:: BORSUK_4 semantic presentation begin registration cluster ($#v1_topreal2 :::"being_simple_closed_curve"::: ) -> ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))); end; theorem :: BORSUK_4:1 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ))))) ; theorem :: BORSUK_4:2 (Bool "for" (Set (Var "X")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "q")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set (Var "p")))))) ; theorem :: BORSUK_4:3 (Bool "for" (Set (Var "T")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "X")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "q")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set (Var "p"))) ")" ))))) ; theorem :: BORSUK_4:4 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "a")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a")) ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; theorem :: BORSUK_4:5 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "a")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g")) ")" ) ($#k2_funct_1 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "g")) ($#k2_funct_1 :::"""::: ) ")" ))))) ; theorem :: BORSUK_4:6 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p")) "," (Set (Var "q")))) "holds" (Bool "not" (Bool (Set (Set (Var "A")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_xboole_0 :::"empty"::: ) ))))) ; theorem :: BORSUK_4:7 (Bool "for" (Set (Var "s1")) "," (Set (Var "s3")) "," (Set (Var "s4")) "," (Set (Var "l")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s3"))) & (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s4"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "l"))) & (Bool (Set (Var "l")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "l")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "s3")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "l")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "s4")) ")" )))) ; theorem :: BORSUK_4:8 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ))))) ; theorem :: BORSUK_4:9 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k4_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k4_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ))))) ; theorem :: BORSUK_4:10 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k3_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ))))) ; theorem :: BORSUK_4:11 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool (Set ($#k6_measure6 :::"Cl"::: ) (Set ($#k4_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k4_rcomp_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ))) ; theorem :: BORSUK_4:12 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool (Set ($#k6_measure6 :::"Cl"::: ) (Set ($#k3_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_rcomp_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ))) ; theorem :: BORSUK_4:13 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) )))) ; theorem :: BORSUK_4:14 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k4_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k4_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) )))) ; theorem :: BORSUK_4:15 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k3_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) )))) ; theorem :: BORSUK_4:16 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))) ; theorem :: BORSUK_4:17 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "c"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k3_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k4_rcomp_1 :::"]."::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k4_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) ))) ; theorem :: BORSUK_4:18 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "holds" (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_rcomp_1 :::".]"::: ) ) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ))) ; theorem :: BORSUK_4:19 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "holds" (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ))) ; begin theorem :: BORSUK_4:20 (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "p")) ($#k1_seq_4 :::"}"::: ) ) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ))) ; theorem :: BORSUK_4:21 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_connsp_1 :::"connected"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) ; theorem :: BORSUK_4:22 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_connsp_1 :::"connected"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "A"))))) ; theorem :: BORSUK_4:23 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: BORSUK_4:24 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p2")))) "holds" (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_rcomp_1 :::".]"::: ) ) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ))) ; theorem :: BORSUK_4:25 (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "X9")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X9")) ($#r1_hidden :::"="::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Var "X9")) "is" ($#v4_xxreal_2 :::"bounded_above"::: ) ) & (Bool (Set (Var "X9")) "is" ($#v3_xxreal_2 :::"bounded_below"::: ) ) ")" ))) ; theorem :: BORSUK_4:26 (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "X9")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X9"))) & (Bool (Set (Var "X9")) ($#r1_hidden :::"="::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "X9"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "x"))) & (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "X9")))) ")" )))) ; theorem :: BORSUK_4:27 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_rcomp_1 :::"closed"::: ) ) "iff" (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ))) ; theorem :: BORSUK_4:28 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "C"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "C"))))) ; theorem :: BORSUK_4:29 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "C9")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "C9"))) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "C9")) ")" ) "," (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "C9")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "C9")))) "holds" (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "C9")) ")" ) "," (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "C9")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "C9"))))) ; theorem :: BORSUK_4:30 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "holds" (Bool (Set (Var "C")) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ))) ; theorem :: BORSUK_4:31 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool "(" (Bool (Set (Var "p1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p2"))) & (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_rcomp_1 :::".]"::: ) )) ")" ))) ; begin definitionfunc :::"I(01)"::: -> ($#v1_pre_topc :::"strict"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) means :: BORSUK_4:def 1 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )); end; :: deftheorem defines :::"I(01)"::: BORSUK_4:def 1 : (Bool "for" (Set (Var "b1")) "being" ($#v1_pre_topc :::"strict"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k1_borsuk_4 :::"I(01)"::: ) )) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b1"))) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) ")" )); registration cluster (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ; end; theorem :: BORSUK_4:32 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) )))) "holds" (Bool (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k5_topmetr :::"I[01]"::: ) ) ($#k1_pre_topc :::"|"::: ) (Set (Var "A"))))) ; theorem :: BORSUK_4:33 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k5_topmetr :::"I[01]"::: ) )) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k2_tarski :::"}"::: ) ))) ; registration cluster (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) -> ($#v1_pre_topc :::"strict"::: ) ($#v1_tsep_1 :::"open"::: ) ; end; theorem :: BORSUK_4:34 (Bool (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) "is" ($#v1_tsep_1 :::"open"::: ) ) ; theorem :: BORSUK_4:35 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) ))) "iff" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) ; theorem :: BORSUK_4:36 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool (Set ($#k4_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k4_rcomp_1 :::".]"::: ) ) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) ))) ; theorem :: BORSUK_4:37 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k3_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_rcomp_1 :::".