:: JORDAN5C semantic presentation begin theorem :: JORDAN5C:1 (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "P")) ")" ) (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "q1"))) & (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"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s1"))) & (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s1")))) "holds" (Bool "not" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) (Set (Var "Q")))) ")" )) "holds" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "P")) ")" ) (Bool "for" (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s2"))) ($#r1_hidden :::"="::: ) (Set (Var "q1"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2"))) & (Bool (Set (Var "s2")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s2")))) "holds" (Bool "not" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) (Set (Var "Q"))))))))))) ; definitionlet "P", "Q" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "p1", "p2" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); assume that (Bool "(" (Bool (Set (Const "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Const "Q"))) & (Bool (Set (Set (Const "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Const "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ) and (Bool (Set (Const "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Const "p1")) "," (Set (Const "p2"))) ; func :::"First_Point"::: "(" "P" "," "p1" "," "p2" "," "Q" ")" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: JORDAN5C:def 1 (Bool "(" (Bool it ($#r2_hidden :::"in"::: ) (Set "P" ($#k9_subset_1 :::"/\"::: ) "Q")) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) "P" ")" ) (Bool "for" (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "p1") & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) "p2") & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s2"))) ($#r1_hidden :::"="::: ) it) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2"))) & (Bool (Set (Var "s2")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s2")))) "holds" (Bool "not" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) "Q")))) ")" ) ")" ); end; :: deftheorem defines :::"First_Point"::: JORDAN5C:def 1 : (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q"))) & (Bool (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "Q")) ")" )) "iff" (Bool "(" (Bool (Set (Var "b5")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q")))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "P")) ")" ) (Bool "for" (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s2"))) ($#r1_hidden :::"="::: ) (Set (Var "b5"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2"))) & (Bool (Set (Var "s2")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s2")))) "holds" (Bool "not" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) (Set (Var "Q")))))) ")" ) ")" ) ")" )))); theorem :: JORDAN5C:2 (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Var "Q")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "p"))))) ; theorem :: JORDAN5C:3 (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set (Var "Q"))) & (Bool (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "p1"))))) ; theorem :: JORDAN5C:4 (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set (Var "P")) ")" ) (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "q1"))) & (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"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s1"))) & (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::">="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "s1")))) "holds" (Bool "not" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) (Set (Var "Q")))) ")" )) "holds" (Bool "for" (Set (Var "g")) "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")) ")" ) (Bool "for" (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s2"))) ($#r1_hidden :::"="::: ) (Set (Var "q1"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2"))) & (Bool (Set (Var "s2")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::">="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "s2")))) "holds" (Bool "not" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) (Set (Var "Q"))))))))))) ; definitionlet "P", "Q" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "p1", "p2" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); assume that (Bool "(" (Bool (Set (Const "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Const "Q"))) & (Bool (Set (Set (Const "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Const "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ) and (Bool (Set (Const "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Const "p1")) "," (Set (Const "p2"))) ; func :::"Last_Point"::: "(" "P" "," "p1" "," "p2" "," "Q" ")" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: JORDAN5C:def 2 (Bool "(" (Bool it ($#r2_hidden :::"in"::: ) (Set "P" ($#k9_subset_1 :::"/\"::: ) "Q")) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) "P" ")" ) (Bool "for" (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "p1") & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) "p2") & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s2"))) ($#r1_hidden :::"="::: ) it) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2"))) & (Bool (Set (Var "s2")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::">="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "s2")))) "holds" (Bool "not" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) "Q")))) ")" ) ")" ); end; :: deftheorem defines :::"Last_Point"::: JORDAN5C:def 2 : (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q"))) & (Bool (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "Q")) ")" )) "iff" (Bool "(" (Bool (Set (Var "b5")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q")))) & (Bool "(" "for" (Set (Var "g")) "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")) ")" ) (Bool "for" (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s2"))) ($#r1_hidden :::"="::: ) (Set (Var "b5"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2"))) & (Bool (Set (Var "s2")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::">="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "s2")))) "holds" (Bool "not" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r2_hidden :::"in"::: ) (Set (Var "Q")))))) ")" ) ")" ) ")" )))); theorem :: JORDAN5C:5 (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Var "Q")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "p"))))) ; theorem :: JORDAN5C:6 (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Var "Q"))) & (Bool (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "p2"))))) ; theorem :: JORDAN5C:7 (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set (Var "Q"))) & (Bool (Set (Var "P")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool "(" (Bool (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) ")" ))) ; begin definitionlet "P" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "p1", "p2", "q1", "q2" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); pred :::"LE"::: "q1" "," "q2" "," "P" "," "p1" "," "p2" means :: JORDAN5C:def 3 (Bool "(" (Bool "q1" ($#r2_hidden :::"in"::: ) "P") & (Bool "q2" ($#r2_hidden :::"in"::: ) "P") & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) "P" ")" ) (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "p1") & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) "p2") & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) "q1") & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s1"))) & (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s2"))) ($#r1_hidden :::"="::: ) "q2") & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2"))) & (Bool (Set (Var "s2")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2")))) ")" ) ")" ); end; :: deftheorem