:: JGRAPH_6 semantic presentation begin theorem :: JGRAPH_6:1 (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) ")" ))) ; theorem :: JGRAPH_6:2 (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" )))) ; theorem :: JGRAPH_6:3 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) ")" ))) ; theorem :: JGRAPH_6:4 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_6:5 (Bool "for" (Set (Var "X")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Z")) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "X")))) ")" ))))) ; theorem :: JGRAPH_6:6 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "X")) ($#k1_pre_topc :::"|"::: ) (Set (Var "B")) ")" ) "," (Set (Var "X")) "st" (Bool "(" (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" (Set (Var "X")) ($#k1_pre_topc :::"|"::: ) (Set (Var "B")) ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ) & (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" )))) ; theorem :: JGRAPH_6:7 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r1")))) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a"))))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))))) ; theorem :: JGRAPH_6:8 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r1")))) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "r1"))))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))))) ; theorem :: JGRAPH_6:9 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p")))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" )))))) ; theorem :: JGRAPH_6:10 (Bool (Set (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) ) ($#k3_funct_2 :::"."::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) )) ; theorem :: JGRAPH_6:11 (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) "}" )) "holds" (Bool (Set (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) ) ($#k3_funct_2 :::"."::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "P"))))) ; theorem :: JGRAPH_6:12 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "g1")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "g2")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "g1")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "g2")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))))) ; theorem :: JGRAPH_6:13 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p2")) ")" ))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" )))))) ; begin theorem :: JGRAPH_6:14 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C0")) "," (Set (Var "KXP")) "," (Set (Var "KXN")) "," (Set (Var "KYP")) "," (Set (Var "KYN")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "O")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "C0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) "}" ) & (Bool (Set (Var "KXP")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q1")) where q1 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ))) ")" ) "}" ) & (Bool (Set (Var "KXN")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q2")) where q2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ))) ")" ) "}" ) & (Bool (Set (Var "KYP")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q3")) where q3 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q3")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "q3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "q3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "q3")) ($#k17_euclid :::"`1"::: ) ")" ))) ")" ) "}" ) & (Bool (Set (Var "KYN")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q4")) where q4 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q4")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "q4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "q4")) ($#k17_euclid :::"`1"::: ) ")" ))) ")" ) "}" ) & (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O"))) ($#r2_hidden :::"in"::: ) (Set (Var "KXP"))) & (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I"))) ($#r2_hidden :::"in"::: ) (Set (Var "KXN"))) & (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "O"))) ($#r2_hidden :::"in"::: ) (Set (Var "KYP"))) & (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "I"))) ($#r2_hidden :::"in"::: ) (Set (Var "KYN"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set (Var "C0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set (Var "C0")))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_6:15 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C0")) "," (Set (Var "KXP")) "," (Set (Var "KXN")) "," (Set (Var "KYP")) "," (Set (Var "KYN")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "O")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "C0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) "}" ) & (Bool (Set (Var "KXP")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q1")) where q1 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ))) ")" ) "}" ) & (Bool (Set (Var "KXN")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q2")) where q2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ))) ")" ) "}" ) & (Bool (Set (Var "KYP")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q3")) where q3 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q3")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "q3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "q3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "q3")) ($#k17_euclid :::"`1"::: ) ")" ))) ")" ) "}" ) & (Bool (Set (Var "KYN")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q4")) where q4 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q4")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "q4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "q4")) ($#k17_euclid :::"`1"::: ) ")" ))) ")" ) "}" ) & (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O"))) ($#r2_hidden :::"in"::: ) (Set (Var "KXP"))) & (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I"))) ($#r2_hidden :::"in"::: ) (Set (Var "KXN"))) & (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "O"))) ($#r2_hidden :::"in"::: ) (Set (Var "KYN"))) & (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "I"))) ($#r2_hidden :::"in"::: ) (Set (Var "KYP"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set (Var "C0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set (Var "C0")))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_6:16 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) "}" ) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p3")) "," (Set (Var "p4")) "," (Set (Var "P")))) "holds" (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "C0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p8")) where p8 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p8")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) "}" ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p3"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set (Var "C0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set (Var "C0")))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g")))))))) ; theorem :: JGRAPH_6:17 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) "}" ) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p3")) "," (Set (Var "p4")) "," (Set (Var "P")))) "holds" (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "C0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p8")) where p8 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p8")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) "}" ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p3"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set (Var "C0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set (Var "C0")))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g")))))))) ; theorem :: JGRAPH_6:18 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) "}" ) & (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set (Var "P")))) "holds" (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "C0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p8")) where p8 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p8")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) "}" ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p3"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set (Var "C0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set (Var "C0")))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g")))))))) ; begin notationlet "a", "b", "c", "d" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; synonym :::"rectangle"::: "(" "a" "," "b" "," "c" "," "d" ")" for :::"[.":::"a" "," "b" "," "c" "," "d":::".]":::; end; theorem :: JGRAPH_6:19 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) ")" ))) ; definitionlet "a", "b", "c", "d" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"inside_of_rectangle"::: "(" "a" "," "b" "," "c" "," "d" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_6:def 1 "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool "a" ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) "b") & (Bool "c" ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) "d") ")" ) "}" ; end; :: deftheorem defines :::"inside_of_rectangle"::: JGRAPH_6:def 1 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_jgraph_6 :::"inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) ")" ) "}" )); definitionlet "a", "b", "c", "d" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"closed_inside_of_rectangle"::: "(" "a" "," "b" "," "c" "," "d" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_6:def 2 "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool "a" ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) "b") & (Bool "c" ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) "d") ")" ) "}" ; end; :: deftheorem defines :::"closed_inside_of_rectangle"::: JGRAPH_6:def 2 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) ")" ) "}" )); definitionlet "a", "b", "c", "d" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"outside_of_rectangle"::: "(" "a" "," "b" "," "c" "," "d" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_6:def 3 "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" "not" (Bool "a" ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) "b") "or" "not" (Bool "c" ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) "d") ")" ) "}" ; end; :: deftheorem defines :::"outside_of_rectangle"::: JGRAPH_6:def 3 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_jgraph_6 :::"outside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" "not" (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) "or" "not" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) ")" ) "}" )); definitionlet "a", "b", "c", "d" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"closed_outside_of_rectangle"::: "(" "a" "," "b" "," "c" "," "d" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_6:def 4 "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" "not" (Bool "a" ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) "b") "or" "not" (Bool "c" ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) "d") ")" ) "}" ; end; :: deftheorem defines :::"closed_outside_of_rectangle"::: JGRAPH_6:def 4 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_jgraph_6 :::"closed_outside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" "not" (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) "or" "not" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) ")" ) "}" )); theorem :: JGRAPH_6:20 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "Kb")) "," (Set (Var "Cb")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "Kb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) "}" ) & (Bool (Set (Var "Cb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p2")) where p2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p2")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Set "(" ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set (Var "r")) "," (Set (Var "a")) "," (Set (Var "r")) "," (Set (Var "b")) ")" ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Kb"))) ($#r1_hidden :::"="::: ) (Set (Var "Cb"))))) ; theorem :: JGRAPH_6:21 (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "P")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "Q")) ")" ) "st" (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) )) & (Bool (Set (Var "P")) "is" ($#v1_topreal2 :::"being_simple_closed_curve"::: ) )) "holds" (Bool (Set (Var "Q")) "is" ($#v1_topreal2 :::"being_simple_closed_curve"::: ) )) ; theorem :: JGRAPH_6:22 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) "is" ($#v1_topreal2 :::"being_simple_closed_curve"::: ) )) "holds" (Bool (Set (Var "P")) "is" ($#v2_compts_1 :::"compact"::: ) )) ; theorem :: JGRAPH_6:23 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "Cb")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "Cb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "r"))) "}" )) "holds" (Bool (Set (Var "Cb")) "is" ($#v1_topreal2 :::"being_simple_closed_curve"::: ) ))) ; definitionlet "a", "b", "r" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"circle"::: "(" "a" "," "b" "," "r" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_6:def 5 "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) "a" "," "b" ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) "r") "}" ; end; :: deftheorem defines :::"circle"::: JGRAPH_6:def 5 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k5_jgraph_6 :::"circle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "r"))) "}" )); registrationlet "a", "b", "r" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set ($#k5_jgraph_6 :::"circle"::: ) "(" "a" "," "b" "," "r" ")" ) -> ($#v2_compts_1 :::"compact"::: ) ; end; registrationlet "a", "b" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "r" be ($#v1_xreal_0 :::"real"::: ) ($#~v3_xxreal_0 "non" ($#v3_xxreal_0 :::"negative"::: ) ) ($#m1_hidden :::"number"::: ) ; cluster (Set ($#k5_jgraph_6 :::"circle"::: ) "(" "a" "," "b" "," "r" ")" ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "a", "b", "r" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"inside_of_circle"::: "(" "a" "," "b" "," "r" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_6:def 6 "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) "a" "," "b" ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) "r") "}" ; end; :: deftheorem defines :::"inside_of_circle"::: JGRAPH_6:def 6 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k6_jgraph_6 :::"inside_of_circle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" )); definitionlet "a", "b", "r" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"closed_inside_of_circle"::: "(" "a" "," "b" "," "r" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_6:def 7 "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) "a" "," "b" ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) "r") "}" ; end; :: deftheorem defines :::"closed_inside_of_circle"::: JGRAPH_6:def 7 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k7_jgraph_6 :::"closed_inside_of_circle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) "}" )); definitionlet "a", "b", "r" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"outside_of_circle"::: "(" "a" "," "b" "," "r" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_6:def 