:: TOPREAL1 semantic presentation begin scheme :: TOPREAL1:sch 1 FraenkelAlt{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) ], P2[ ($#m1_hidden :::"set"::: ) ] } : (Bool "{" (Set (Var "v")) where v "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) : (Bool "(" (Bool P1[(Set (Var "v"))]) "or" (Bool P2[(Set (Var "v"))]) ")" ) "}" ($#r1_hidden :::"="::: ) (Set "{" (Set (Var "v1")) where v1 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) : (Bool P1[(Set (Var "v1"))]) "}" ($#k2_xboole_0 :::"\/"::: ) "{" (Set (Var "v2")) where v2 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) : (Bool P2[(Set (Var "v2"))]) "}" )) proof end; definitionlet "T" be ($#l1_pre_topc :::"TopSpace":::); let "p1", "p2" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); let "P" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); pred "P" :::"is_an_arc_of"::: "p1" "," "p2" means :: TOPREAL1:def 1 (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" "T" ($#k1_pre_topc :::"|"::: ) "P" ")" ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "p1") & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) "p2") ")" )); end; :: deftheorem defines :::"is_an_arc_of"::: TOPREAL1:def 1 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set (Var "T")) ($#k1_pre_topc :::"|"::: ) (Set (Var "P")) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) ")" )) ")" )))); theorem :: TOPREAL1:1 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool "(" (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) ")" )))) ; theorem :: TOPREAL1:2 (Bool "for" (Set (Var "T")) "being" ($#v8_pre_topc :::"T_2"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "q1"))) & (Bool (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p2")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "P")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Q"))) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "q1")))))) ; definitionfunc :::"R^2-unit_square"::: -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: TOPREAL1:def 2 (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ")" )); end; :: deftheorem defines :::"R^2-unit_square"::: TOPREAL1:def 2 : (Bool (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ")" ))); theorem :: TOPREAL1:3 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) ")" )) ; theorem :: TOPREAL1:4 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) ")" )) ; theorem :: TOPREAL1:5 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p")) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "p2")) ")" ")" ))))) ; theorem :: TOPREAL1:6 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" )) & (Bool (Set (Var "q2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q1")) "," (Set (Var "q2")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" )))) ; theorem :: TOPREAL1:7 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" )) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p")) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ")" ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q")) "," (Set (Var "p2")) ")" ")" ))))) ; theorem :: TOPREAL1:8 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "p2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )))) ; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) "n" ")" )) -> ($#v4_funct_1 :::"functional"::: ) ; end; theorem :: TOPREAL1:9 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2")))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))))) ; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster (Set ($#k15_euclid :::"TOP-REAL"::: ) "n") -> ($#v8_pre_topc :::"T_2"::: ) ; end; theorem :: TOPREAL1:10 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p2")) "," (Set (Var "q1")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p2")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "P")) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p2")) "," (Set (Var "q1")) ")" ")" )) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "q1")))))) ; theorem :: TOPREAL1:11 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p2")) "," (Set (Var "p1"))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q1")) "," (Set (Var "p2")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p2")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q1")) "," (Set (Var "p2")) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set (Var "P"))) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "q1")) "," (Set (Var "p1")))))) ; theorem :: TOPREAL1:12 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool "(" (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2"))) "or" (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set (Var "q1"))) ")" ) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p2")) "," (Set (Var "q1")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p2")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p2")) "," (Set (Var "q1")) ")" ")" )) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "q1"))))) ; theorem :: TOPREAL1:13 (Bool "(" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p1")) where p1 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p2")) where p2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) "}" ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "q1")) where q1 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "q2")) where q2 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q2")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "}" ) ")" ) ; theorem :: TOPREAL1:14 (Bool (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) "}" ) ; registration cluster (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: TOPREAL1:15 (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) )) ; theorem :: TOPREAL1:16 (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) )) ; theorem :: TOPREAL1:17 (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) )) ; theorem :: TOPREAL1:18 (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) )) ; theorem :: TOPREAL1:19 (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#r1_subset_1 :::"misses"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" )) ; theorem :: TOPREAL1:20 (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ($#r1_subset_1 :::"misses"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" )) ; definitionlet "n" be ($#m1_hidden :::"Nat":::); let "f" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); let "i" be ($#m1_hidden :::"Nat":::); func :::"LSeg"::: "(" "f" "," "i" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) "n" ")" ) equals :: TOPREAL1:def 3 (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" "f" ($#k7_partfun1 :::"/."::: ) "i" ")" ) "," (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set "(" "i" ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) if (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) "i") & (Bool (Set "i" ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "f")) ")" ) otherwise (Set ($#k1_xboole_0 :::"{}"::: ) ); end; :: deftheorem defines :::"LSeg"::: TOPREAL1:def 3 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "(" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "implies" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" )) ")" & "(" (Bool (Bool "(" "not" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) "or" "not" (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" )) "implies" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" )))); theorem :: TOPREAL1:21 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" )) ")" ))) ; definitionlet "n" be ($#m1_hidden :::"Nat":::); let "f" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); func :::"L~"::: "f" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) "n" ")" ) equals :: TOPREAL1:def 4 (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" "f" "," (Set (Var "i")) ")" ")" ) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "f")) ")" ) "}" ); end; :: deftheorem defines :::"L~"::: TOPREAL1:def 4 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" ) "}" )))); theorem :: TOPREAL1:22 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) "iff" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))) ; theorem :: TOPREAL1:23 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; definitionlet "IT" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); attr "IT" is :::"special"::: means :: TOPREAL1:def 5 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "IT")) & (Bool (Bool "not" (Set (Set "(" "IT" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" "IT" ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k17_euclid :::"`1"::: ) )))) "holds" (Bool (Set (Set "(" "IT" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" "IT" ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ))); attr "IT" is :::"unfolded"::: means :: TOPREAL1:def 6 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "IT"))) "holds" (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "i")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" "IT" ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_tarski :::"}"::: ) ))); attr "IT" is :::"s.n.c."::: means :: TOPREAL1:def 7 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "i")) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "j")) ")" ))); end; :: deftheorem defines :::"special"::: TOPREAL1:def 5 : (Bool "for" (Set (Var "IT")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_topreal1 :::"special"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "IT")))) & (Bool (Bool "not" (Set (Set "(" (Set (Var "IT")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "IT")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k17_euclid :::"`1"::: ) )))) "holds" (Bool (Set (Set "(" (Set (Var "IT")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "IT")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ))) ")" )); :: deftheorem defines :::"unfolded"::: TOPREAL1:def 6 : (Bool "for" (Set (Var "IT")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "IT"))))) "holds" (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "i")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "IT")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_tarski :::"}"::: ) ))) ")" )); :: deftheorem defines :::"s.n.c."::: TOPREAL1:def 7 : (Bool "for" (Set (Var "IT")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "i")) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "j")) ")" ))) ")" )); definitionlet "f" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); attr "f" is :::"being_S-Seq"::: means :: TOPREAL1:def 8 (Bool "(" (Bool "f" "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) "f") ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool "f" "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool "f" "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool "f" "is" ($#v1_topreal1 :::"special"::: ) ) ")" ); end; :: deftheorem defines :::"being_S-Seq"::: TOPREAL1:def 8 : (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) "iff" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) ")" ) ")" )); theorem :: TOPREAL1:24 (Bool "ex" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "f1")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ($#k2_tarski :::"}"::: ) )) & (Bool (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) )) ")" )) ; theorem :: TOPREAL1:25 (Bool "for" (Set (Var "h")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "h")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) )) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h"))) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")) ")" )))) ; definitionlet "P" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); attr "P" is :::"being_S-P_arc"::: means :: TOPREAL1:def 9 (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool "P" ($#r1_hidden :::"="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) ")" )); end; :: deftheorem defines :::"being_S-P_arc"::: TOPREAL1:def 9 : (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "P")) "is" ($#v5_topreal1 :::"being_S-P_arc"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) ")" )) ")" )); theorem :: TOPREAL1:26 (Bool "for" (Set (Var "P1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P1")) "is" ($#v5_topreal1 :::"being_S-P_arc"::: ) )) "holds" (Bool (Set (Var "P1")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; registration cluster ($#v5_topreal1 :::"being_S-P_arc"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))); end; theorem :: TOPREAL1:27 (Bool "ex" (Set (Var "P1")) "," (Set (Var "P2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "P1")) "is" ($#v5_topreal1 :::"being_S-P_arc"::: ) ) & (Bool (Set (Var "P2")) "is" ($#v5_topreal1 :::"being_S-P_arc"::: ) ) & (Bool (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "P1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "P2")))) & (Bool (Set (Set (Var "P1")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "P2"))) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ($#k2_tarski :::"}"::: ) )) ")" )) ; theorem :: TOPREAL1:28 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) "is" ($#v5_topreal1 :::"being_S-P_arc"::: ) )) "holds" (Bool "ex" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))))) ; theorem :: TOPREAL1:29 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) "is" ($#v5_topreal1 :::"being_S-P_arc"::: ) )) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "P")) ")" ) "st" (Bool (Set (Var "f")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ))) ; scheme :: TOPREAL1:sch 2 TRSubsetEx{ F1() -> ($#m1_hidden :::"Nat":::), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set F1 "(" ")" ) ")" ) "st" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set F1 "(" ")" ) ")" ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "iff" (Bool P1[(Set (Var "p"))]) ")" ))) proof end; scheme :: TOPREAL1:sch 