:: JGRAPH_7 semantic presentation begin theorem :: JGRAPH_7:1 (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")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p")) ($#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"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p")) ($#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")) ($#k19_euclid :::"]|"::: ) ) ")" )))) ; theorem :: JGRAPH_7:2 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "P"))) & (Bool (Set (Set (Var "f")) ($#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 "p2"))) ")" ))))) ; theorem :: JGRAPH_7:3 (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 "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:4 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "b")) "," (Set (Var "c")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ")" )))) ; theorem :: JGRAPH_7:5 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:6 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "b")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set (Var "b")) "," (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:7 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (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")) ")" )))) ; theorem :: JGRAPH_7:8 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (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")) ")" )))) ; theorem :: JGRAPH_7:9 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (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")) ")" )))) ; theorem :: JGRAPH_7:10 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:11 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool ($#r1_jordan6 :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:12 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (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")) ")" )))) ; theorem :: JGRAPH_7:13 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (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")) ")" )))) ; theorem :: JGRAPH_7:14 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:15 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:16 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:17 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:18 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:19 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:20 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:21 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:22 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:23 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:24 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:25 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:26 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:27 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:28 (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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ))) "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 "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ")" ) "," (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ")" ))) ; theorem :: JGRAPH_7:29 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:30 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:31 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:32 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:33 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:34 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:35 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:36 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:37 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:38 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:39 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:40 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:41 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:42 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:43 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:44 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:45 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:46 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "a")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:47 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "a")))) "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 (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:48 (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 "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"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "a")))) "holds" (Bool (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) ($#r1_jordan17 :::"are_in_this_order_on"::: ) (Set ($#k1_sppol_2 :::"rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )))) ; theorem :: JGRAPH_7:49 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "," (Set (Var "D")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "h")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "C")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "," (Set (Var "D")) ")" )) & (Bool (Set (Var "g")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "A")) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "B")) ($#k7_xcmplx_0 :::"/"::: ) (Set (Var "A")) ")" ) ")" ) "," (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "C")) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "D")) ($#k7_xcmplx_0 :::"/"::: ) (Set (Var "C")) ")" ) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "h")) ($#k2_funct_1 :::"""::: ) )) & (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k2_funct_1 :::"""::: ) )) ")" ))) ; theorem :: JGRAPH_7:50 (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 "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "C")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "," (Set (Var "D")) ")" ))) "holds" (Bool "(" (Bool (Set (Var "h")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool "(" "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "p1")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "p2")) ")" ) ($#k17_euclid :::"`1"::: ) )) ")" ) ")" ))) ; theorem :: JGRAPH_7:51 (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 "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "C")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "," (Set (Var "D")) ")" ))) "holds" (Bool "(" (Bool (Set (Var "h")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool "(" "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "p1")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "p2")) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" ))) ; theorem :: JGRAPH_7:52 (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 "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#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 "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ")" ))))) ; theorem :: JGRAPH_7:53 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )))) ; theorem :: JGRAPH_7:54 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "O")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))))))) ; theorem :: JGRAPH_7:55 (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 "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)))))) ; theorem :: JGRAPH_7:56 (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 "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)))))) ; theorem :: JGRAPH_7:57 (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 "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))))))) ; theorem :: JGRAPH_7:58 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "O")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))))) ; theorem :: JGRAPH_7:59 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "O")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))))) ; theorem :: JGRAPH_7:60 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "O")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))))) ; theorem :: JGRAPH_7:61 (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 "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#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 "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))))) ; theorem :: JGRAPH_7:62 (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 "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#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 "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))))) ; theorem :: JGRAPH_7:63 (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 "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#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 "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d")))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))))) ; theorem :: JGRAPH_7:64 (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 "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#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 "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))))) ; theorem :: JGRAPH_7:65 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "O")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c")))) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) ")" ))))) ; theorem :: JGRAPH_7:66 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "O")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))))) ; theorem :: JGRAPH_7:67 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "O")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set ($#k2_jgraph_2 :::"AffineMap"::: ) "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "a")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "d")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "c")) ")" ) ($#k7_xcmplx_0 :::"/"::: ) (Set "(" (Set (Var "d")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "c")) ")" ) ")" ) ")" ) ")" )) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))))) ; theorem :: JGRAPH_7:68 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:69 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:70 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:71 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:72 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:73 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:74 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:75 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:76 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:77 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:78 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:79 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:80 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:81 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:82 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:83 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:84 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:85 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:86 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:87 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:88 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:89 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:90 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:91 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:92 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:93 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:94 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:95 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:96 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:97 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:98 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:99 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:100 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:101 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:102 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:103 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:104 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:105 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:106 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:107 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:108 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:109 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:110 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:111 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:112 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:113 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:114 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:115 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:116 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:117 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:118 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:119 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:120 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:121 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:122 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:123 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:124 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:125 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:126 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:127 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:128 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:129 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:130 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:131 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:132 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "a"))) & (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 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:133 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Var "c"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "a"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:134 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:135 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "a"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ; theorem :: JGRAPH_7:136 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p4")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "f")) ($#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)) ($#r1_hidden :::"="::: ) (Set (Var "p4"))) & (Bool (Set (Var "f")) "is" ($#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" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ; theorem :: JGRAPH_7:137 (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 "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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 (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p3")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set (Var "p4")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p3")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p4")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "a"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p3"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p4"))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jgraph_6 :::"closed_inside_of_rectangle"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) ")" ))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q")))))) ;