:: BORSUK_1 semantic presentation begin theorem :: BORSUK_1:1 (Bool "for" (Set (Var "X")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "Y")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))))) ; definitionlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "F" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "Y")); redefine attr "F" is :::"continuous"::: means :: BORSUK_1:def 1 (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Point":::) "of" "X" (Bool "for" (Set (Var "G")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set "F" ($#k3_funct_2 :::"."::: ) (Set (Var "W"))) (Bool "ex" (Set (Var "H")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "W")) "st" (Bool (Set "F" ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))) ($#r1_tarski :::"c="::: ) (Set (Var "G")))))); end; :: deftheorem defines :::"continuous"::: BORSUK_1:def 1 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "G")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "W"))) (Bool "ex" (Set (Var "H")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "W")) "st" (Bool (Set (Set (Var "F")) ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))) ($#r1_tarski :::"c="::: ) (Set (Var "G")))))) ")" ))); registrationlet "X", "Y", "Z" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "F" be ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "Y")); let "G" be ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "Y")) "," (Set (Const "Z")); cluster (Set "F" ($#k3_relat_1 :::"*"::: ) "G") -> ($#v5_pre_topc :::"continuous"::: ) for ($#m1_subset_1 :::"Function":::) "of" "X" "," "Z"; end; theorem :: BORSUK_1:2 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool (Set (Set (Var "A")) ($#k8_relset_1 :::"""::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "G")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" (Set (Var "A")) ($#k8_relset_1 :::"""::: ) (Set (Var "G")) ")" )))))) ; theorem :: BORSUK_1:3 (Bool "for" (Set (Var "Y")) "," (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) (Bool "for" (Set (Var "A")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "G")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "W")) "holds" (Bool (Set (Set (Var "A")) ($#k8_relset_1 :::"""::: ) (Set (Var "G"))) "is" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Set (Var "A")) ($#k8_relset_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "W")) ($#k6_domain_1 :::"}"::: ) ))))))) ; definitionlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "W" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "Y")); let "A" be ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "Y")); let "G" be ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Const "W")); :: original: :::"""::: redefine func "A" :::"""::: "G" -> ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set "A" ($#k8_relset_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) "W" ($#k6_domain_1 :::"}"::: ) )); end; theorem :: BORSUK_1:4 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "U")) "being" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "B")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "U")) "is" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "A")))))) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "X")); cluster (Set ($#k1_tarski :::"{"::: ) "x" ($#k1_tarski :::"}"::: ) ) -> ($#v2_compts_1 :::"compact"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "X"; end; begin definitionlet "X", "Y" be ($#l1_pre_topc :::"TopSpace":::); func :::"[:":::"X" "," "Y":::":]"::: -> ($#v1_pre_topc :::"strict"::: ) ($#l1_pre_topc :::"TopSpace":::) means :: BORSUK_1:def 2 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" it) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "A")) ")" ) where A "is" ($#m1_subset_1 :::"Subset-Family":::) "of" it : (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) "{" (Set ($#k8_mcart_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k8_mcart_1 :::":]"::: ) ) where X1 "is" ($#m1_subset_1 :::"Subset":::) "of" "X", Y1 "is" ($#m1_subset_1 :::"Subset":::) "of" "Y" : (Bool "(" (Bool (Set (Var "X1")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" "X")) & (Bool (Set (Var "Y1")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" "Y")) ")" ) "}" ) "}" ) ")" ); end; :: deftheorem defines :::"[:"::: BORSUK_1:def 2 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "b3")) "being" ($#v1_pre_topc :::"strict"::: ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) )) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "A")) ")" ) where A "is" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "b3")) : (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) "{" (Set ($#k8_mcart_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k8_mcart_1 :::":]"::: ) ) where X1 "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")), Y1 "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) : (Bool "(" (Bool (Set (Var "X1")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "X")))) & (Bool (Set (Var "Y1")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y")))) ")" ) "}" ) "}" ) ")" ) ")" ))); registrationlet "T1" be ($#l1_pre_topc :::"TopSpace":::); let "T2" be ($#v2_struct_0 :::"empty"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k2_borsuk_1 :::"[:"::: ) "T1" "," "T2" ($#k2_borsuk_1 :::":]"::: ) ) -> ($#v2_struct_0 :::"empty"::: ) ($#v1_pre_topc :::"strict"::: ) ; cluster (Set ($#k2_borsuk_1 :::"[:"::: ) "T2" "," "T1" ($#k2_borsuk_1 :::":]"::: ) ) -> ($#v2_struct_0 :::"empty"::: ) ($#v1_pre_topc :::"strict"::: ) ; end; registrationlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k2_borsuk_1 :::"[:"::: ) "X" "," "Y" ($#k2_borsuk_1 :::":]"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ; end; theorem :: BORSUK_1:5 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) ) "iff" (Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool "(" (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "A")))) & (Bool "(" "for" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "X1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X"))(Bool "ex" (Set (Var "Y1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool "(" (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set ($#k8_mcart_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k8_mcart_1 :::":]"::: ) )) & (Bool (Set (Var "X1")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "Y1")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ))) ")" ) ")" )) ")" ))) ; definitionlet "X", "Y" be ($#l1_pre_topc :::"TopSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "X")); let "B" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "Y")); :: original: :::"[:"::: redefine func :::"[:":::"A" "," "B":::":]"::: -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) "X" "," "Y" ($#k2_borsuk_1 :::":]"::: ) ); end; definitionlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "X")); let "y" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "Y")); :: original: :::"["::: redefine func :::"[":::"x" "," "y":::"]"::: -> ($#m1_subset_1 :::"Point":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) "X" "," "Y" ($#k2_borsuk_1 :::":]"::: ) ); end; theorem :: BORSUK_1:6 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "V")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "V")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "W")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "V")) "," (Set (Var "W")) ($#k3_borsuk_1 :::":]"::: ) ) "is" ($#v3_pre_topc :::"open"::: ) )))) ; theorem :: BORSUK_1:7 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "V")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "V")) "," (Set (Var "W")) ($#k3_borsuk_1 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_borsuk_1 :::"[:"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "V")) ")" ) "," (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "W")) ")" ) ($#k3_borsuk_1 :::":]"::: ) ))))) ; theorem :: BORSUK_1:8 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) (Bool "for" (Set (Var "V")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "x")) (Bool "for" (Set (Var "W")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "y")) "holds" (Bool (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "V")) "," (Set (Var "W")) ($#k3_borsuk_1 :::":]"::: ) ) "is" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set ($#k4_borsuk_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_borsuk_1 :::"]"::: ) ))))))) ; theorem :: BORSUK_1:9 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) (Bool "for" (Set (Var "V")) "being" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "W")) "being" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "B")) "holds" (Bool (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "V")) "," (Set (Var "W")) ($#k3_borsuk_1 :::":]"::: ) ) "is" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "A")) "," (Set (Var "B")) ($#k3_borsuk_1 :::":]"::: ) ))))))) ; definitionlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "X")); let "y" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "Y")); let "V" be ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Const "x")); let "W" be ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Const "y")); :: original: :::"[:"::: redefine func :::"[:":::"V" "," "W":::":]"::: -> ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set ($#k4_borsuk_1 :::"["::: ) "x" "," "y" ($#k4_borsuk_1 :::"]"::: ) ); end; theorem :: BORSUK_1:10 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "XT")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) (Bool "ex" (Set (Var "W")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X"))(Bool "ex" (Set (Var "T")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) "st" (Bool (Set (Var "XT")) ($#r1_hidden :::"="::: ) (Set ($#k4_borsuk_1 :::"["::: ) (Set (Var "W")) "," (Set (Var "T")) ($#k4_borsuk_1 :::"]"::: ) )))))) ; definitionlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "X")); let "t" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "Y")); let "V" be ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Const "A")); let "W" be ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Const "t")); :: original: :::"[:"::: redefine func :::"[:":::"V" "," "W":::":]"::: -> ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set ($#k3_borsuk_1 :::"[:"::: ) "A" "," (Set ($#k6_domain_1 :::"{"::: ) "t" ($#k6_domain_1 :::"}"::: ) ) ($#k3_borsuk_1 :::":]"::: ) ); end; definitionlet "X", "Y" be ($#l1_pre_topc :::"TopSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Const "X")) "," (Set (Const "Y")) ($#k2_borsuk_1 :::":]"::: ) ); func :::"Base-Appr"::: "A" -> ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) "X" "," "Y" ($#k2_borsuk_1 :::":]"::: ) ) equals :: BORSUK_1:def 3 "{" (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k3_borsuk_1 :::":]"::: ) ) where X1 "is" ($#m1_subset_1 :::"Subset":::) "of" "X", Y1 "is" ($#m1_subset_1 :::"Subset":::) "of" "Y" : (Bool "(" (Bool (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k3_borsuk_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) "A") & (Bool (Set (Var "X1")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "Y1")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ) "}" ; end; :: deftheorem defines :::"Base-Appr"::: BORSUK_1:def 3 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "holds" (Bool (Set ($#k7_borsuk_1 :::"Base-Appr"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) "{" (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k3_borsuk_1 :::":]"::: ) ) where X1 "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")), Y1 "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) : (Bool "(" (Bool (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k3_borsuk_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set (Var "X1")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "Y1")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ) "}" ))); registrationlet "X", "Y" be ($#l1_pre_topc :::"TopSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Const "X")) "," (Set (Const "Y")) ($#k2_borsuk_1 :::":]"::: ) ); cluster (Set ($#k7_borsuk_1 :::"Base-Appr"::: ) "A") -> ($#v1_tops_2 :::"open"::: ) ; end; theorem :: BORSUK_1:11 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k7_borsuk_1 :::"Base-Appr"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k7_borsuk_1 :::"Base-Appr"::: ) (Set (Var "B")))))) ; theorem :: BORSUK_1:12 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k7_borsuk_1 :::"Base-Appr"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "A"))))) ; theorem :: BORSUK_1:13 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k7_borsuk_1 :::"Base-Appr"::: ) (Set (Var "A")) ")" ))))) ; theorem :: BORSUK_1:14 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k7_borsuk_1 :::"Base-Appr"::: ) (Set (Var "A")) ")" ))))) ; definitionlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); func :::"Pr1"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) "X" "," "Y" ($#k2_borsuk_1 :::":]"::: ) )) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) equals :: BORSUK_1:def 4 (Set ($#k2_funct_3 :::".:"::: ) (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ")" ")" )); func :::"Pr2"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) "X" "," "Y" ($#k2_borsuk_1 :::":]"::: ) )) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ")" ) equals :: BORSUK_1:def 5 (Set ($#k2_funct_3 :::".:"::: ) (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ")" ")" )); end; :: deftheorem defines :::"Pr1"::: BORSUK_1:def 4 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k8_borsuk_1 :::"Pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_funct_3 :::".:"::: ) (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ")" )))); :: deftheorem defines :::"Pr2"::: BORSUK_1:def 5 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k9_borsuk_1 :::"Pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_funct_3 :::".:"::: ) (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ")" )))); theorem :: BORSUK_1:15 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set (Var "H")))) "holds" (Bool "(" (Bool (Set (Var "e")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool "ex" (Set (Var "X1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X"))(Bool "ex" (Set (Var "Y1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k3_borsuk_1 :::":]"::: ) )))) ")" ) ")" )) "holds" (Bool (Set ($#k3_borsuk_1 :::"[:"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set "(" (Set "(" ($#k8_borsuk_1 :::"Pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set "(" (Set "(" ($#k9_borsuk_1 :::"Pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H")) ")" ) ")" ) ($#k3_borsuk_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "A")))))) ; theorem :: BORSUK_1:16 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "C")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k8_borsuk_1 :::"Pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))))) "holds" (Bool "ex" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool "(" (Bool (Set (Var "D")) ($#r2_hidden :::"in"::: ) (Set (Var "H"))) & (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "D")))) ")" ))))) ; theorem :: BORSUK_1:17 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "C")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_borsuk_1 :::"Pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))))) "holds" (Bool "ex" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool "(" (Bool (Set (Var "D")) ($#r2_hidden :::"in"::: ) (Set (Var "H"))) & (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "D")))) ")" ))))) ; theorem :: BORSUK_1:18 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "D")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool "for" (Set (Var "X1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "Y1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "X1")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "D"))))) "implies" (Bool (Set (Var "X1")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" & "(" (Bool (Bool (Set (Var "Y1")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "D"))))) "implies" (Bool (Set (Var "Y1")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ")" ))))) ; theorem :: BORSUK_1:19 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "H")) "is" ($#v1_tops_2 :::"open"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k8_borsuk_1 :::"Pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))) "is" ($#v1_tops_2 :::"open"::: ) ) & (Bool (Set (Set "(" ($#k9_borsuk_1 :::"Pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))) "is" ($#v1_tops_2 :::"open"::: ) ) ")" ))) ; theorem :: BORSUK_1:20 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool (Bool "(" (Bool (Set (Set "(" ($#k8_borsuk_1 :::"Pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool (Set (Set "(" ($#k9_borsuk_1 :::"Pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) "holds" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: BORSUK_1:21 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) (Bool "for" (Set (Var "X1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "Y1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "H")) "is" ($#m1_setfam_1 :::"Cover"::: ) "of" (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k3_borsuk_1 :::":]"::: ) ))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "Y1")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Set "(" ($#k8_borsuk_1 :::"Pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))) "is" ($#m1_setfam_1 :::"Cover"::: ) "of" (Set (Var "X1"))) ")" & "(" (Bool (Bool (Set (Var "X1")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Set "(" ($#k9_borsuk_1 :::"Pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))) "is" ($#m1_setfam_1 :::"Cover"::: ) "of" (Set (Var "Y1"))) ")" ")" ))))) ; theorem :: BORSUK_1:22 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "H")) "is" ($#m1_setfam_1 :::"Cover"::: ) "of" (Set (Var "Y")))) "holds" (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "H"))) & (Bool (Set (Var "F")) "is" ($#m1_setfam_1 :::"Cover"::: ) "of" (Set (Var "Y"))) & (Bool "(" "for" (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "C")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "C")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Y"))) ")" ) ")" ))))) ; theorem :: BORSUK_1:23 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_finset_1 :::"finite"::: ) ) & (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k8_borsuk_1 :::"Pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "H"))))) "holds" (Bool "ex" (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "st" (Bool "(" (Bool (Set (Var "G")) ($#r1_tarski :::"c="::: ) (Set (Var "H"))) & (Bool (Set (Var "G")) "is" ($#v1_finset_1 :::"finite"::: ) ) & (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_borsuk_1 :::"Pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "G")))) ")" ))))) ; theorem :: BORSUK_1:24 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "Y1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k3_borsuk_1 :::":]"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k8_borsuk_1 :::"Pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k3_borsuk_1 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "X1"))) & (Bool (Set (Set "(" ($#k9_borsuk_1 :::"Pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "X1")) "," (Set (Var "Y1")) ($#k3_borsuk_1 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "Y1"))) ")" )))) ; theorem :: BORSUK_1:25 (Bool "for" (Set (Var "Y")) "," (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v2_compts_1 :::"compact"::: ) )) "holds" (Bool "for" (Set (Var "G")) "being" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set ($#k3_borsuk_1 :::"[:"::: ) (Set (Var "A")) "," (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "t")) ($#k6_domain_1 :::"}"::: ) ) ($#k3_borsuk_1 :::":]"::: ) ) (Bool "ex" (Set (Var "V")) "being" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "A"))(Bool "ex" (Set (Var "W")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "t")) "st" (Bool (Set ($#k6_borsuk_1 :::"[:"::: ) (Set (Var "V")) "," (Set (Var "W")) ($#k6_borsuk_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "G"))))))))) ; begin definitionlet "X" be ($#l1_struct_0 :::"1-sorted"::: ) ; func :::"TrivDecomp"::: "X" -> ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") equals :: BORSUK_1:def 6 (Set ($#k8_eqrel_1 :::"Class"::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" )); end; :: deftheorem defines :::"TrivDecomp"::: BORSUK_1:def 6 : (Bool "for" (Set (Var "X")) "being" ($#l1_struct_0 :::"1-sorted"::: ) "holds" (Bool (Set ($#k10_borsuk_1 :::"TrivDecomp"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k8_eqrel_1 :::"Class"::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" )))); registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster (Set ($#k10_borsuk_1 :::"TrivDecomp"::: ) "X") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: BORSUK_1:26 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set ($#k10_borsuk_1 :::"TrivDecomp"::: ) (Set (Var "X"))))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ))))) ; definitionlet "X" be ($#l1_pre_topc :::"TopSpace":::); let "D" be ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); func :::"space"::: "D" -> ($#v1_pre_topc :::"strict"::: ) ($#l1_pre_topc :::"TopSpace":::) means :: BORSUK_1:def 7 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) "D") & (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" it) ($#r1_hidden :::"="::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" "D" : (Bool (Set ($#k3_tarski :::"union"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" "X")) "}" ) ")" ); end; :: deftheorem defines :::"space"::: BORSUK_1:def 7 : (Bool "for" (Set (Var "X")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "b3")) "being" ($#v1_pre_topc :::"strict"::: ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k11_borsuk_1 :::"space"::: ) (Set (Var "D")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "D")) : (Bool (Set ($#k3_tarski :::"union"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "X")))) "}" ) ")" ) ")" )))); registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "D" be ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); cluster (Set ($#k11_borsuk_1 :::"space"::: ) "D") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ; end; theorem :: BORSUK_1:27 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k3_tarski :::"union"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "X")))) "iff" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set "(" ($#k11_borsuk_1 :::"space"::: ) (Set (Var "D")) ")" ))) ")" )))) ; definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); func :::"Proj"::: "D" -> ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set "(" ($#k11_borsuk_1 :::"space"::: ) "D" ")" ) equals :: BORSUK_1:def 8 (Set ($#k13_eqrel_1 :::"proj"::: ) "D"); end; :: deftheorem defines :::"Proj"::: BORSUK_1:def 8 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "holds" (Bool (Set ($#k12_borsuk_1 :::"Proj"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set ($#k13_eqrel_1 :::"proj"::: ) (Set (Var "D")))))); theorem :: BORSUK_1:28 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set (Var "D")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "W"))))))) ; theorem :: BORSUK_1:29 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k11_borsuk_1 :::"space"::: ) (Set (Var "D")) ")" ) (Bool "ex" (Set (Var "W9")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set (Var "D")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "W9"))) ($#r1_hidden :::"="::: ) (Set (Var "W"))))))) ; theorem :: BORSUK_1:30 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set (Var "D")) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k11_borsuk_1 :::"space"::: ) (Set (Var "D")) ")" ))))) ; definitionlet "XX" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Const "XX")); let "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "X"))); func :::"TrivExt"::: "D" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "XX") equals :: BORSUK_1:def 9 (Set "D" ($#k2_xboole_0 :::"\/"::: ) "{" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) where p "is" ($#m1_subset_1 :::"Point":::) "of" "XX" : (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X"))) "}" ); end; :: deftheorem defines :::"TrivExt"::: BORSUK_1:def 9 : (Bool "for" (Set (Var "XX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "XX")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "holds" (Bool (Set ($#k13_borsuk_1 :::"TrivExt"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "D")) ($#k2_xboole_0 :::"\/"::: ) "{" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "XX")) : (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) "}" ))))); theorem :: BORSUK_1:31 (Bool "for" (Set (Var "XX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "XX")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "XX")) "holds" (Bool "(" "not" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set ($#k13_borsuk_1 :::"TrivExt"::: ) (Set (Var "D")))) "or" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) "or" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "XX")) "st" (Bool "(" (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "X"))))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) ")" )) ")" ))))) ; theorem :: BORSUK_1:32 (Bool "for" (Set (Var "XX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "XX")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "XX")) "st" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))))) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k13_borsuk_1 :::"TrivExt"::: ) (Set (Var "D")))))))) ; theorem :: BORSUK_1:33 (Bool "for" (Set (Var "XX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "XX")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "XX")) "st" (Bool (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) "holds" (Bool (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set "(" ($#k13_borsuk_1 :::"TrivExt"::: ) (Set (Var "D")) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set (Var "D")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "W")))))))) ; theorem :: BORSUK_1:34 (Bool "for" (Set (Var "XX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "XX")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "XX")) "st" (Bool (Bool (Bool "not" (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))))) "holds" (Bool (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set "(" ($#k13_borsuk_1 :::"TrivExt"::: ) (Set (Var "D")) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "W")) ($#k6_domain_1 :::"}"::: ) )))))) ; theorem :: BORSUK_1:35 (Bool "for" (Set (Var "XX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "XX")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "W")) "," (Set (Var "W9")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "XX")) "st" (Bool (Bool (Bool "not" (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) & (Bool (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set "(" ($#k13_borsuk_1 :::"TrivExt"::: ) (Set (Var "D")) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set "(" ($#k13_borsuk_1 :::"TrivExt"::: ) (Set (Var "D")) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "W9"))))) "holds" (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "W9"))))))) ; theorem :: BORSUK_1:36 (Bool "for" (Set (Var "XX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "XX")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "XX")) "st" (Bool (Bool (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set "(" ($#k13_borsuk_1 :::"TrivExt"::: ) (Set (Var "D")) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "e"))) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k11_borsuk_1 :::"space"::: ) (Set (Var "D")) ")" )))) "holds" (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))))))) ; theorem :: BORSUK_1:37 (Bool "for" (Set (Var "XX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "XX")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) (Bool "for" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) "holds" (Bool (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set "(" ($#k13_borsuk_1 :::"TrivExt"::: ) (Set (Var "D")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "e"))) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k11_borsuk_1 :::"space"::: ) (Set (Var "D")) ")" ))))))) ; begin definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); mode :::"u.s.c._decomposition"::: "of" "X" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") means :: BORSUK_1:def 10 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "X" "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) it)) "holds" (Bool "for" (Set (Var "V")) "being" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "A")) (Bool "ex" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" "X" "st" (Bool "(" (Bool (Set (Var "W")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "W"))) & (Bool (Set (Var "W")) ($#r1_tarski :::"c="::: ) (Set (Var "V"))) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" "X" "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) it) & (Bool (Set (Var "B")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "W")))) "holds" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "W"))) ")" ) ")" )))); end; :: deftheorem defines :::"u.s.c._decomposition"::: BORSUK_1:def 10 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "b2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m1_borsuk_1 :::"u.s.c._decomposition"::: ) "of" (Set (Var "X"))) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "b2")))) "holds" (Bool "for" (Set (Var "V")) "being" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "A")) (Bool "ex" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "W")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "W"))) & (Bool (Set (Var "W")) ($#r1_tarski :::"c="::: ) (Set (Var "V"))) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) & (Bool (Set (Var "B")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "W")))) "holds" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "W"))) ")" ) ")" )))) ")" ))); theorem :: BORSUK_1:38 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#m1_borsuk_1 :::"u.s.c._decomposition"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k11_borsuk_1 :::"space"::: ) (Set (Var "D")) ")" ) (Bool "for" (Set (Var "G")) "being" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set (Var "D")) ")" ) ($#k8_relset_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "t")) ($#k6_domain_1 :::"}"::: ) )) "holds" (Bool (Set (Set "(" ($#k12_borsuk_1 :::"Proj"::: ) (Set (Var "D")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "G"))) "is" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "t"))))))) ; theorem :: BORSUK_1:39 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k10_borsuk_1 :::"TrivDecomp"::: ) (Set (Var "X"))) "is" ($#m1_borsuk_1 :::"u.s.c._decomposition"::: ) "of" (Set (Var "X")))) ; definitionlet "X" be ($#l1_pre_topc :::"TopSpace":::); let "IT" be ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Const "X")); attr "IT" is :::"closed"::: means :: BORSUK_1:def 11 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "X" "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "IT"))) "holds" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) )); end; :: deftheorem defines :::"closed"::: BORSUK_1:def 11 : (Bool "for" (Set (Var "X")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "IT")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_borsuk_1 :::"closed"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "IT"))))) "holds" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) )) ")" ))); registrationlet "X" be ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_borsuk_1 :::"closed"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_borsuk_1 :::"closed"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; definitionlet "XX" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_borsuk_1 :::"closed"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Const "XX")); let "D" be ($#m1_borsuk_1 :::"u.s.c._decomposition"::: ) "of" (Set (Const "X")); :: original: :::"TrivExt"::: redefine func :::"TrivExt"::: "D" -> ($#m1_borsuk_1 :::"u.s.c._decomposition"::: ) "of" "XX"; end; definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "IT" be ($#m1_borsuk_1 :::"u.s.c._decomposition"::: ) "of" (Set (Const "X")); attr "IT" is :::"DECOMPOSITION-like"::: means :: BORSUK_1:def 12 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "X" "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) "IT")) "holds" (Bool (Set (Var "A")) "is" ($#v2_compts_1 :::"compact"::: ) )); end; :: deftheorem defines :::"DECOMPOSITION-like"::: BORSUK_1:def 12 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "IT")) "being" ($#m1_borsuk_1 :::"u.s.c._decomposition"::: ) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v2_borsuk_1 :::"DECOMPOSITION-like"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "IT")))) "holds" (Bool (Set (Var "A")) "is" ($#v2_compts_1 :::"compact"::: ) )) ")" ))); registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ($#v2_borsuk_1 :::"DECOMPOSITION-like"::: ) for ($#m1_borsuk_1 :::"u.s.c._decomposition"::: ) "of" "X"; end; definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); mode DECOMPOSITION of "X" is ($#v2_borsuk_1 :::"DECOMPOSITION-like"::: ) ($#m1_borsuk_1 :::"u.s.c._decomposition"::: ) "of" "X"; end; definitionlet "XX" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_borsuk_1 :::"closed"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Const "XX")); let "D" be ($#m1_borsuk_1 :::"DECOMPOSITION":::) "of" (Set (Const "X")); :: original: :::"TrivExt"::: redefine func :::"TrivExt"::: "D" -> ($#m1_borsuk_1 :::"DECOMPOSITION":::) "of" "XX"; end; definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_borsuk_1 :::"closed"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Const "X")); let "D" be ($#m1_borsuk_1 :::"DECOMPOSITION":::) "of" (Set (Const "Y")); :: original: :::"space"::: redefine func :::"space"::: "D" -> ($#v1_pre_topc :::"strict"::: ) ($#v1_borsuk_1 :::"closed"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set ($#k11_borsuk_1 :::"space"::: ) (Set "(" ($#k15_borsuk_1 :::"TrivExt"::: ) "D" ")" )); end; begin definitionfunc :::"I[01]"::: -> ($#l1_pre_topc :::"TopStruct"::: ) means :: BORSUK_1:def 13 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set ($#k8_metric_1 :::"RealSpace"::: ) ) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool it ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set ($#k8_metric_1 :::"RealSpace"::: ) ) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "P"))))); end; :: deftheorem defines :::"I[01]"::: BORSUK_1:def 13 : (Bool "for" (Set (Var "b1")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k17_borsuk_1 :::"I[01]"::: ) )) "iff" (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set ($#k8_metric_1 :::"RealSpace"::: ) ) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set ($#k8_metric_1 :::"RealSpace"::: ) ) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "P"))))) ")" )); registration cluster (Set ($#k17_borsuk_1 :::"I[01]"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ; end; theorem :: BORSUK_1:40 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k17_borsuk_1 :::"I[01]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) )) ; definitionfunc :::"0[01]"::: -> ($#m1_subset_1 :::"Point":::) "of" (Set ($#k17_borsuk_1 :::"I[01]"::: ) ) equals :: BORSUK_1:def 14 (Set ($#k6_numbers :::"0"::: ) ); func :::"1[01]"::: -> ($#m1_subset_1 :::"Point":::) "of" (Set ($#k17_borsuk_1 :::"I[01]"::: ) ) equals :: BORSUK_1:def 15 (Num 1); end; :: deftheorem defines :::"0[01]"::: BORSUK_1:def 14 : (Bool (Set ($#k18_borsuk_1 :::"0[01]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )); :: deftheorem defines :::"1[01]"::: BORSUK_1:def 15 : (Bool (Set ($#k19_borsuk_1 :::"1[01]"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)); definitionlet "A" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "B" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Const "A")); let "F" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set (Const "B")); attr "F" is :::"being_a_retraction"::: means :: BORSUK_1:def 16 (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Point":::) "of" "A" "st" (Bool (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "B"))) "holds" (Bool (Set "F" ($#k3_funct_2 :::"."