:: JGRAPH_4 semantic presentation begin theorem :: JGRAPH_4:1 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) : (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "a"))) "}" )) "holds" (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) ))))) ; theorem :: JGRAPH_4:2 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) : (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) "}" )) "holds" (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) ))))) ; theorem :: JGRAPH_4:3 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "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 ($#k2_struct_0 :::"[#]"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool "(" "for" (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "ex" (Set (Var "K")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "K")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "K")))) & (Bool "ex" (Set (Var "V2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) & (Bool (Set (Var "V2")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "V2")) ($#r1_tarski :::"c="::: ) (Set (Var "K"))) & (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) ")" )) ")" )) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) )) ; theorem :: JGRAPH_4:4 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r1"))) & (Bool (Set (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r2")))) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "r1")) ($#k13_complex1 :::"/"::: ) (Set (Var "r2")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ($#k13_complex1 :::"/"::: ) (Set (Var "b"))))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))))) ; theorem :: JGRAPH_4:5 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r1"))) & (Bool (Set (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r2")))) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r2")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "r1")) ($#k13_complex1 :::"/"::: ) (Set (Var "r2")) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "a")) ")" ) ($#k13_complex1 :::"/"::: ) (Set (Var "b")) ")" )))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))))) ; theorem :: JGRAPH_4:6 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r1")))) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r1")) ($#k3_square_1 :::"^2"::: ) ))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" )))) ; theorem :: JGRAPH_4:7 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r1")))) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k18_complex1 :::"abs"::: ) (Set (Var "r1"))))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" )))) ; theorem :: JGRAPH_4:8 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r1")))) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "r1"))))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" )))) ; theorem :: JGRAPH_4:9 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r1"))) & (Bool (Set (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r2")))) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r2")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "r1")) ($#k13_complex1 :::"/"::: ) (Set (Var "r2")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ($#k13_complex1 :::"/"::: ) (Set (Var "b")) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" )))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))))) ; theorem :: JGRAPH_4:10 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r1"))) & (Bool (Set (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "r2")))) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r2")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "r1")) ($#k13_complex1 :::"/"::: ) (Set (Var "r2")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ($#k13_complex1 :::"/"::: ) (Set (Var "b")) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" )))) ")" ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))))) ; definitionlet "n" be ($#m1_hidden :::"Nat":::); func "n" :::"NormF"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) "n" ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) means :: JGRAPH_4:def 1 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) "n" ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ))); end; :: deftheorem defines :::"NormF"::: JGRAPH_4:def 1 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k1_jgraph_4 :::"NormF"::: ) )) "iff" (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ))) ")" ))); theorem :: JGRAPH_4:11 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "n")) ($#k1_jgraph_4 :::"NormF"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ))) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "n")) ($#k1_jgraph_4 :::"NormF"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n")))) ")" )) ; theorem :: JGRAPH_4:12 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "n")) ($#k1_jgraph_4 :::"NormF"::: ) ) "is" ($#v5_pre_topc :::"continuous"::: ) )) ; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster (Set "n" ($#k1_jgraph_4 :::"NormF"::: ) ) -> ($#v5_pre_topc :::"continuous"::: ) ; end; theorem :: JGRAPH_4:13 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k1_jgraph_4 :::"NormF"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p")))) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:14 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k10_metric_1 :::"cl_Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ))) "holds" (Bool "(" (Bool (Set (Var "B")) "is" ($#v9_rltopsp1 :::"bounded"::: ) ) & (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ))))) ; theorem :: JGRAPH_4:15 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k10_metric_1 :::"cl_Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ))) "holds" (Bool (Set (Var "B")) "is" ($#v2_compts_1 :::"compact"::: ) )))) ; begin definitionlet "s" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "q" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); func :::"FanW"::: "(" "s" "," "q" ")" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_4:def 2 (Set (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) "s" ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) "s" ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) if (Bool "(" (Bool (Set (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) "s") & (Bool (Set "q" ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) (Set (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) "s" ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) "s" ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) if (Bool "(" (Bool (Set (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) "s") & (Bool (Set "q" ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) otherwise "q"; end; :: deftheorem defines :::"FanW"::: JGRAPH_4:def 2 : (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k2_jgraph_4 :::"FanW"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k2_jgraph_4 :::"FanW"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "s")) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))) ")" & "(" (Bool (Bool "(" "not" (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "s"))) "or" "not" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "(" "not" (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) "or" "not" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "implies" (Bool (Set ($#k2_jgraph_4 :::"FanW"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ")" ))); definitionlet "s" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func "s" :::"-FanMorphW"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: JGRAPH_4:def 3 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k2_jgraph_4 :::"FanW"::: ) "(" "s" "," (Set (Var "q")) ")" ))); end; :: deftheorem defines :::"-FanMorphW"::: JGRAPH_4:def 3 : (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) )) "iff" (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k2_jgraph_4 :::"FanW"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ))) ")" ))); theorem :: JGRAPH_4:16 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "sn")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ")" ))) ; theorem :: JGRAPH_4:17 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )))) ; theorem :: JGRAPH_4:18 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))) "implies" (Bool (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))) "implies" (Bool (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" ")" ))) ; theorem :: JGRAPH_4:19 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:20 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:21 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:22 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "sn"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:23 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" ) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:24 