:: JORDAN23 semantic presentation begin theorem :: JORDAN23:1 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_jordan3 :::"L_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_xxreal_0 :::">="::: ) (Num 1)))) ; theorem :: JORDAN23:2 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_jordan3 :::"R_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_xxreal_0 :::">="::: ) (Num 1)))) ; theorem :: JORDAN23:3 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; registrationlet "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k5_finseq_1 :::"<*"::: ) "x" ($#k5_finseq_1 :::"*>"::: ) ) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; definitionlet "f" be ($#m1_hidden :::"FinSequence":::); attr "f" is :::"almost-one-to-one"::: means :: JORDAN23:def 1 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "f")) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "f")) & (Bool "(" (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 1)) "or" (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) (Set ($#k3_finseq_1 :::"len"::: ) "f")) ")" ) & (Bool "(" (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set ($#k3_finseq_1 :::"len"::: ) "f")) "or" (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) (Num 1)) ")" ) & (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "j"))))) "holds" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))); end; :: deftheorem defines :::"almost-one-to-one"::: JORDAN23:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) ) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 1)) "or" (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" ) & (Bool "(" (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) "or" (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) (Num 1)) ")" ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))))) "holds" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))) ")" )); definitionlet "f" be ($#m1_hidden :::"FinSequence":::); attr "f" is :::"weakly-one-to-one"::: means :: JORDAN23:def 2 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) "f"))) "holds" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))); end; :: deftheorem defines :::"weakly-one-to-one"::: JORDAN23:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) ")" )); definitionlet "f" be ($#m1_hidden :::"FinSequence":::); attr "f" is :::"poorly-one-to-one"::: means :: JORDAN23:def 3 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) "f"))) "holds" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) if (Bool (Set ($#k3_finseq_1 :::"len"::: ) "f") ($#r1_hidden :::"<>"::: ) (Num 2)) otherwise (Bool verum); end; :: deftheorem defines :::"poorly-one-to-one"::: JORDAN23:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 2))) "implies" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 2)))) "implies" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) ) "iff" (Bool verum) ")" ) ")" ")" )); theorem :: JORDAN23:4 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) ) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 1)) "or" (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" ) & (Bool "(" (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) "or" (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) (Num 1)) ")" ) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j"))))) "holds" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))) ")" ))) ; theorem :: JORDAN23:5 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) ")" ))) ; theorem :: JORDAN23:6 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) ) "iff" "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 2))) "implies" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) ")" ")" ))) ; registration cluster bbbadV1_RELAT_1() ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) -> ($#v1_jordan23 :::"almost-one-to-one"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster bbbadV1_RELAT_1() ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v1_jordan23 :::"almost-one-to-one"::: ) -> ($#v3_jordan23 :::"poorly-one-to-one"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: JORDAN23:7 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 2))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) ) "iff" (Bool (Set (Var "f")) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) ) ")" )) ; registration cluster ($#v1_xboole_0 :::"empty"::: ) bbbadV1_RELAT_1() ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) -> ($#v2_jordan23 :::"weakly-one-to-one"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k5_finseq_1 :::"<*"::: ) "x" ($#k5_finseq_1 :::"*>"::: ) ) -> ($#v2_jordan23 :::"weakly-one-to-one"::: ) ; end; registrationlet "x", "y" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) -> ($#v3_jordan23 :::"poorly-one-to-one"::: ) ; end; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set ($#k5_numbers :::"NAT"::: ) )) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v2_jordan23 :::"weakly-one-to-one"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set ($#k5_numbers :::"NAT"::: ) )) bbbadV5_RELAT_1("D") ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_finseq_6 :::"circular"::: ) ($#v2_jordan23 :::"weakly-one-to-one"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" "D"; end; theorem :: JORDAN23:8 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) )) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f"))) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) )) ; theorem :: JORDAN23:9 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f"))) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )) ; theorem :: JORDAN23:10 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) )) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f"))) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) )) ; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set ($#k5_numbers :::"NAT"::: ) )) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "f" be ($#v1_jordan23 :::"almost-one-to-one"::: ) ($#m1_hidden :::"FinSequence":::); cluster (Set ($#k3_finseq_5 :::"Rev"::: ) "f") -> ($#v1_jordan23 :::"almost-one-to-one"::: ) ; end; registrationlet "f" be ($#v2_jordan23 :::"weakly-one-to-one"::: ) ($#m1_hidden :::"FinSequence":::); cluster (Set ($#k3_finseq_5 :::"Rev"::: ) "f") -> ($#v2_jordan23 :::"weakly-one-to-one"::: ) ; end; registrationlet "f" be ($#v3_jordan23 :::"poorly-one-to-one"::: ) ($#m1_hidden :::"FinSequence":::); cluster (Set ($#k3_finseq_5 :::"Rev"::: ) "f") -> ($#v3_jordan23 :::"poorly-one-to-one"::: ) ; end; theorem :: JORDAN23:11 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) )))) ; theorem :: JORDAN23:12 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_finseq_6 :::"circular"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )))) ; theorem :: JORDAN23:13 