problem 9},
    SECTION10 = {Problem 10},
    EXTERNALREFS = {Mizar-State-2015; BancerekJAR:2018; Sierpinski:1970; },
    INTERNALREFS = {NAT_5.ABS; AFINSQ_1.ABS; INT_5.ABS; },
    KEYWORDS = {number theory; recreational mathematics; },
    SUBMITTED = {February 26, 2020}}


@ARTICLE{FUZIMPL3.MIZ,
    JOURNAL = {Formalized Mathematics},
	ISSN = {1426-2630},
	EISSN = {1898-9934},
      VOLUME = {28},
    NUMBER = {{\bf 1}},
    PAGES = {121--128},
    YEAR = {2020},
    DOI = {10.2478/forma-2020-0011},
    VERSION = {8.1.09 5.60.1374},
    TITLE = {{O}n Fuzzy Negations Generated by Fuzzy Implications},
    AUTHOR = {Grabowski, Adam},
    ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Poland},
      SUMMARY = {We continue in the Mizar system \cite{Mizar-State-2015} the formalization of fuzzy implications according to the book of Baczy\'nski and Jayaram {``Fuzzy Implications''} \cite{Baczynski:2008}. In this article we define fuzzy negations and show their connections with previously defined fuzzy implications \cite{FUZIMPL1.ABS} and \cite{FUZIMPL2.ABS} and triangular norms and conorms \cite{FUZNORM1.ABS}. This can be seen as a step towards building a formal framework of fuzzy connectives \cite{Hajek:1998}. We introduce formally Sugeno negation, boundary negations and show how these operators are pointwise ordered. This work is a continuation of the development of fuzzy sets \cite{Zadeh:1965}, \cite{DuboisPrade:1980} in Mizar \cite{GrabowskiFuzzy:2013} started in \cite{FUZZY_1.ABS} and partially described in \cite{Grabowski2018}. This submission can be treated also as a part of a formal comparison of fuzzy and rough approaches to incomplete or uncertain information within the Mizar Mathematical Library \cite{GrabowskiMitsuishi:2015}.   },
    MSC2010 = {03B52 68V20 03B35},
    SECTION1 = {Preliminaries},
    SECTION2 = {Conjugate Fuzzy Implications},
    SECTION3 = {Fuzzy Negations},
    SECTION4 = {Generating Fuzzy Negations from Fuzzy Implications},
    SECTION5 = {Boundary Fuzzy Negations},
    SECTION6 = {Fuzzy Negations Generated by Nine Fuzzy Implications},
    SECTION7 = {Sugeno Negation},
    SECTION8 = {Conjugate Fuzzy Negations},
    EXTERNALREFS = {Mizar-State-2015; Baczynski:2008; Hajek:1998; Zadeh:1965; DuboisPrade:1980; GrabowskiFuzzy:2013; Grabowski2018; 
                    GrabowskiMitsuishi:2015; },
    INTERNALREFS = {FUZIMPL1.ABS; FUZIMPL2.ABS; FUZNORM1.ABS; FUZZY_1.ABS; },
    KEYWORDS = {fuzzy set; fuzzy negation; fuzzy implication; },
    SUBMITTED = {February 26, 2020}}