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" ) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:25 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K03")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "K03")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "sn")) ($#k8_real_1 :::"*"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" )) "holds" (Bool (Set (Var "K03")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:26 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K03")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "K03")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "sn")) ($#k8_real_1 :::"*"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" )) "holds" (Bool (Set (Var "K03")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:27 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:28 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:29 (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "K0")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:30 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))) ; theorem :: JGRAPH_4:31 (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "K0")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:32 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))) ; theorem :: JGRAPH_4:33 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )))) ; theorem :: JGRAPH_4:34 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "," (Set (Var "K0")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))))) ; theorem :: JGRAPH_4:35 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "," (Set (Var "K0")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))))) ; scheme :: JGRAPH_4:sch 1 InclSub{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool P1[(Set (Var "p"))]) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set F1 "(" ")" ) ")" ))) provided (Bool (Set F1 "(" ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) proof end; theorem :: JGRAPH_4:36 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "D")) ($#k3_subset_1 :::"`"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ")" ) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "D")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "D")) ")" ) "st" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "D")))) & (Bool (Set (Var "h")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" )))) ; theorem :: JGRAPH_4:37 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) )) & (Bool (Set (Var "h")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))) ; theorem :: JGRAPH_4:38 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: JGRAPH_4:39 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" (Bool (Set (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" )) ; theorem :: JGRAPH_4:40 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "K")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "K")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "K")))) & (Bool "ex" (Set (Var "V2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) & (Bool (Set (Var "V2")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "V2")) ($#r1_tarski :::"c="::: ) (Set (Var "K"))) & (Bool (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) ")" )) ")" )))) ; theorem :: JGRAPH_4:41 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) )) & (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) ")" ))) ; theorem :: JGRAPH_4:42 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:43 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:44 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:45 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:46 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:47 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "sn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:48 (Bool "for" (Set (Var "sn")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k3_jgraph_4 :::"-FanMorphW"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ")" )))) ; begin definitionlet "s" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "q" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); func :::"FanN"::: "(" "s" "," "q" ")" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_4:def 4 (Set (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) "s" ")" ) ")" ) "," (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) "s" ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) if (Bool "(" (Bool (Set (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) "s") & (Bool (Set "q" ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) (Set (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) "s" ")" ) ")" ) "," (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) "s" ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) if (Bool "(" (Bool (Set (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) "s") & (Bool (Set "q" ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) otherwise "q"; end; :: deftheorem defines :::"FanN"::: JGRAPH_4:def 4 : (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k4_jgraph_4 :::"FanN"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" ) ")" ) "," (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k4_jgraph_4 :::"FanN"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "s")) ")" ) ")" ) "," (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))) ")" & "(" (Bool (Bool "(" "not" (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "s"))) "or" "not" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "(" "not" (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) "or" "not" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "implies" (Bool (Set ($#k4_jgraph_4 :::"FanN"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ")" ))); definitionlet "c" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func "c" :::"-FanMorphN"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: JGRAPH_4:def 5 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k4_jgraph_4 :::"FanN"::: ) "(" "c" "," (Set (Var "q")) ")" ))); end; :: deftheorem defines :::"-FanMorphN"::: JGRAPH_4:def 5 : (Bool "for" (Set (Var "c")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) )) "iff" (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k4_jgraph_4 :::"FanN"::: ) "(" (Set (Var "c")) "," (Set (Var "q")) ")" ))) ")" ))); theorem :: JGRAPH_4:49 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "cn")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ")" ))) ; theorem :: JGRAPH_4:50 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )))) ; theorem :: JGRAPH_4:51 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))) "implies" (Bool (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))) "implies" (Bool (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" ")" ))) ; theorem :: JGRAPH_4:52 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:53 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:54 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:55 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "cn"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:56 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" ) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:57 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" ) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:58 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K03")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "K03")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "cn")) ($#k8_real_1 :::"*"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" )) "holds" (Bool (Set (Var "K03")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:59 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K03")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "K03")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "cn")) ($#k8_real_1 :::"*"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" )) "holds" (Bool (Set (Var "K03")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:60 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:61 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:62 (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "K0")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:63 (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "K0")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:64 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))) ; theorem :: JGRAPH_4:65 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))) ; theorem :: JGRAPH_4:66 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )))) ; theorem :: JGRAPH_4:67 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "," (Set (Var "K0")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))))) ; theorem :: JGRAPH_4:68 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "," (Set (Var "K0")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))))) ; theorem :: JGRAPH_4:69 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "D")) ($#k3_subset_1 :::"`"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ")" ) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "D")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "D")) ")" ) "st" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "D")))) & (Bool (Set (Var "h")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" )))) ; theorem :: JGRAPH_4:70 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) )) & (Bool (Set (Var "h")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))) ; theorem :: JGRAPH_4:71 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: JGRAPH_4:72 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" (Bool (Set (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" )) ; theorem :: JGRAPH_4:73 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "K")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "K")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "K")))) & (Bool "ex" (Set (Var "V2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) & (Bool (Set (Var "V2")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "V2")) ($#r1_tarski :::"c="::: ) (Set (Var "K"))) & (Bool (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) ")" )) ")" )))) ; theorem :: JGRAPH_4:74 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) )) & (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) ")" ))) ; theorem :: JGRAPH_4:75 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:76 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:77 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:78 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:79 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:80 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "cn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:81 (Bool "for" (Set (Var "cn")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k5_jgraph_4 :::"-FanMorphN"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ")" )))) ; begin definitionlet "s" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "q" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); func :::"FanE"::: "(" "s" "," "q" ")" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_4:def 6 (Set (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) "s" ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) "s" ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) if (Bool "(" (Bool (Set (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) "s") & (Bool (Set "q" ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) (Set (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) "s" ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) "s" ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) if (Bool "(" (Bool (Set (Set "(" "q" ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) "s") & (Bool (Set "q" ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) otherwise "q"; end; :: deftheorem defines :::"FanE"::: JGRAPH_4:def 6 : (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k6_jgraph_4 :::"FanE"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k6_jgraph_4 :::"FanE"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "s")) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))) ")" & "(" (Bool (Bool "(" "not" (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "s"))) "or" "not" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "(" "not" (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) "or" "not" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "implies" (Bool (Set ($#k6_jgraph_4 :::"FanE"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ")" ))); definitionlet "s" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func "s" :::"-FanMorphE"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: JGRAPH_4:def 7 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k6_jgraph_4 :::"FanE"::: ) "(" "s" "," (Set (Var "q")) ")" ))); end; :: deftheorem defines :::"-FanMorphE"::: JGRAPH_4:def 7 : (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) )) "iff" (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k6_jgraph_4 :::"FanE"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ))) ")" ))); theorem :: JGRAPH_4:82 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "sn")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ")" ))) ; theorem :: JGRAPH_4:83 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )))) ; theorem :: JGRAPH_4:84 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))) "implies" (Bool (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))) "implies" (Bool (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" ")" ))) ; theorem :: JGRAPH_4:85 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:86 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:87 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:88 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "sn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "sn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "sn"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:89 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" ) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:90 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" ) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:91 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K03")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "K03")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "sn")) ($#k8_real_1 :::"*"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" )) "holds" (Bool (Set (Var "K03")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:92 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K03")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "K03")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "sn")) ($#k8_real_1 :::"*"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" )) "holds" (Bool (Set (Var "K03")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:93 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:94 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:95 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))) ; theorem :: JGRAPH_4:96 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))) ; theorem :: JGRAPH_4:97 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )))) ; theorem :: JGRAPH_4:98 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "," (Set (Var "K0")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))))) ; theorem :: JGRAPH_4:99 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "," (Set (Var "K0")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))))) ; theorem :: JGRAPH_4:100 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "D")) ($#k3_subset_1 :::"`"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ")" ) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "D")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "D")) ")" ) "st" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "D")))) & (Bool (Set (Var "h")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" )))) ; theorem :: JGRAPH_4:101 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "h")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) )) & (Bool (Set (Var "h")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" ))) ; theorem :: JGRAPH_4:102 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: JGRAPH_4:103 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" (Bool (Set (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" )) ; theorem :: JGRAPH_4:104 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "K")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "K")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "K")))) & (Bool "ex" (Set (Var "V2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) & (Bool (Set (Var "V2")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "V2")) ($#r1_tarski :::"c="::: ) (Set (Var "K"))) & (Bool (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) ")" )) ")" )))) ; theorem :: JGRAPH_4:105 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) )) & (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) ")" ))) ; theorem :: JGRAPH_4:106 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:107 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:108 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "sn"))) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:109 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:110 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "sn"))) & (Bool (Set (Var "sn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:111 (Bool "for" (Set (Var "sn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "sn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:112 (Bool "for" (Set (Var "sn")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "sn")) ($#k7_jgraph_4 :::"-FanMorphE"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ")" )))) ; begin definitionlet "s" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "q" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); func :::"FanS"::: "(" "s" "," "q" ")" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JGRAPH_4:def 8 (Set (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) "s" ")" ) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) "s" ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) if (Bool "(" (Bool (Set (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) "s") & (Bool (Set "q" ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) (Set (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) "s" ")" ) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "s" ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) "s" ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) if (Bool "(" (Bool (Set (Set "(" "q" ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) "q" ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) "s") & (Bool (Set "q" ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) otherwise "q"; end; :: deftheorem defines :::"FanS"::: JGRAPH_4:def 8 : (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k8_jgraph_4 :::"FanS"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" ) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k8_jgraph_4 :::"FanS"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k1_rlvect_1 :::"*"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "s")) ")" ) ")" ) "," (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))) ")" & "(" (Bool (Bool "(" "not" (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "s"))) "or" "not" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "(" "not" (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) "or" "not" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "implies" (Bool (Set ($#k8_jgraph_4 :::"FanS"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ")" ))); definitionlet "c" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func "c" :::"-FanMorphS"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: JGRAPH_4:def 9 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k8_jgraph_4 :::"FanS"::: ) "(" "c" "," (Set (Var "q")) ")" ))); end; :: deftheorem defines :::"-FanMorphS"::: JGRAPH_4:def 9 : (Bool "for" (Set (Var "c")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) )) "iff" (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k8_jgraph_4 :::"FanS"::: ) "(" (Set (Var "c")) "," (Set (Var "q")) ")" ))) ")" ))); theorem :: JGRAPH_4:113 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "cn")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ")" ))) ; theorem :: JGRAPH_4:114 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )))) ; theorem :: JGRAPH_4:115 (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))) "implies" (Bool (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))) "implies" (Bool (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ) ")" ) "," (Set "(" (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" ")" ))) ; theorem :: JGRAPH_4:116 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:117 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:118 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:119 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ")" ) ($#k5_real_1 :::"-"::: ) (Set (Var "cn")) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "cn")) ")" ) ")" ) ($#k4_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K1")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "cn"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:120 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" ) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:121 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" ) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:122 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K03")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "K03")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "cn")) ($#k8_real_1 :::"*"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" )) "holds" (Bool (Set (Var "K03")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:123 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K03")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "K03")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "cn")) ($#k8_real_1 :::"*"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" )) "holds" (Bool (Set (Var "K03")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: JGRAPH_4:124 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:125 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "K0")) "," (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: JGRAPH_4:126 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))) ; theorem :: JGRAPH_4:127 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "B0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "K0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "K0")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "B0")) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "K0")))) & (Bool (Set (Var "B0")) ($#r1_hidden :::"="::: ) (Set ($#k8_struct_0 :::"NonZero"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))) ; theorem :: JGRAPH_4:128 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )))) ; theorem :: JGRAPH_4:129 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "," (Set (Var "K0")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))))) ; theorem :: JGRAPH_4:130 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "," (Set (Var "K0")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))) & (Bool (Set (Var "K0")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "K0"))))) ; theorem :: JGRAPH_4:131 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "D")) ($#k3_subset_1 :::"`"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ")" ) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "D")) ")" ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "D")) ")" ) "st" (Bool "(" (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "D")))) & (Bool (Set (Var "h")) "is" ($#v5_pre_topc :::"continuous"::: ) ) ")" )))) ; theorem :: JGRAPH_4:132 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ) "is" ($#v5_pre_topc :::"continuous"::: ) )) ; theorem :: JGRAPH_4:133 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: JGRAPH_4:134 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" (Bool (Set (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ")" )) ; theorem :: JGRAPH_4:135 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "K")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "K")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "K")))) & (Bool "ex" (Set (Var "V2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) & (Bool (Set (Var "V2")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "V2")) ($#r1_tarski :::"c="::: ) (Set (Var "K"))) & (Bool (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p2"))) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) ")" )) ")" )))) ; theorem :: JGRAPH_4:136 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) )) & (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) ")" ))) ; theorem :: JGRAPH_4:137 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:138 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:139 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "cn"))) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:140 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:141 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_real_1 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "cn"))) & (Bool (Set (Var "cn")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q2")) ($#k12_euclid :::".|"::: ) )))) "holds" (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q1")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q2"))))) "holds" (Bool (Set (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p1")) ($#k12_euclid :::".|"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p2")) ($#k12_euclid :::".|"::: ) )))))) ; theorem :: JGRAPH_4:142 (Bool "for" (Set (Var "cn")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k13_complex1 :::"/"::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "q")) ($#k12_euclid :::".|"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "cn")))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "cn")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: JGRAPH_4:143 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k9_jgraph_4 :::"-FanMorphS"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ")" )))) ;