:: JORDAN12 semantic presentation begin theorem :: JORDAN12:1 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1)))) ; theorem :: JORDAN12:2 (Bool (Num 1) "is" ($#v1_abian :::"odd"::: ) ) ; theorem :: JORDAN12:3 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) ")" )))) ; registration cluster ($#v3_topreal1 :::"s.n.c."::: ) -> ($#v1_goboard5 :::"s.c.c."::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )); end; theorem :: JORDAN12:4 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g"))) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g"))) "is" ($#v1_goboard5 :::"s.c.c."::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) ")" )) ; theorem :: JORDAN12:5 (Bool "for" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g1"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "g1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g2")) ")" )))) ; begin definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f1", "f2" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); pred "f1" :::"is_in_general_position_wrt"::: "f2" means :: JORDAN12:def 1 (Bool "(" (Bool (Set ($#k3_topreal1 :::"L~"::: ) "f1") ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "f2")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) "f2"))) "holds" (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) "f1" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" "f2" "," (Set (Var "i")) ")" ")" )) "is" ($#v1_zfmisc_1 :::"trivial"::: ) ) ")" ) ")" ); end; :: deftheorem defines :::"is_in_general_position_wrt"::: JORDAN12:def 1 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f1")) ($#r1_jordan12 :::"is_in_general_position_wrt"::: ) (Set (Var "f2"))) "iff" (Bool "(" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f2")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2"))))) "holds" (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f2")) "," (Set (Var "i")) ")" ")" )) "is" ($#v1_zfmisc_1 :::"trivial"::: ) ) ")" ) ")" ) ")" ))); definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f1", "f2" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); pred "f1" "," "f2" :::"are_in_general_position"::: means :: JORDAN12:def 2 (Bool "(" (Bool "f1" ($#r1_jordan12 :::"is_in_general_position_wrt"::: ) "f2") & (Bool "f2" ($#r1_jordan12 :::"is_in_general_position_wrt"::: ) "f1") ")" ); symmetry (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_jordan12 :::"is_in_general_position_wrt"::: ) (Set (Var "f2"))) & (Bool (Set (Var "f2")) ($#r1_jordan12 :::"is_in_general_position_wrt"::: ) (Set (Var "f1")))) "holds" (Bool "(" (Bool (Set (Var "f2")) ($#r1_jordan12 :::"is_in_general_position_wrt"::: ) (Set (Var "f1"))) & (Bool (Set (Var "f1")) ($#r1_jordan12 :::"is_in_general_position_wrt"::: ) (Set (Var "f2"))) ")" )) ; end; :: deftheorem defines :::"are_in_general_position"::: JORDAN12:def 2 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f1")) "," (Set (Var "f2")) ($#r2_jordan12 :::"are_in_general_position"::: ) ) "iff" (Bool "(" (Bool (Set (Var "f1")) ($#r1_jordan12 :::"is_in_general_position_wrt"::: ) (Set (Var "f2"))) & (Bool (Set (Var "f2")) ($#r1_jordan12 :::"is_in_general_position_wrt"::: ) (Set (Var "f1"))) ")" ) ")" ))); theorem :: JORDAN12:6 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f1")) "," (Set (Var "f2")) ($#r2_jordan12 :::"are_in_general_position"::: ) )) "holds" (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" )))) "holds" (Bool (Set (Var "f1")) "," (Set (Var "f")) ($#r2_jordan12 :::"are_in_general_position"::: ) )))) ; theorem :: JORDAN12:7 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "f2"))) "," (Set (Set (Var "g1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g2"))) ($#r2_jordan12 :::"are_in_general_position"::: ) )) "holds" (Bool (Set (Set (Var "f1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "f2"))) "," (Set (Var "g1")) ($#r2_jordan12 :::"are_in_general_position"::: ) )) ; theorem :: JORDAN12:8 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) "," (Set (Var "g")) ($#r2_jordan12 :::"are_in_general_position"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) ")" ))) ; theorem :: JORDAN12:9 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f1")) "," (Set (Var "f2")) ($#r2_jordan12 :::"are_in_general_position"::: ) )) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2"))))) "holds" (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f1")) "," (Set (Var "i")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f2")) "," (Set (Var "j")) ")" ")" )) "is" ($#v1_zfmisc_1 :::"trivial"::: ) ))) ; theorem :: JORDAN12:10 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k3_setfam_1 :::"INTERSECTION"::: ) "(" "{" (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" ) "}" "," "{" (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "g")) "," (Set (Var "j")) ")" ")" ) where j "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) ")" ) "}" ")" ) "is" ($#v1_finset_1 :::"finite"::: ) )) ; theorem :: JORDAN12:11 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "," (Set (Var "g")) ($#r2_jordan12 :::"are_in_general_position"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )) "is" ($#v1_finset_1 :::"finite"::: ) )) ; theorem :: JORDAN12:12 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "," (Set (Var "g")) ($#r2_jordan12 :::"are_in_general_position"::: ) )) "holds" (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "g")) "," (Set (Var "k")) ")" ")" )) "is" ($#v1_finset_1 :::"finite"::: ) ))) ; begin theorem :: JORDAN12:13 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )))) ; theorem :: JORDAN12:14 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool "(" (Bool "(" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) ")" ) "or" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) ")" ) ")" ) "iff" (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )) ")" ))) ; theorem :: JORDAN12:15 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool "(" "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) "or" "not" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "or" "not" (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" ) ")" ) ")" ) "iff" (Bool "(" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) ")" ) "or" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) ")" ) ")" ) ")" ))) ; theorem :: JORDAN12:16 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )) & (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" ))) "holds" (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )))) ; theorem :: JORDAN12:17 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool "(" "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) "or" "not" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "or" "not" (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) "or" "not" (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "or" "not" (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" ) ")" )) "holds" (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )))) ; begin theorem :: JORDAN12:18 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G"))))) "holds" (Bool (Set ($#k1_goboard5 :::"v_strip"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) ")" ) "is" ($#v1_convex1 :::"convex"::: ) ))) ; theorem :: JORDAN12:19 (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set ($#k2_goboard5 :::"h_strip"::: ) "(" (Set (Var "G")) "," (Set (Var "j")) ")" ) "is" ($#v1_convex1 :::"convex"::: ) ))) ; theorem :: JORDAN12:20 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#v1_convex1 :::"convex"::: ) ))) ; theorem :: JORDAN12:21 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k5_goboard5 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ) "is" ($#v1_convex1 :::"convex"::: ) ))) ; theorem :: JORDAN12:22 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "is" ($#v1_convex1 :::"convex"::: ) ) & (Bool (Set ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "is" ($#v1_convex1 :::"convex"::: ) ) ")" ))) ; begin theorem :: JORDAN12:23 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" )) & (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) & (Bool (Bool "not" (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool (Bool "not" (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "r")) ")" )))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "r")) "," (Set (Var "p2")) ")" ))))) ; theorem :: JORDAN12:24 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) "is" ($#v2_sppol_1 :::"vertical"::: ) ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "r")) "," (Set (Var "s")) ")" ) "is" ($#v2_sppol_1 :::"vertical"::: ) ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "r")) "," (Set (Var "s")) ")" ))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k17_euclid :::"`1"::: ) ))) ; theorem :: JORDAN12:25 (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Bool "not" (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "p2")) ")" )))) "holds" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "p1")) ")" ))) ; theorem :: JORDAN12:26 (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Bool "not" (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "p2")) ")" )))) "holds" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "p1")) ")" ))) ; theorem :: JORDAN12:27 (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "p1"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) "holds" (Bool "not" (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "p2")) ")" )))) ; theorem :: JORDAN12:28 (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "p1"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2"))) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" ) & (Bool (Bool "not" (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q")) "," (Set (Var "p")) ")" )))) "holds" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q")) "," (Set (Var "p")) ")" ))) ; theorem :: JORDAN12:29 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool "(" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" ) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p3")) "," (Set (Var "p4")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "p1")))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "p2"))))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "p3")))) ; begin theorem :: JORDAN12:30 (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) & (Bool (Bool "not" (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool (Bool "not" (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k17_euclid :::"`1"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" ) & (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool "(" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" )) "or" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" )) ")" ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" )) ")" )) & (Bool (Bool "not" (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool "(" "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) "or" "not" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "or" "not" (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" ) ")" )) "holds" (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" ))))) ; theorem :: JORDAN12:31 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool "for" (Set (Var "rl")) "," (Set (Var "rp")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool (Bool "not" (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "rl")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "rl")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "rp")) ($#k17_euclid :::"`1"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "rl")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "rl")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "rp")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" ) & (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "rl")) ($#r2_hidden :::"in"::: ) (Set ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" )) & (Bool (Set (Var "rp")) ($#r2_hidden :::"in"::: ) (Set ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" )) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" )) ")" )) & (Bool (Bool "not" (Set (Var "rl")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool (Bool "not" (Set (Var "rp")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) "holds" (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) "or" "not" (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "or" "not" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" ))))) ; theorem :: JORDAN12:32 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) & (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) "or" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) ")" ) & (Bool (Bool "not" (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool (Bool "not" (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_subset_1 :::"misses"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) "holds" (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) "or" "not" (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "or" "not" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" ))))) ; theorem :: JORDAN12:33 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "g")) "being" ($#v1_topreal1 :::"special"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "," (Set (Var "g")) ($#r2_jordan12 :::"are_in_general_position"::: ) )) "holds" (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "g")) "," (Set (Var "k")) ")" ")" ) ")" )) "is" ($#v1_abian :::"even"::: ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) "iff" (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )) ")" )))) ; theorem :: JORDAN12:34 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g1")) "being" ($#v1_topreal1 :::"special"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "f2"))) "is" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::)) & (Bool (Set (Set (Var "f1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "f2"))) "," (Set (Var "g1")) ($#r2_jordan12 :::"are_in_general_position"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g1"))) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Var "g1")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "g1")) "is" ($#v3_topreal1 :::"s.n.c."::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "f1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "f2")) ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g1")) ")" ) ")" )) "is" ($#v1_abian :::"even"::: ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) "iff" (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "f1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "f2")) ")" ) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Set (Var "g1")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Set (Var "g1")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g1")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )) ")" )) ; theorem :: JORDAN12:35 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#v1_topreal1 :::"special"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "f2"))) "is" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::)) & (Bool (Set (Set (Var "g1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g2"))) "is" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::)) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g2")))) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f2"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g1")))) & (Bool (Set (Set (Var "f1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "f2"))) "," (Set (Set (Var "g1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g2"))) ($#r2_jordan12 :::"are_in_general_position"::: ) )) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "g1")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "q1"))) & (Bool (Set (Set (Var "g1")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "q2"))) & (Bool (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Num 1))) & (Bool (Set (Set (Var "g1")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g2")) ($#k7_partfun1 :::"/."::: ) (Num 1))) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g2")) ")" ))) & (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "f1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "f2")) ")" ) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "q2")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" ))) "holds" (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "g1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g2")) ")" ) ")" ) ($#k3_subset_1 :::"`"::: ) )) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )))) ;