3 TRSubsetUniq{ F1() -> ($#m1_hidden :::"Nat":::), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set F1 "(" ")" ) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set F1 "(" ")" ) ")" ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "iff" (Bool P1[(Set (Var "p"))]) ")" ) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set F1 "(" ")" ) ")" ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "B"))) "iff" (Bool P1[(Set (Var "p"))]) ")" ) ")" )) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) proof end; definitionlet "p" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); func :::"north_halfline"::: "p" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: TOPREAL1:def 10 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Set (Var "x")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set "p" ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "x")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set "p" ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )); func :::"east_halfline"::: "p" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: TOPREAL1:def 11 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Set (Var "x")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set "p" ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "x")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set "p" ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )); func :::"south_halfline"::: "p" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: TOPREAL1:def 12 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Set (Var "x")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set "p" ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "x")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set "p" ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )); func :::"west_halfline"::: "p" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: TOPREAL1:def 13 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Set (Var "x")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set "p" ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "x")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set "p" ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )); end; :: deftheorem defines :::"north_halfline"::: TOPREAL1:def 10 : (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_topreal1 :::"north_halfline"::: ) (Set (Var "p")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "(" (Bool (Set (Set (Var "x")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "x")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )) ")" ))); :: deftheorem defines :::"east_halfline"::: TOPREAL1:def 11 : (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_topreal1 :::"east_halfline"::: ) (Set (Var "p")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "(" (Bool (Set (Set (Var "x")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "x")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )) ")" ))); :: deftheorem defines :::"south_halfline"::: TOPREAL1:def 12 : (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k6_topreal1 :::"south_halfline"::: ) (Set (Var "p")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "(" (Bool (Set (Set (Var "x")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "x")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )) ")" ))); :: deftheorem defines :::"west_halfline"::: TOPREAL1:def 13 : (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k7_topreal1 :::"west_halfline"::: ) (Set (Var "p")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "(" (Bool (Set (Set (Var "x")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "x")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )) ")" ))); theorem :: TOPREAL1:30 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k4_topreal1 :::"north_halfline"::: ) (Set (Var "p"))) ($#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_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) "}" )) ; theorem :: TOPREAL1:31 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k4_topreal1 :::"north_halfline"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) "{" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set (Var "r")) ($#k19_euclid :::"]|"::: ) ) where r "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) : (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) "}" )) ; theorem :: TOPREAL1:32 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k5_topreal1 :::"east_halfline"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) "}" )) ; theorem :: TOPREAL1:33 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k5_topreal1 :::"east_halfline"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) "{" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) where r "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) : (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "}" )) ; theorem :: TOPREAL1:34 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k6_topreal1 :::"south_halfline"::: ) (Set (Var "p"))) ($#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_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) "}" )) ; theorem :: TOPREAL1:35 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k6_topreal1 :::"south_halfline"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) "{" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set (Var "r")) ($#k19_euclid :::"]|"::: ) ) where r "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) : (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) "}" )) ; theorem :: TOPREAL1:36 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k7_topreal1 :::"west_halfline"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) "}" )) ; theorem :: TOPREAL1:37 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k7_topreal1 :::"west_halfline"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) "{" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) where r "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) : (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "}" )) ; registrationlet "p" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); cluster (Set ($#k4_topreal1 :::"north_halfline"::: ) "p") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set ($#k5_topreal1 :::"east_halfline"::: ) "p") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set ($#k6_topreal1 :::"south_halfline"::: ) "p") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set ($#k7_topreal1 :::"west_halfline"::: ) "p") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: TOPREAL1:38 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k7_topreal1 :::"west_halfline"::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k5_topreal1 :::"east_halfline"::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k4_topreal1 :::"north_halfline"::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k6_topreal1 :::"south_halfline"::: ) (Set (Var "p")))) ")" )) ;