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Var "W")))); end; :: deftheorem defines :::"being_a_retraction"::: BORSUK_1:def 16 : (Bool "for" (Set (Var "A")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set (Var "B")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v3_borsuk_1 :::"being_a_retraction"::: ) ) "iff" (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "B"))))) "holds" (Bool (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Var "W")))) ")" )))); definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Const "X")); pred "Y" :::"is_a_retract_of"::: "X" means :: BORSUK_1:def 17 (Bool "ex" (Set (Var "F")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" "X" "," "Y" "st" (Bool (Set (Var "F")) "is" ($#v3_borsuk_1 :::"being_a_retraction"::: ) )); pred "Y" :::"is_an_SDR_of"::: "X" means :: BORSUK_1:def 18 (Bool "ex" (Set (Var "H")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) "X" "," (Set ($#k17_borsuk_1 :::"I[01]"::: ) ) ($#k2_borsuk_1 :::":]"::: ) ) "," "X" "st" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Point":::) "of" "X" "holds" (Bool "(" (Bool (Set (Set (Var "H")) ($#k3_funct_2 :::"."::: ) (Set ($#k4_borsuk_1 :::"["::: ) (Set (Var "A")) "," (Set ($#k18_borsuk_1 :::"0[01]"::: ) ) ($#k4_borsuk_1 :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "H")) ($#k3_funct_2 :::"."::: ) (Set ($#k4_borsuk_1 :::"["::: ) (Set (Var "A")) "," (Set ($#k19_borsuk_1 :::"1[01]"::: ) ) ($#k4_borsuk_1 :::"]"::: ) )) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")) & "(" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y"))) "implies" (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k17_borsuk_1 :::"I[01]"::: ) ) "holds" (Bool (Set (Set (Var "H")) ($#k3_funct_2 :::"."::: ) (Set ($#k4_borsuk_1 :::"["::: ) (Set (Var "A")) "," (Set (Var "T")) ($#k4_borsuk_1 :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "A")))) ")" ")" ))); end; :: deftheorem defines :::"is_a_retract_of"::: BORSUK_1:def 17 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "Y")) ($#r1_borsuk_1 :::"is_a_retract_of"::: ) (Set (Var "X"))) "iff" (Bool "ex" (Set (Var "F")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "st" (Bool (Set (Var "F")) "is" ($#v3_borsuk_1 :::"being_a_retraction"::: ) )) ")" ))); :: deftheorem defines :::"is_an_SDR_of"::: BORSUK_1:def 18 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "Y")) ($#r2_borsuk_1 :::"is_an_SDR_of"::: ) (Set (Var "X"))) "iff" (Bool "ex" (Set (Var "H")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set ($#k17_borsuk_1 :::"I[01]"::: ) ) ($#k2_borsuk_1 :::":]"::: ) ) "," (Set (Var "X")) "st" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Set (Var "H")) ($#k3_funct_2 :::"."::: ) (Set ($#k4_borsuk_1 :::"["::: ) (Set (Var "A")) "," (Set ($#k18_borsuk_1 :::"0[01]"::: ) ) ($#k4_borsuk_1 :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "H")) ($#k3_funct_2 :::"."::: ) (Set ($#k4_borsuk_1 :::"["::: ) (Set (Var "A")) "," (Set ($#k19_borsuk_1 :::"1[01]"::: ) ) ($#k4_borsuk_1 :::"]"::: ) )) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y")))) & "(" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))))) "implies" (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k17_borsuk_1 :::"I[01]"::: ) ) "holds" (Bool (Set (Set (Var "H")) ($#k3_funct_2 :::"."::: ) (Set ($#k4_borsuk_1 :::"["::: ) (Set (Var "A")) "," (Set (Var "T")) ($#k4_borsuk_1 :::"]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "A")))) ")" ")" ))) ")" ))); theorem :: BORSUK_1:41 (Bool "for" (Set (Var "XX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_borsuk_1 :::"closed"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "XX")) (Bool "for" (Set (Var "D")) "being" ($#m1_borsuk_1 :::"DECOMPOSITION":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "X")) ($#r1_borsuk_1 :::"is_a_retract_of"::: ) (Set (Var "XX")))) "holds" (Bool (Set ($#k16_borsuk_1 :::"space"::: ) (Set (Var "D"))) ($#r1_borsuk_1 :::"is_a_retract_of"::: ) (Set ($#k11_borsuk_1 :::"space"::: ) (Set "(" ($#k15_borsuk_1 :::"TrivExt"::: ) (Set (Var "D")) ")" )))))) ; theorem :: BORSUK_1:42 (Bool "for" (Set (Var "XX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_borsuk_1 :::"closed"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "XX")) (Bool "for" (Set (Var "D")) "being" ($#m1_borsuk_1 :::"DECOMPOSITION":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "X")) ($#r2_borsuk_1 :::"is_an_SDR_of"::: ) (Set (Var "XX")))) "holds" (Bool (Set ($#k16_borsuk_1 :::"space"::: ) (Set (Var "D"))) ($#r2_borsuk_1 :::"is_an_SDR_of"::: ) (Set ($#k11_borsuk_1 :::"space"::: ) (Set "(" ($#k15_borsuk_1 :::"TrivExt"::: ) (Set (Var "D")) ")" )))))) ; theorem :: BORSUK_1:43 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) "iff" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k17_borsuk_1 :::"I[01]"::: ) ))) ")" )) ;