["::: ) ) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) ))) ; theorem :: BORSUK_4:38 (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) )) "," (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "D"))) ($#r2_borsuk_3 :::"are_homeomorphic"::: ) )) ; theorem :: BORSUK_4:39 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "D")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) "," (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" (Set (Var "D")) ($#k7_subset_1 :::"\"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k7_domain_1 :::"}"::: ) ) ")" )) ($#r1_borsuk_3 :::"are_homeomorphic"::: ) )))) ; theorem :: BORSUK_4:40 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "D")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "D"))) ($#r1_borsuk_3 :::"are_homeomorphic"::: ) )))) ; theorem :: BORSUK_4:41 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2")))) "holds" (Bool (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ")" )) ($#r1_borsuk_3 :::"are_homeomorphic"::: ) ))) ; theorem :: BORSUK_4:42 (Bool "for" (Set (Var "E")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) "st" (Bool (Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool "(" (Bool (Set (Var "p1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p2"))) & (Bool (Set (Var "E")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_rcomp_1 :::".]"::: ) )) ")" ))) "holds" (Bool (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) ($#k1_pre_topc :::"|"::: ) (Set (Var "E"))) ($#r1_borsuk_3 :::"are_homeomorphic"::: ) )) ; theorem :: BORSUK_4:43 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p")) "," (Set (Var "q"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b")))) "holds" (Bool "ex" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) )(Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set ($#k5_topmetr :::"I[01]"::: ) ) ($#k1_pre_topc :::"|"::: ) (Set (Var "E")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "A")) ")" ) "st" (Bool "(" (Bool (Set (Var "E")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) )) & (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ))))))) ; theorem :: BORSUK_4:44 (Bool "for" (Set (Var "A")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "C")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "C")) "is" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "B")))) "holds" (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "A")) ($#k1_pre_topc :::"|"::: ) (Set (Var "X")) ")" ) "," (Set (Var "C")) "st" (Bool (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "h")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))))) ; theorem :: BORSUK_4:45 (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set (Var "X")) "is" ($#v3_pre_topc :::"open"::: ) ))) ; theorem :: BORSUK_4:46 (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set (Var "X")) "is" ($#v3_pre_topc :::"open"::: ) ))) ; theorem :: BORSUK_4:47 (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k4_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "a")) ($#k4_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Var "X")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: BORSUK_4:48 (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k3_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Num 1) ($#k3_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set (Var "X")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: BORSUK_4:49 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p")) "," (Set (Var "q"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) )(Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) ($#k1_pre_topc :::"|"::: ) (Set (Var "E")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" (Set (Var "A")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) "st" (Bool "(" (Bool (Set (Var "E")) ($#r1_hidden :::"="::: ) (Set ($#k4_rcomp_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k4_rcomp_1 :::".]"::: ) )) & (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ))))))) ; theorem :: BORSUK_4:50 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p")) "," (Set (Var "q"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) )(Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) ($#k1_pre_topc :::"|"::: ) (Set (Var "E")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" (Set (Var "A")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "q")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) "st" (Bool "(" (Bool (Set (Var "E")) ($#r1_hidden :::"="::: ) (Set ($#k3_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k3_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ))))))) ; theorem :: BORSUK_4:51 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p")) "," (Set (Var "q"))) & (Bool (Set (Var "B")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "q")) "," (Set (Var "p"))) & (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k7_domain_1 :::"}"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) "," (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" (Set "(" (Set (Var "A")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set (Var "B")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" )) ($#r1_borsuk_3 :::"are_homeomorphic"::: ) )))) ; theorem :: BORSUK_4:52 (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" (Set (Var "D")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) ")" )) "," (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) ($#r1_borsuk_3 :::"are_homeomorphic"::: ) ))) ; theorem :: BORSUK_4:53 (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" (Set (Var "D")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) ")" )) "," (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" (Set (Var "D")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "q")) ($#k6_domain_1 :::"}"::: ) ) ")" )) ($#r1_borsuk_3 :::"are_homeomorphic"::: ) ))) ; theorem :: BORSUK_4:54 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "E")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) "st" (Bool (Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool "(" (Bool (Set (Var "p1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p2"))) & (Bool (Set (Var "E")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_rcomp_1 :::".]"::: ) )) ")" )) & (Bool (Set (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) ($#k1_pre_topc :::"|"::: ) (Set (Var "E"))) "," (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "C"))) ($#r1_borsuk_3 :::"are_homeomorphic"::: ) )) "holds" (Bool "ex" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Set (Var "C")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "s1")) "," (Set (Var "s2"))))))) ; theorem :: BORSUK_4:55 (Bool "for" (Set (Var "Dp")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "Dp")) ")" ) "," (Set ($#k1_borsuk_4 :::"I(01)"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "Dp"))) & (Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool "(" (Bool (Set (Var "p1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_rcomp_1 :::".]"::: ) )) ")" ))) "holds" (Bool "ex" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Set (Var "C")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "s1")) "," (Set (Var "s2"))))))) ; theorem :: BORSUK_4:56 (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "D"))) "or" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "D"))) "or" (Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Set (Var "C")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "or" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ))) ")" ))) ; begin theorem :: BORSUK_4:57 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "ex" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "D"))) & (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set ($#k3_seq_4 :::"lower_bound"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "D")))) & (Bool (Set ($#k2_seq_4 :::"upper_bound"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "D")))) ")" ))) ; theorem :: BORSUK_4:58 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool "(" (Bool (Set (Var "p1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p2"))) & (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_rcomp_1 :::".]"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) ")" ))) ; theorem :: BORSUK_4:59 (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "C")) "being" ($#v4_pre_topc :::"closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "C")) ($#r2_xboole_0 :::"c<"::: ) (Set (Var "D")))) "holds" (Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )(Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "B")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "D"))) ")" ))))) ;