8 "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) "a" "," "b" ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::">"::: ) "r") "}" ; end; :: deftheorem defines :::"outside_of_circle"::: JGRAPH_6:def 8 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k8_jgraph_6 :::"outside_of_circle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "r"))) "}" )); definitionlet "a", "b", "r" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"closed_outside_of_circle"::: "(" "a" "," "b" "," "r" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_6:def 9 "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) "a" "," "b" ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::">="::: ) "r") "}" ; end; :: deftheorem defines :::"closed_outside_of_circle"::: JGRAPH_6:def 9 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k9_jgraph_6 :::"closed_outside_of_circle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "r"))) "}" )); theorem :: JGRAPH_6:24 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set ($#k6_jgraph_6 :::"inside_of_circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" ) & "(" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k5_jgraph_6 :::"circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p2")) where p2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "r"))) "}" ) ")" & (Bool (Set ($#k8_jgraph_6 :::"outside_of_circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p3")) where p3 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p3")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "r"))) "}" ) & (Bool (Set ($#k7_jgraph_6 :::"closed_inside_of_circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) "}" ) & (Bool (Set ($#k9_jgraph_6 :::"closed_outside_of_circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "q2")) where q2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "r"))) "}" ) ")" )) ; theorem :: JGRAPH_6:25 (Bool "for" (Set (Var "Kb")) "," (Set (Var "Cb")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "Kb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) "}" ) & (Bool (Set (Var "Cb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p2")) where p2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) "}" )) "holds" (Bool (Set (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Kb"))) ($#r1_hidden :::"="::: ) (Set (Var "Cb")))) ; theorem :: JGRAPH_6:26 (Bool "for" (Set (Var "Kb")) "," (Set (Var "Cb")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "Kb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" "not" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) "or" "not" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) "}" ) & (Bool (Set (Var "Cb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p2")) where p2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Num 1)) "}" )) "holds" (Bool (Set (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Kb"))) ($#r1_hidden :::"="::: ) (Set (Var "Cb")))) ; theorem :: JGRAPH_6:27 (Bool "for" (Set (Var "Kb")) "," (Set (Var "Cb")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "Kb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) "}" ) & (Bool (Set (Var "Cb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p2")) where p2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) "}" )) "holds" (Bool (Set (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Kb"))) ($#r1_hidden :::"="::: ) (Set (Var "Cb")))) ; theorem :: JGRAPH_6:28 (Bool "for" (Set (Var "Kb")) "," (Set (Var "Cb")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "Kb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" "not" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) "or" "not" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) "}" ) & (Bool (Set (Var "Cb")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p2")) where p2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Num 1)) "}" )) "holds" (Bool (Set (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Kb"))) ($#r1_hidden :::"="::: ) (Set (Var "Cb")))) ; theorem :: JGRAPH_6:29 (Bool "for" (Set (Var "P0")) "," (Set (Var "P1")) "," (Set (Var "P01")) "," (Set (Var "P11")) "," (Set (Var "K0")) "," (Set (Var "K1")) "," (Set (Var "K01")) "," (Set (Var "K11")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k5_jgraph_6 :::"circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) & (Bool (Set (Var "P0")) ($#r1_hidden :::"="::: ) (Set ($#k6_jgraph_6 :::"inside_of_circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) & (Bool (Set (Var "P1")) ($#r1_hidden :::"="::: ) (Set ($#k8_jgraph_6 :::"outside_of_circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) & (Bool (Set (Var "P01")) ($#r1_hidden :::"="::: ) (Set ($#k7_jgraph_6 :::"closed_inside_of_circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) & (Bool (Set (Var "P11")) ($#r1_hidden :::"="::: ) (Set ($#k9_jgraph_6 :::"closed_outside_of_circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) (Set ($#k1_jgraph_6 :::"inside_of_rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool (Set (Var "K1")) ($#r1_hidden :::"="::: ) (Set ($#k3_jgraph_6 :::"outside_of_rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool (Set (Var "K01")) ($#r1_hidden :::"="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool (Set (Var "K11")) ($#r1_hidden :::"="::: ) (Set ($#k4_jgraph_6 :::"closed_outside_of_rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "P"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "K0"))) ($#r1_hidden :::"="::: ) (Set (Var "P0"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "P0"))) ($#r1_hidden :::"="::: ) (Set (Var "K0"))) & (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "K1"))) ($#r1_hidden :::"="::: ) (Set (Var "P1"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "P1"))) ($#r1_hidden :::"="::: ) (Set (Var "K1"))) & (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "K01"))) ($#r1_hidden :::"="::: ) (Set (Var "P01"))) & (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "K11"))) ($#r1_hidden :::"="::: ) (Set (Var "P11"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "P01"))) ($#r1_hidden :::"="::: ) (Set (Var "K01"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "P11"))) ($#r1_hidden :::"="::: ) (Set (Var "K11"))) ")" )))) ; begin theorem :: JGRAPH_6:30 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool "(" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p1")) where p1 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) ")" ) "}" ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p2")) where p2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) ")" ) "}" ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "q1")) where q1 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) ")" ) "}" ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "q2")) where q2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) ")" ) "}" ) ")" )) ; theorem :: JGRAPH_6:31 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: JGRAPH_6:32 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: JGRAPH_6:33 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: JGRAPH_6:34 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: JGRAPH_6:35 (Bool "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) "or" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) "or" (Bool "(" (Bool (Num 1) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) ")" ) "}" ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ) ")" ) "}" ) ; theorem :: JGRAPH_6:36 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "a")))) ; theorem :: JGRAPH_6:37 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "d")))) ; theorem :: JGRAPH_6:38 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "b")))) ; theorem :: JGRAPH_6:39 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "c")))) ; theorem :: JGRAPH_6:40 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k11_pscomp_1 :::"NW-corner"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ))) ; theorem :: JGRAPH_6:41 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k12_pscomp_1 :::"NE-corner"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ))) ; theorem :: JGRAPH_6:42 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k10_pscomp_1 :::"SW-corner"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ))) ; theorem :: JGRAPH_6:43 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k13_pscomp_1 :::"SE-corner"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ))) ; theorem :: JGRAPH_6:44 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) ; theorem :: JGRAPH_6:45 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) ; theorem :: JGRAPH_6:46 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool "(" (Bool (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) )) ")" )) ; theorem :: JGRAPH_6:47 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" )) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) "," (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" )) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) "," (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) ")" )) ; theorem :: JGRAPH_6:48 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p0")) "," (Set (Var "p1")) "," (Set (Var "p01")) "," (Set (Var "p10")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p0")) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Var "p01")) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Var "p10")) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Var "f1")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p0")) "," (Set (Var "p01")) "," (Set (Var "p1")) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p0")) "," (Set (Var "p10")) "," (Set (Var "p1")) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f1")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p0")) "," (Set (Var "p01")) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p01")) "," (Set (Var "p1")) ")" ")" ))) & (Bool (Set (Var "f2")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p0")) "," (Set (Var "p10")) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p10")) "," (Set (Var "p1")) ")" ")" ))) & (Bool (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "p0")) "," (Set (Var "p1")) ($#k2_tarski :::"}"::: ) )) & (Bool (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p0"))) & (Bool (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p0"))) & (Bool (Set (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) ")" )))) ; theorem :: JGRAPH_6:49 (Bool "for" (Set (Var "P1")) "," (Set (Var "P2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (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 (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Var "f1")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Var "P1")) ($#r1_hidden :::"="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1")))) & (Bool (Set (Var "P2")) ($#r1_hidden :::"="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f2"))))) "holds" (Bool "(" (Bool (Set (Var "P1")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Var "P2")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Bool "not" (Set (Var "P1")) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Bool "not" (Set (Var "P2")) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "P1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "P2")))) & (Bool (Set (Set (Var "P1")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "P2"))) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k2_tarski :::"}"::: ) )) ")" ))))) ; theorem :: JGRAPH_6:50 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ) "is" ($#v1_topreal2 :::"being_simple_closed_curve"::: ) )) ; theorem :: JGRAPH_6:51 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" )))) ; theorem :: JGRAPH_6:52 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ")" )))) ; theorem :: JGRAPH_6:53 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d")))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ")" ) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "r")) ")" ) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "c")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2))) & (Bool (Set (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "c")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "c")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ))) & (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ) ")" ) ")" ))) ; theorem :: JGRAPH_6:54 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d")))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ")" ) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "r")) ")" ) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Num 1) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "d")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "c")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "d")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2))) & (Bool (Set (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "d")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "c")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "d")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "d")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "c")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "d")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "b")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "b")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ))) & (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "b")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "b")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "b")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "a")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "b")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ) ")" ) ")" ))) ; theorem :: JGRAPH_6:55 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) "iff" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) ")" ))) ; theorem :: JGRAPH_6:56 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (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 (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) "iff" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) ")" ))) ; theorem :: JGRAPH_6:57 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) "iff" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) ")" ))) ; theorem :: JGRAPH_6:58 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (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 (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) "iff" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) ")" ) ")" ))) ; theorem :: JGRAPH_6:59 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) "iff" (Bool "(" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) ")" ) "or" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" )) "or" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )) "or" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) ")" ) ")" ) ")" ))) ; theorem :: JGRAPH_6:60 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) "iff" (Bool "(" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) ")" ) "or" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )) "or" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) ")" ) ")" ) ")" ))) ; theorem :: JGRAPH_6:61 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) "iff" (Bool "(" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) ")" ) ")" ) ")" ))) ; theorem :: JGRAPH_6:62 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" )))) "holds" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) "iff" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ")" ))) ")" ) ")" ))) ; theorem :: JGRAPH_6:63 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) & (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) ")" ))) & (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "a")) "," (Set (Var "c")) ($#k19_euclid :::"]|"::: ) ) ")" )))) ; begin theorem :: JGRAPH_6:64 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool "not" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) ")" )) & (Bool (Bool "not" (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ))) & (Bool (Bool "not" (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" )) ; theorem :: JGRAPH_6:65 (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 "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k5_jgraph_6 :::"circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) & (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) )) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" ))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p1"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) "," (Set (Var "P")))))) ; theorem :: JGRAPH_6:66 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k5_jgraph_6 :::"circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) & (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) )) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p3")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" ))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p3"))) "," (Set (Var "P")))))) ; theorem :: JGRAPH_6:67 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: JGRAPH_6:68 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) "iff" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_6:69 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) "iff" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_6:70 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) )) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "q")) ")" ) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: JGRAPH_6:71 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) )) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "q")) ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JGRAPH_6:72 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k5_jgraph_6 :::"circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) & (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) )) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p3")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p3")) "," (Set (Var "p4")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p1"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p3"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p4"))) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set (Var "P")))))) ; theorem :: JGRAPH_6:73 (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 "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) "is" ($#v1_topreal2 :::"being_simple_closed_curve"::: ) ) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Bool "not" ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "P"))))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p1")) "," (Set (Var "P"))))) ; theorem :: JGRAPH_6:74 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) "is" ($#v1_topreal2 :::"being_simple_closed_curve"::: ) ) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "p3")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool "not" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "P"))) ")" )) & (Bool "not" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p3")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p3")) "," (Set (Var "p2")) "," (Set (Var "P"))) ")" )) & (Bool "not" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p1")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p3")) "," (Set (Var "P"))) ")" )) & (Bool "not" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p3")) "," (Set (Var "p1")) "," (Set (Var "P"))) ")" )) & (Bool "not" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p3")) "," (Set (Var "p1")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "P"))) ")" ))) "holds" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p3")) "," (Set (Var "p2")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p1")) "," (Set (Var "P"))) ")" ))) ; theorem :: JGRAPH_6:75 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) "is" ($#v1_topreal2 :::"being_simple_closed_curve"::: ) ) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "p3")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "P"))) & (Bool (Bool "not" ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "P")))) & (Bool "not" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p1")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p3")) "," (Set (Var "P"))) ")" ))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p3")) "," (Set (Var "p1")) "," (Set (Var "P"))))) ; theorem :: JGRAPH_6:76 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) "is" ($#v1_topreal2 :::"being_simple_closed_curve"::: ) ) & (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "p3")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "p4")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p3")) "," (Set (Var "p4")) "," (Set (Var "P"))) & (Bool (Bool "not" ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "P")))) & (Bool "not" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p2")) "," (Set (Var "p1")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p3")) "," (Set (Var "P"))) ")" )) & (Bool "not" (Bool "(" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p3")) "," (Set (Var "p1")) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p4")) "," (Set (Var "P"))) ")" ))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p4")) "," (Set (Var "p1")) "," (Set (Var "P"))))) ; theorem :: JGRAPH_6:77 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k5_jgraph_6 :::"circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) & (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) )) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p1"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p3"))) "," (Set (Var "P"))) & (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p3"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p4"))) "," (Set (Var "P")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" ))))) ; theorem :: JGRAPH_6:78 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k5_jgraph_6 :::"circle"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ")" )) & (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_jgraph_3 :::"Sq_Circ"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) "iff" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p1"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p3"))) "," (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p4"))) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set (Var "P"))) ")" )))) ; theorem :: JGRAPH_6:79 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" ))) "holds" (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p3"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g")))))) ;