:: TOPALG_6 semantic presentation begin registrationlet "S" be ($#l1_pre_topc :::"TopSpace":::); let "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v3_funct_1 :::"constant"::: ) ($#v1_funct_2 :::"quasi_total"::: ) -> ($#v5_pre_topc :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: TOPALG_6:1 (Bool (Set ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" ) ($#r2_funct_2 :::"="::: ) (Set ($#k3_struct_0 :::"id"::: ) (Set "(" ($#k4_topmetr :::"Closed-Interval-TSpace"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" ")" ))) ; theorem :: TOPALG_6:2 (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "," (Set (Var "r3")) "," (Set (Var "r4")) "," (Set (Var "r5")) "," (Set (Var "r6")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "r3")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r4"))) & (Bool (Set (Var "r5")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r6")))) "holds" (Bool (Set (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set (Var "r1")) "," (Set (Var "r2")) "," (Set (Var "r3")) "," (Set (Var "r4")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set (Var "r5")) "," (Set (Var "r6")) "," (Set (Var "r1")) "," (Set (Var "r2")) ")" ")" )) ($#r2_funct_2 :::"="::: ) (Set ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set (Var "r5")) "," (Set (Var "r6")) "," (Set (Var "r3")) "," (Set (Var "r4")) ")" ))) ; registrationlet "n" be ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"Nat":::); cluster (Set ($#k15_euclid :::"TOP-REAL"::: ) "n") -> ($#v8_struct_0 :::"infinite"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) "n" ($#v2_mfold_1 :::"-locally_euclidean"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v8_struct_0 :::"infinite"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: TOPALG_6:3 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal9 :::"Sphere"::: ) "(" (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ")" ) "," (Num 1) ")" ))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k3_topreal9 :::"Sphere"::: ) "(" (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ")" ) "," (Num 1) ")" ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: TOPALG_6:4 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "t1")) "," (Set (Var "t2")) "holds" (Bool "(" (Bool (Set ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ")" )))) ; theorem :: TOPALG_6:5 (Bool "for" (Set (Var "T")) "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 "T")) (Bool "for" (Set (Var "C1")) "," (Set (Var "C2")) "being" ($#v3_funct_1 :::"constant"::: ) ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Var "t")) "holds" (Bool (Set (Var "C1")) "," (Set (Var "C2")) ($#r3_borsuk_2 :::"are_homotopic"::: ) )))) ; theorem :: TOPALG_6:6 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "T")) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "t1")) "," (Set (Var "t2")) (Bool "for" (Set (Var "C")) "," (Set (Var "D")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "s1")) "," (Set (Var "s2")) "st" (Bool (Bool (Set (Var "s1")) "," (Set (Var "s2")) ($#r1_borsuk_6 :::"are_connected"::: ) ) & (Bool (Set (Var "t1")) "," (Set (Var "t2")) ($#r1_borsuk_6 :::"are_connected"::: ) ) & (Bool (Set (Var "A")) ($#r1_funct_2 :::"="::: ) (Set (Var "C"))) & (Bool (Set (Var "B")) ($#r1_funct_2 :::"="::: ) (Set (Var "D"))) & (Bool (Set (Var "C")) "," (Set (Var "D")) ($#r3_borsuk_2 :::"are_homotopic"::: ) )) "holds" (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r3_borsuk_2 :::"are_homotopic"::: ) ))))))) ; theorem :: TOPALG_6:7 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "T")) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "t1")) "," (Set (Var "t2")) (Bool "for" (Set (Var "C")) "," (Set (Var "D")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "s1")) "," (Set (Var "s2")) "st" (Bool (Bool (Set (Var "s1")) "," (Set (Var "s2")) ($#r1_borsuk_6 :::"are_connected"::: ) ) & (Bool (Set (Var "t1")) "," (Set (Var "t2")) ($#r1_borsuk_6 :::"are_connected"::: ) ) & (Bool (Set (Var "A")) ($#r1_funct_2 :::"="::: ) (Set (Var "C"))) & (Bool (Set (Var "B")) ($#r1_funct_2 :::"="::: ) (Set (Var "D"))) & (Bool (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k3_topalg_1 :::"EqRel"::: ) "(" (Set (Var "S")) "," (Set (Var "s1")) "," (Set (Var "s2")) ")" ")" ) "," (Set (Var "C")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k3_topalg_1 :::"EqRel"::: ) "(" (Set (Var "S")) "," (Set (Var "s1")) "," (Set (Var "s2")) ")" ")" ) "," (Set (Var "D")) ")" ))) "holds" (Bool (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k3_topalg_1 :::"EqRel"::: ) "(" (Set (Var "T")) "," (Set (Var "t1")) "," (Set (Var "t2")) ")" ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k3_topalg_1 :::"EqRel"::: ) "(" (Set (Var "T")) "," (Set (Var "t1")) "," (Set (Var "t2")) ")" ")" ) "," (Set (Var "B")) ")" )))))))) ; theorem :: TOPALG_6:8 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "L")) "being" ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Var "t")) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k5_topalg_1 :::"pi_1"::: ) "(" (Set (Var "T")) "," (Set (Var "t")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k4_topalg_1 :::"EqRel"::: ) "(" (Set (Var "T")) "," (Set (Var "t")) ")" ")" ) "," (Set (Var "L")) ")" ")" ) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: TOPALG_6:9 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )))) "holds" (Bool (Set (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "S"))) "is" ($#v1_borsuk_2 :::"pathwise_connected"::: ) )))) ; theorem :: TOPALG_6:10 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "t")) ($#k1_tarski :::"}"::: ) ))) & (Bool (Set ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n"))) "," (Set (Var "T")) ($#r1_mfold_2 :::"are_homeomorphic"::: ) )) "holds" (Bool (Set (Set (Var "T")) ($#k1_pre_topc :::"|"::: ) (Set (Var "S"))) "is" ($#v1_borsuk_2 :::"pathwise_connected"::: ) ))))) ; registrationlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "p", "q" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); cluster (Set ($#k3_mfold_2 :::"TPlane"::: ) "(" "p" "," "q" ")" ) -> ($#v1_topalg_2 :::"convex"::: ) ; end; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); attr "T" is :::"having_trivial_Fundamental_Group"::: means :: TOPALG_6:def 1 (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" "T" "holds" (Bool (Set ($#k5_topalg_1 :::"pi_1"::: ) "(" "T" "," (Set (Var "t")) ")" ) "is" ($#v7_struct_0 :::"trivial"::: ) )); end; :: deftheorem defines :::"having_trivial_Fundamental_Group"::: TOPALG_6:def 1 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) ) "iff" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_topalg_1 :::"pi_1"::: ) "(" (Set (Var "T")) "," (Set (Var "t")) ")" ) "is" ($#v7_struct_0 :::"trivial"::: ) )) ")" )); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); attr "T" is :::"simply_connected"::: means :: TOPALG_6:def 2 (Bool "(" (Bool "T" "is" ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) ) & (Bool "T" "is" ($#v1_borsuk_2 :::"pathwise_connected"::: ) ) ")" ); end; :: deftheorem defines :::"simply_connected"::: TOPALG_6:def 2 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v2_topalg_6 :::"simply_connected"::: ) ) "iff" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) ) & (Bool (Set (Var "T")) "is" ($#v1_borsuk_2 :::"pathwise_connected"::: ) ) ")" ) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v2_topalg_6 :::"simply_connected"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_borsuk_2 :::"pathwise_connected"::: ) ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_borsuk_2 :::"pathwise_connected"::: ) ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_topalg_6 :::"simply_connected"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: TOPALG_6:11 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) )) "holds" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Var "t")) "holds" (Bool (Set (Var "P")) "," (Set (Var "Q")) ($#r3_borsuk_2 :::"are_homotopic"::: ) )))) ; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster (Set ($#k15_euclid :::"TOP-REAL"::: ) "n") -> ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: TOPALG_6:12 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v2_topalg_6 :::"simply_connected"::: ) ) "iff" (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "t1")) "," (Set (Var "t2")) ($#r1_borsuk_6 :::"are_connected"::: ) ) & (Bool "(" "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "t1")) "," (Set (Var "t2")) "holds" (Bool (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k3_topalg_1 :::"EqRel"::: ) "(" (Set (Var "T")) "," (Set (Var "t1")) "," (Set (Var "t2")) ")" ")" ) "," (Set (Var "P")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k3_topalg_1 :::"EqRel"::: ) "(" (Set (Var "T")) "," (Set (Var "t1")) "," (Set (Var "t2")) ")" ")" ) "," (Set (Var "Q")) ")" )) ")" ) ")" )) ")" )) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) ($#l1_pre_topc :::"TopSpace":::); let "t" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); cluster (Set ($#k5_topalg_1 :::"FundamentalGroup"::: ) "(" "T" "," "t" ")" ) -> ($#v7_struct_0 :::"trivial"::: ) ; end; theorem :: TOPALG_6:13 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "S")) "," (Set (Var "T")) ($#r1_mfold_2 :::"are_homeomorphic"::: ) ) & (Bool (Set (Var "S")) "is" ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) )) "holds" (Bool (Set (Var "T")) "is" ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) )) ; theorem :: TOPALG_6:14 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "S")) "," (Set (Var "T")) ($#r1_mfold_2 :::"are_homeomorphic"::: ) ) & (Bool (Set (Var "S")) "is" ($#v2_topalg_6 :::"simply_connected"::: ) )) "holds" (Bool (Set (Var "T")) "is" ($#v2_topalg_6 :::"simply_connected"::: ) )) ; theorem :: TOPALG_6:15 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "P1")) "," (Set (Var "P2")) "being" ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Var "t")) "holds" (Bool (Set (Var "P1")) "," (Set (Var "P2")) ($#r3_borsuk_2 :::"are_homotopic"::: ) )))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "t" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); let "l" be ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Const "t")); attr "l" is :::"nullhomotopic"::: means :: TOPALG_6:def 3 (Bool "ex" (Set (Var "c")) "being" ($#v3_funct_1 :::"constant"::: ) ($#m1_borsuk_2 :::"Loop":::) "of" "t" "st" (Bool "l" "," (Set (Var "c")) ($#r3_borsuk_2 :::"are_homotopic"::: ) )); end; :: deftheorem defines :::"nullhomotopic"::: TOPALG_6:def 3 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "l")) "being" ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Var "t")) "holds" (Bool "(" (Bool (Set (Var "l")) "is" ($#v3_topalg_6 :::"nullhomotopic"::: ) ) "iff" (Bool "ex" (Set (Var "c")) "being" ($#v3_funct_1 :::"constant"::: ) ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Var "t")) "st" (Bool (Set (Var "l")) "," (Set (Var "c")) ($#r3_borsuk_2 :::"are_homotopic"::: ) )) ")" )))); registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "t" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); cluster ($#v3_funct_1 :::"constant"::: ) -> ($#v3_topalg_6 :::"nullhomotopic"::: ) for ($#m1_borsuk_2 :::"Path"::: ) "of" "t" "," "t"; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "t" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k5_topmetr :::"I[01]"::: ) )) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v3_funct_1 :::"constant"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v5_pre_topc :::"continuous"::: ) for ($#m1_borsuk_2 :::"Path"::: ) "of" "t" "," "t"; end; theorem :: TOPALG_6:16 (Bool "for" (Set (Var "T")) "," (Set (Var "U")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "f")) "being" ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Var "t")) (Bool "for" (Set (Var "g")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set (Var "U")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_topalg_6 :::"nullhomotopic"::: ) )) "holds" (Bool (Set (Set (Var "g")) ($#k2_topalg_3 :::"*"::: ) (Set (Var "f"))) "is" ($#v3_topalg_6 :::"nullhomotopic"::: ) ))))) ; registrationlet "T", "U" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "t" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); let "f" be ($#v3_topalg_6 :::"nullhomotopic"::: ) ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Const "t")); let "g" be ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "T")) "," (Set (Const "U")); cluster (Set "f" ($#k3_relat_1 :::"*"::: ) "g") -> ($#v3_topalg_6 :::"nullhomotopic"::: ) for ($#m1_borsuk_2 :::"Loop":::) "of" (Set "g" ($#k3_funct_2 :::"."::: ) "t"); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) ($#l1_pre_topc :::"TopSpace":::); let "t" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); cluster -> ($#v3_topalg_6 :::"nullhomotopic"::: ) for ($#m1_borsuk_2 :::"Path"::: ) "of" "t" "," "t"; end; theorem :: TOPALG_6:17 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool "(" "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "f")) "being" ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Var "t")) "holds" (Bool (Set (Var "f")) "is" ($#v3_topalg_6 :::"nullhomotopic"::: ) )) ")" )) "holds" (Bool (Set (Var "T")) "is" ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) )) ; registrationlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "p", "q" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); cluster (Set ($#k3_mfold_2 :::"TPlane"::: ) "(" "p" "," "q" ")" ) -> ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) ; end; theorem :: TOPALG_6:18 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "T")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Var "t")) (Bool "for" (Set (Var "g")) "being" ($#m1_borsuk_2 :::"Loop":::) "of" (Set (Var "s")) "st" (Bool (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool (Set (Var "f")) ($#r1_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) "is" ($#v3_topalg_6 :::"nullhomotopic"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v3_topalg_6 :::"nullhomotopic"::: ) ))))))) ; begin definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) ) "," (Set (Const "T")); attr "f" is :::"parametrized-curve"::: means :: TOPALG_6:def 4 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) "f") "is" ($#v6_xxreal_2 :::"interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "ex" (Set (Var "S")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) )(Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," "T" "st" (Bool "(" (Bool "f" ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_topalg_2 :::"R^1"::: ) ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ))) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))) ")" ); end; :: deftheorem defines :::"parametrized-curve"::: TOPALG_6:def 4 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) ) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v4_topalg_6 :::"parametrized-curve"::: ) ) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) "is" ($#v6_xxreal_2 :::"interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "ex" (Set (Var "S")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) )(Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_topalg_2 :::"R^1"::: ) ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))) ")" ) ")" ))); registrationlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) )) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v4_topalg_6 :::"parametrized-curve"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) )) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: TOPALG_6:19 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) "is" ($#v4_topalg_6 :::"parametrized-curve"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) ) "," (Set (Var "T")))) ; definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; func :::"Curves"::: "T" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_partfun1 :::"PFuncs"::: ) "(" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k2_struct_0 :::"[#]"::: ) "T" ")" ) ")" ")" ) equals :: TOPALG_6:def 5 "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_partfun1 :::"PFuncs"::: ) "(" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k2_struct_0 :::"[#]"::: ) "T" ")" ) ")" ) : (Bool (Set (Var "f")) "is" ($#v4_topalg_6 :::"parametrized-curve"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) ) "," "T") "}" ; end; :: deftheorem defines :::"Curves"::: TOPALG_6:def 5 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_partfun1 :::"PFuncs"::: ) "(" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) : (Bool (Set (Var "f")) "is" ($#v4_topalg_6 :::"parametrized-curve"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) ) "," (Set (Var "T"))) "}" )); registrationlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; cluster (Set ($#k1_topalg_6 :::"Curves"::: ) "T") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; mode Curve of "T" is ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) "T"); end; theorem :: TOPALG_6:20 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#v4_topalg_6 :::"parametrized-curve"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) ) "," (Set (Var "T")) "holds" (Bool (Set (Var "f")) "is" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T"))))) ; theorem :: TOPALG_6:21 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) "is" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")))) ; theorem :: TOPALG_6:22 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "t1")) "," (Set (Var "t2")) "st" (Bool (Bool (Set (Var "t1")) "," (Set (Var "t2")) ($#r1_borsuk_2 :::"are_connected"::: ) )) "holds" (Bool (Set (Var "p")) "is" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")))))) ; theorem :: TOPALG_6:23 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool (Set (Var "c")) "is" ($#v4_topalg_6 :::"parametrized-curve"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) ) "," (Set (Var "T"))))) ; theorem :: TOPALG_6:24 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "c"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")))) ")" ))) ; registrationlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "c" be ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")); cluster (Set ($#k9_xtuple_0 :::"dom"::: ) "c") -> ($#v3_membered :::"real-membered"::: ) ; end; definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "c" be ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")); attr "c" is :::"with_first_point"::: means :: TOPALG_6:def 6 (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) "c") "is" ($#v1_xxreal_2 :::"left_end"::: ) ); attr "c" is :::"with_last_point"::: means :: TOPALG_6:def 7 (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) "c") "is" ($#v2_xxreal_2 :::"right_end"::: ) ); end; :: deftheorem defines :::"with_first_point"::: TOPALG_6:def 6 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "c")) "is" ($#v5_topalg_6 :::"with_first_point"::: ) ) "iff" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c"))) "is" ($#v1_xxreal_2 :::"left_end"::: ) ) ")" ))); :: deftheorem defines :::"with_last_point"::: TOPALG_6:def 7 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "c")) "is" ($#v6_topalg_6 :::"with_last_point"::: ) ) "iff" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c"))) "is" ($#v2_xxreal_2 :::"right_end"::: ) ) ")" ))); definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "c" be ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")); attr "c" is :::"with_endpoints"::: means :: TOPALG_6:def 8 (Bool "(" (Bool "c" "is" ($#v5_topalg_6 :::"with_first_point"::: ) ) & (Bool "c" "is" ($#v6_topalg_6 :::"with_last_point"::: ) ) ")" ); end; :: deftheorem defines :::"with_endpoints"::: TOPALG_6:def 8 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "c")) "is" ($#v7_topalg_6 :::"with_endpoints"::: ) ) "iff" (Bool "(" (Bool (Set (Var "c")) "is" ($#v5_topalg_6 :::"with_first_point"::: ) ) & (Bool (Set (Var "c")) "is" ($#v6_topalg_6 :::"with_last_point"::: ) ) ")" ) ")" ))); registrationlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#v5_topalg_6 :::"with_first_point"::: ) ($#v6_topalg_6 :::"with_last_point"::: ) -> ($#v7_topalg_6 :::"with_endpoints"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) "T"); cluster ($#v7_topalg_6 :::"with_endpoints"::: ) -> ($#v5_topalg_6 :::"with_first_point"::: ) ($#v6_topalg_6 :::"with_last_point"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) "T"); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v7_topalg_6 :::"with_endpoints"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) "T"); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "c" be ($#v5_topalg_6 :::"with_first_point"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")); cluster (Set ($#k9_xtuple_0 :::"dom"::: ) "c") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "c" ")" )) -> ($#v1_xreal_0 :::"real"::: ) ; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "c" be ($#v6_topalg_6 :::"with_last_point"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")); cluster (Set ($#k9_xtuple_0 :::"dom"::: ) "c") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "c" ")" )) -> ($#v1_xreal_0 :::"real"::: ) ; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#v5_topalg_6 :::"with_first_point"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) "T"); cluster ($#v6_topalg_6 :::"with_last_point"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) "T"); end; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "c" be ($#v5_topalg_6 :::"with_first_point"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")); func :::"the_first_point_of"::: "c" -> ($#m1_subset_1 :::"Point":::) "of" "T" equals :: TOPALG_6:def 9 (Set "c" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "c" ")" ) ")" )); end; :: deftheorem defines :::"the_first_point_of"::: TOPALG_6:def 9 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#v5_topalg_6 :::"with_first_point"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ) ")" ))))); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "c" be ($#v6_topalg_6 :::"with_last_point"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")); func :::"the_last_point_of"::: "c" -> ($#m1_subset_1 :::"Point":::) "of" "T" equals :: TOPALG_6:def 10 (Set "c" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "c" ")" ) ")" )); end; :: deftheorem defines :::"the_last_point_of"::: TOPALG_6:def 10 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#v6_topalg_6 :::"with_last_point"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ) ")" ))))); theorem :: TOPALG_6:25 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "t1")) "," (Set (Var "t2")) "st" (Bool (Bool (Set (Var "t1")) "," (Set (Var "t2")) ($#r1_borsuk_2 :::"are_connected"::: ) )) "holds" (Bool (Set (Var "p")) "is" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")))))) ; theorem :: TOPALG_6:26 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "c")) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "r1")) "," (Set (Var "r2")) ($#k1_rcomp_1 :::".]"::: ) )) "is" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")))))) ; theorem :: TOPALG_6:27 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )))) ; theorem :: TOPALG_6:28 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Var "c")) "is" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c"))) "," (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c")))))) ; theorem :: TOPALG_6:29 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "c")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ) ")" ) ")" ")" )) "is" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c"))) "," (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c")))))) ; theorem :: TOPALG_6:30 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "c")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ) ")" ) ")" ")" )) "is" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "t1")) "," (Set (Var "t2"))) & (Bool (Set (Var "t1")) "," (Set (Var "t2")) ($#r1_borsuk_2 :::"are_connected"::: ) )) "holds" (Bool "(" (Bool (Set (Var "t1")) ($#r1_hidden :::"="::: ) (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c")))) & (Bool (Set (Var "t2")) ($#r1_hidden :::"="::: ) (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c")))) ")" )))) ; theorem :: TOPALG_6:31 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "c")))) & (Bool (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "c")))) ")" ))) ; theorem :: TOPALG_6:32 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p1")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "t1")) "," (Set (Var "t2")) "st" (Bool (Bool (Set (Var "t1")) "," (Set (Var "t2")) ($#r1_borsuk_2 :::"are_connected"::: ) ) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool (Set (Set (Var "p1")) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set (Var "r1")) "," (Set (Var "r2")) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" ")" )) "is" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T"))))))) ; theorem :: TOPALG_6:33 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c"))) "," (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c"))) ($#r1_borsuk_2 :::"are_connected"::: ) ))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "c1", "c2" be ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")); pred "c1" "," "c2" :::"are_homotopic"::: means :: TOPALG_6:def 11 (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" "T"(Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool "(" (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set "c1" ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "c1" ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "c1" ")" ) ")" ) ")" ")" ))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set "c2" ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "c2" ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "c2" ")" ) ")" ) ")" ")" ))) & (Bool (Set (Var "p1")) "," (Set (Var "p2")) ($#r3_borsuk_2 :::"are_homotopic"::: ) ) ")" ))); symmetry (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")) "st" (Bool (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T"))(Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool "(" (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c1")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1")) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1")) ")" ) ")" ) ")" ")" ))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c2")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")) ")" ) ")" ) ")" ")" ))) & (Bool (Set (Var "p1")) "," (Set (Var "p2")) ($#r3_borsuk_2 :::"are_homotopic"::: ) ) ")" )))) "holds" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T"))(Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool "(" (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c2")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")) ")" ) ")" ) ")" ")" ))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c1")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1")) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1")) ")" ) ")" ) ")" ")" ))) & (Bool (Set (Var "p1")) "," (Set (Var "p2")) ($#r3_borsuk_2 :::"are_homotopic"::: ) ) ")" )))) ; end; :: deftheorem defines :::"are_homotopic"::: TOPALG_6:def 11 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "c1")) "," (Set (Var "c2")) ($#r1_topalg_6 :::"are_homotopic"::: ) ) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T"))(Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool "(" (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c1")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1")) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1")) ")" ) ")" ) ")" ")" ))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c2")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_borsuk_6 :::"L[01]"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) "," (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")) ")" ) ")" ) ")" ")" ))) & (Bool (Set (Var "p1")) "," (Set (Var "p2")) ($#r3_borsuk_2 :::"are_homotopic"::: ) ) ")" ))) ")" ))); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "c1", "c2" be ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")); :: original: :::"are_homotopic"::: redefine pred "c1" "," "c2" :::"are_homotopic"::: ; reflexivity (Bool "for" (Set (Var "c1")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")) "holds" (Bool ((Set (Const "T")) "," (Set (Var "b1")) "," (Set (Var "b1"))))) ; symmetry (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")) "st" (Bool (Bool ((Set (Const "T")) "," (Set (Var "b1")) "," (Set (Var "b2"))))) "holds" (Bool ((Set (Const "T")) "," (Set (Var "b2")) "," (Set (Var "b1"))))) ; end; theorem :: TOPALG_6:34 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_borsuk_2 :::"Path"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "c1")) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Var "c2")) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Var "a")) "," (Set (Var "b")) ($#r1_borsuk_2 :::"are_connected"::: ) )) "holds" (Bool "(" (Bool (Set (Var "c1")) "," (Set (Var "c2")) ($#r1_topalg_6 :::"are_homotopic"::: ) ) "iff" (Bool (Set (Var "p1")) "," (Set (Var "p2")) ($#r3_borsuk_2 :::"are_homotopic"::: ) ) ")" ))))) ; theorem :: TOPALG_6:35 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "c1")) "," (Set (Var "c2")) ($#r1_topalg_6 :::"are_homotopic"::: ) )) "holds" (Bool "(" (Bool (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c1"))) ($#r1_hidden :::"="::: ) (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c2")))) & (Bool (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c2")))) ")" ))) ; theorem :: TOPALG_6:36 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_topalg_2 :::"R^1"::: ) ) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")))) & (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1"))))) "holds" (Bool "(" (Bool (Set (Var "c1")) "," (Set (Var "c2")) ($#r2_topalg_6 :::"are_homotopic"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set "(" (Set ($#k2_topalg_2 :::"R^1"::: ) ) ($#k1_pre_topc :::"|"::: ) (Set (Var "S")) ")" ) "," (Set ($#k5_topmetr :::"I[01]"::: ) ) ($#k2_borsuk_1 :::":]"::: ) ) "," (Set (Var "T"))(Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool "(" "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" (Set ($#k2_topalg_2 :::"R^1"::: ) ) ($#k1_pre_topc :::"|"::: ) (Set (Var "S")) ")" ) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "t")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "c1")) ($#k1_funct_1 :::"."::: ) (Set (Var "t")))) & (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "t")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "c2")) ($#k1_funct_1 :::"."::: ) (Set (Var "t")))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set (Var "S")) ")" ) "," (Set (Var "t")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set (Var "S")) ")" ) "," (Set (Var "t")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) ")" ) ")" ) ")" ))) ")" )))) ; definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "c1", "c2" be ($#m2_subset_1 :::"Curve":::) "of" (Set (Const "T")); func "c1" :::"+"::: "c2" -> ($#m2_subset_1 :::"Curve":::) "of" "T" equals :: TOPALG_6:def 12 (Set "c1" ($#k2_xboole_0 :::"\/"::: ) "c2") if (Bool (Set "c1" ($#k2_xboole_0 :::"\/"::: ) "c2") "is" ($#m2_subset_1 :::"Curve":::) "of" "T") otherwise (Set ($#k1_xboole_0 :::"{}"::: ) ); end; :: deftheorem defines :::"+"::: TOPALG_6:def 12 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "c1")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "c2"))) "is" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")))) "implies" (Bool (Set (Set (Var "c1")) ($#k4_topalg_6 :::"+"::: ) (Set (Var "c2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c1")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "c2")))) ")" & "(" (Bool (Bool (Set (Set (Var "c1")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "c2"))) "is" (Bool "not" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T"))))) "implies" (Bool (Set (Set (Var "c1")) ($#k4_topalg_6 :::"+"::: ) (Set (Var "c2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" ))); theorem :: TOPALG_6:37 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Var "T"))) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c1")) ($#k4_topalg_6 :::"+"::: ) (Set (Var "c2")))) & (Bool (Set (Var "c1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set (Var "r")) ($#k1_rcomp_1 :::".]"::: ) ))) & (Bool (Set (Var "c2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "r")) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) ")" ))))) ; theorem :: TOPALG_6:38 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")) ")" ))) & (Bool (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c1"))) ($#r1_hidden :::"="::: ) (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c2"))))) "holds" (Bool "(" (Bool (Set (Set (Var "c1")) ($#k4_topalg_6 :::"+"::: ) (Set (Var "c2"))) "is" ($#v7_topalg_6 :::"with_endpoints"::: ) ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "c1")) ($#k4_topalg_6 :::"+"::: ) (Set (Var "c2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1")) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) )) & (Bool (Set (Set "(" (Set (Var "c1")) ($#k4_topalg_6 :::"+"::: ) (Set (Var "c2")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c1")))) & (Bool (Set (Set "(" (Set (Var "c1")) ($#k4_topalg_6 :::"+"::: ) (Set (Var "c2")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c2")))) ")" ))) ; theorem :: TOPALG_6:39 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "," (Set (Var "c3")) "," (Set (Var "c4")) "," (Set (Var "c5")) "," (Set (Var "c6")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "c1")) "," (Set (Var "c2")) ($#r2_topalg_6 :::"are_homotopic"::: ) ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c2")))) & (Bool (Set (Var "c3")) "," (Set (Var "c4")) ($#r2_topalg_6 :::"are_homotopic"::: ) ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c3"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c4")))) & (Bool (Set (Var "c5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c1")) ($#k4_topalg_6 :::"+"::: ) (Set (Var "c3")))) & (Bool (Set (Var "c6")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c2")) ($#k4_topalg_6 :::"+"::: ) (Set (Var "c4")))) & (Bool (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c1"))) ($#r1_hidden :::"="::: ) (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c3")))) & (Bool (Set ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c3")) ")" )))) "holds" (Bool (Set (Var "c5")) "," (Set (Var "c6")) ($#r2_topalg_6 :::"are_homotopic"::: ) ))) ; definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Const "T"))); func :::"Partial_Sums"::: "f" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) "T") means :: TOPALG_6:def 13 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "f") ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) it)) & (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set it ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) "f"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" it ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k4_topalg_6 :::"+"::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"Partial_Sums"::: TOPALG_6:def 13 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "b3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Var "T"))) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k5_topalg_6 :::"Partial_Sums"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b3")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "b3")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k4_topalg_6 :::"+"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" ) ")" ) ")" ))); definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Const "T"))); func :::"Sum"::: "f" -> ($#m2_subset_1 :::"Curve":::) "of" "T" equals :: TOPALG_6:def 14 (Set (Set "(" ($#k5_topalg_6 :::"Partial_Sums"::: ) "f" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) "f" ")" )) if (Bool (Set ($#k3_finseq_1 :::"len"::: ) "f") ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) otherwise (Set ($#k1_xboole_0 :::"{}"::: ) ); end; :: deftheorem defines :::"Sum"::: TOPALG_6:def 14 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Var "T"))) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k6_topalg_6 :::"Sum"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_topalg_6 :::"Partial_Sums"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) ")" & "(" (Bool (Bool (Bool "not" (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )))) "implies" (Bool (Set ($#k6_topalg_6 :::"Sum"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" ))); theorem :: TOPALG_6:40 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_topalg_6 :::"Sum"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "c")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "c"))))) ; theorem :: TOPALG_6:41 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Var "T"))) "holds" (Bool (Set ($#k6_topalg_6 :::"Sum"::: ) (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "c")) ($#k12_finseq_1 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_topalg_6 :::"Sum"::: ) (Set (Var "f")) ")" ) ($#k4_topalg_6 :::"+"::: ) (Set (Var "c"))))))) ; theorem :: TOPALG_6:42 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Var "T"))) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) ")" )) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k6_topalg_6 :::"Sum"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "X")))))) ; theorem :: TOPALG_6:43 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Var "T"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ")" ))) & (Bool (Set ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) "is" ($#v7_topalg_6 :::"with_endpoints"::: ) ) ")" )) "holds" (Bool "ex" (Set (Var "c")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set ($#k6_topalg_6 :::"Sum"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ) ")" ) "," (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ")" ) ")" ) ($#k1_xxreal_1 :::".]"::: ) )) & (Bool (Set ($#k2_topalg_6 :::"the_first_point_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ) ")" ))) & (Bool (Set ($#k3_topalg_6 :::"the_last_point_of"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ")" ) ")" ))) ")" )))) ; theorem :: TOPALG_6:44 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Var "T"))) (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2")))) & (Bool (Set ($#k6_topalg_6 :::"Sum"::: ) (Set (Var "f1"))) ($#r1_hidden :::"="::: ) (Set (Var "c1"))) & (Bool (Set ($#k6_topalg_6 :::"Sum"::: ) (Set (Var "f2"))) ($#r1_hidden :::"="::: ) (Set (Var "c2"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ")" ))) & (Bool (Set ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ")" ))) & (Bool (Set ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1"))))) "holds" (Bool "ex" (Set (Var "c3")) "," (Set (Var "c4")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "c3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")))) & (Bool (Set (Var "c4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")))) & (Bool (Set (Var "c3")) "," (Set (Var "c4")) ($#r2_topalg_6 :::"are_homotopic"::: ) ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c3"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c4")))) ")" )) ")" )) "holds" (Bool (Set (Var "c1")) "," (Set (Var "c2")) ($#r2_topalg_6 :::"are_homotopic"::: ) )))) ; theorem :: TOPALG_6:45 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "c")) "being" ($#v7_topalg_6 :::"with_endpoints"::: ) ($#m2_subset_1 :::"Curve":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "h")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h"))) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_2 :::"inf"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ))) & (Bool (Set (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_2 :::"sup"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "c")) ")" ))) & (Bool (Set (Var "h")) "is" ($#v5_valued_0 :::"increasing"::: ) )) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_topalg_6 :::"Curves"::: ) (Set (Var "T"))) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) & (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set ($#k6_topalg_6 :::"Sum"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k5_relat_1 :::"|"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) ")" ) ")" ))))) ; theorem :: TOPALG_6:46 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set ($#k4_mfold_2 :::"TUnitSphere"::: ) (Set (Var "n"))) "is" ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) )) ; theorem :: TOPALG_6:47 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 3))) "holds" (Bool (Set ($#k7_toprealb :::"Tcircle"::: ) "(" (Set (Var "x")) "," (Set (Var "r")) ")" ) "is" ($#v1_topalg_6 :::"having_trivial_Fundamental_Group"::: ) )))) ;