defines :::"LE"::: JORDAN5C:def 3 : (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))) "iff" (Bool "(" (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "q2")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool "(" "for" (Set (Var "g")) "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")) ")" ) (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "q1"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s1"))) & (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s2"))) ($#r1_hidden :::"="::: ) (Set (Var "q2"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2"))) & (Bool (Set (Var "s2")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2")))) ")" ) ")" ) ")" ))); theorem :: JORDAN5C:8 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "g")) "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")) ")" ) (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Var "g")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p2"))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "q1"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s1"))) & (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "s2"))) ($#r1_hidden :::"="::: ) (Set (Var "q2"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2"))) & (Bool (Set (Var "s2")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Var "s1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s2")))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))))))) ; theorem :: JORDAN5C:9 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q1")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))))) ; theorem :: JORDAN5C:10 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool "(" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "q1")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "p2")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))) ")" ))) ; theorem :: JORDAN5C:11 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))))) ; theorem :: JORDAN5C:12 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q2")) "," (Set (Var "q1")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool (Set (Var "q1")) ($#r1_hidden :::"="::: ) (Set (Var "q2"))))) ; theorem :: JORDAN5C:13 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "," (Set (Var "q2")) "," (Set (Var "q3")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q2")) "," (Set (Var "q3")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q3")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))))) ; theorem :: JORDAN5C:14 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "q2")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Set (Var "q1")) ($#r1_hidden :::"<>"::: ) (Set (Var "q2"))) & (Bool "not" (Bool "(" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Bool "not" ($#r1_jordan5c :::"LE"::: ) (Set (Var "q2")) "," (Set (Var "q1")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")))) ")" ))) "holds" (Bool "(" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q2")) "," (Set (Var "q1")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Bool "not" ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")))) ")" ))) ; begin theorem :: JORDAN5C:15 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "Q")))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) "," (Set (Var "q")) "," (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))))) ; theorem :: JORDAN5C:16 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "Q")))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q")) "," (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) "," (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))))) ; theorem :: JORDAN5C:17 (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2"))) & (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ) "," (Set (Var "p1")) "," (Set (Var "p2")))) "holds" (Bool ($#r2_jordan3 :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set (Var "p1")) "," (Set (Var "p2")))) ; theorem :: JORDAN5C:18 (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q"))) & (Bool (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p2")) "," (Set (Var "p1")) "," (Set (Var "Q")) ")" )) & (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set (Var "P")) "," (Set (Var "p2")) "," (Set (Var "p1")) "," (Set (Var "Q")) ")" )) ")" ))) ; theorem :: JORDAN5C:19 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q"))) & (Bool (Set (Var "Q")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ))) "holds" (Bool (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) "," (Set (Var "Q")) ")" ))))) ; theorem :: JORDAN5C:20 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q"))) & (Bool (Set (Var "Q")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ))) "holds" (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) "," (Set (Var "Q")) ")" ))))) ; theorem :: JORDAN5C:21 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ))) "holds" (Bool (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")))))) ; theorem :: JORDAN5C:22 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ))) "holds" (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))) ; theorem :: JORDAN5C:23 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) "," (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))))) ; theorem :: JORDAN5C:24 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j"))) "," (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))))) ; theorem :: JORDAN5C:25 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) "," (Set (Var "q")) "," (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))))) ; theorem :: JORDAN5C:26 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q")) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) "," (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))))) ; theorem :: JORDAN5C:27 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q"))) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Var "Q")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "j")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "Q"))) & (Bool (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "q")))) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))) "implies" (Bool ($#r2_jordan3 :::"LE"::: ) (Set ($#k1_jordan5c :::"First_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) "," (Set (Var "q")) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ")" ))))) ; theorem :: JORDAN5C:28 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Q"))) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Var "Q")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "j")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "Q"))) & (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "q")))) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "j"))) & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))) "implies" (Bool ($#r2_jordan3 :::"LE"::: ) (Set (Var "q")) "," (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "Q")) ")" ) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ")" ))))) ; theorem :: JORDAN5C:29 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" )) & (Bool (Set (Var "q2")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" )) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))))) ; theorem :: JORDAN5C:30 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q2")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Var "q1")) ($#r1_hidden :::"<>"::: ) (Set (Var "q2")))) "holds" (Bool "(" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "q1")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" )) & (Bool (Set (Var "q2")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "j")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))) "implies" (Bool ($#r2_jordan3 :::"LE"::: ) (Set (Var "q1")) "," (Set (Var "q2")) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ")" )) ")" ))) ;