@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}{${\bf SCM}_{\rm FSA}$}"}
@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 Abelian 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 ${\BbbR}$ to ${\BbbR}$ 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 ${\BbbR}$ to ${\BbbR}$},
      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 $\BbbR$ into $\BbbR$},
      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 Kornilowicz, 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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      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},
      YEAR    = {2012},
      DOI     = {10.2478/v10037-012-0039-4},
      VOLUME  = 20,
      PAGES   = {349--357},
      NUMBER  = {{\bf 4}})
    
@ARTICLE(AOFA_A01.ABS,
    JOURNAL = {Formalized Mathematics},
    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},
    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},
    NOTE = {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},
    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},
    NOTE = {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},
    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},
    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},
    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},
    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},
    NOTE = {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},
    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; },
    ENVIRONMENT = {hidden:not; subset_1:not; relat_1:not; funct_1:not; ordinal1:not; relset_1:not; partfun1:not; funct_2:not; binop_1:not; card_1:not; xxreal_0:not; xcmplx_0:not; nat_1:not; finseq_1:not; finseq_2:not; margrel1:not; struct_0:not; rlvect_1:not; vectsp_1:not; vectsp_4:not; vectsp_6:not; vectsp_7:not; algstr_0:not; xboole_0:not; zfmisc_1:not; finset_1:not; xboolean:not; descip_1:not; group_1:not; matrlin:not; tarski:not; funct_4:not; numbers:not; binarith:not; vectsp_9:not; bspace:not; },
    SUBMITTED = {April 17, 2013})    
    
@ARTICLE(TOPGEN_6.ABS,
    JOURNAL = {Formalized Mathematics},
    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},
    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},
    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},
    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},
    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},
    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},
    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},
    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},
    NOTE = {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},
    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},
     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},
     NOTE = {This work was supported by JSPS KAKENHI 23500029.},
     AUTHOR = {Endou, Noboru and Okazaki, Hiroyuki and Shidama, Yasunari},
     ADDRESS1 = {Gifu National College of Thechnology\\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},
     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}},
     NOTE = {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},
     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},
     NOTE = {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},
     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},
     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},
     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},
     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},
    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},
    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},
    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},
    NOTE = {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},
    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},
    NOTE = {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},
    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},
    NOTE = {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},
    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},
    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},
    NOTE = {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},
    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},
    NOTE = {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.MIZ,
    JOURNAL = {Formalized Mathematics},
    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},
    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},
    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},
    NOTE = {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},
      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},
    NOTE = {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},
    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},
    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},
    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},
    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},
    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},
    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},
    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},
    NOTE = {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(MSAFREE4.ABS,
    JOURNAL = {Formalized Mathematics},
    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},
    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},
    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},
    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},
    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})