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_finseq_6 :::"circular"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) )))) ; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set ($#k5_numbers :::"NAT"::: ) )) bbbadV5_RELAT_1("D") ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_finseq_6 :::"circular"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" "D"; end; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#v1_jordan23 :::"almost-one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> ($#v1_jordan23 :::"almost-one-to-one"::: ) ; end; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#v1_finseq_6 :::"circular"::: ) ($#v2_jordan23 :::"weakly-one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> ($#v2_jordan23 :::"weakly-one-to-one"::: ) ; end; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#v1_finseq_6 :::"circular"::: ) ($#v3_jordan23 :::"poorly-one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> ($#v3_jordan23 :::"poorly-one-to-one"::: ) ; end; theorem :: JORDAN23:14 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) ) "iff" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Num 1)) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ) ")" ))) ; registrationlet "C" be ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k1_jordan9 :::"Cage"::: ) "(" "C" "," "n" ")" ) -> ($#v1_jordan23 :::"almost-one-to-one"::: ) ; end; registrationlet "C" be ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k1_jordan9 :::"Cage"::: ) "(" "C" "," "n" ")" ) -> ($#v2_jordan23 :::"weakly-one-to-one"::: ) ; end; theorem :: JORDAN23:15 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )) "holds" (Bool (Set ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))) ; theorem :: JORDAN23:16 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )) ; registration cluster bbbadV1_RELAT_1() ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) -> ($#v2_jordan23 :::"weakly-one-to-one"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: JORDAN23:17 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )) "holds" (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "q")) "," (Set (Var "p")) ")" ")" ))))) ; theorem :: JORDAN23:18 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i1"))))) "holds" (Bool (Set (Set "(" ($#k1_jordan3 :::"Index"::: ) "(" (Set (Var "p")) "," (Set (Var "f")) ")" ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "i1")))))) ; theorem :: JORDAN23:19 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )) "holds" (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "q")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p"))))) ; theorem :: JORDAN23:20 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )) "holds" (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "q")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "q")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "q"))))) ; theorem :: JORDAN23:21 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan3 :::"L_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: JORDAN23:22 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "q")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: JORDAN23:23 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k3_graph_2 :::"^'"::: ) (Set (Var "g")) ")" )))) ; theorem :: JORDAN23:24 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k3_graph_2 :::"^'"::: ) (Set (Var "g")) ")" ))))) ; theorem :: JORDAN23:25 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k3_graph_2 :::"^'"::: ) (Set (Var "g"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v3_funct_1 :::"constant"::: ) )) ; theorem :: JORDAN23:26 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k3_graph_2 :::"^'"::: ) (Set (Var "g"))) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "f")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "g")) "is" ($#v3_funct_1 :::"constant"::: ) )) ; theorem :: JORDAN23:27 (Bool "for" (Set (Var "f")) "being" ($#v1_topreal1 :::"special"::: ) ($#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 ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#v1_topreal1 :::"special"::: ) ))) ; theorem :: JORDAN23:28 (Bool "for" (Set (Var "f")) "being" ($#v2_topreal1 :::"unfolded"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#v2_topreal1 :::"unfolded"::: ) ))) ; theorem :: JORDAN23:29 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_jordan3 :::"L_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) "is" ($#v1_topreal1 :::"special"::: ) ))) ; theorem :: JORDAN23:30 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_jordan3 :::"R_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) "is" ($#v1_topreal1 :::"special"::: ) ))) ; theorem :: JORDAN23:31 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )) "holds" (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "q")) ")" ) "is" ($#v1_topreal1 :::"special"::: ) ))) ; theorem :: JORDAN23:32 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_jordan3 :::"L_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) "is" ($#v2_topreal1 :::"unfolded"::: ) ))) ; theorem :: JORDAN23:33 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_jordan3 :::"R_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) "is" ($#v2_topreal1 :::"unfolded"::: ) ))) ; theorem :: JORDAN23:34 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) )) "holds" (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "q")) ")" ) "is" ($#v2_topreal1 :::"unfolded"::: ) ))) ; theorem :: JORDAN23:35 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Num 1) "," (Set "(" ($#k1_jordan3 :::"Index"::: ) "(" (Set (Var "p")) "," (Set (Var "f")) ")" ")" ) ")" ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool (Set (Var "g")) ($#r1_jordan3 :::"is_S-Seq_joining"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Var "p"))))) ; theorem :: JORDAN23:36 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k1_jordan3 :::"Index"::: ) "(" (Set (Var "p")) "," (Set (Var "f")) ")" ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan3 :::"Index"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ")" ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k1_jordan3 :::"Index"::: ) "(" (Set (Var "p")) "," (Set (Var "f")) ")" ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))))) ; theorem :: JORDAN23:37 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_jordan23 :::"weakly-one-to-one"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set ($#k2_jordan3 :::"L_Cut"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f"))))) ; theorem :: JORDAN23:38 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_jordan23 :::"poorly-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k2_jordan3 :::"L_Cut"::: ) "(" (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" ($#k3_jordan3 :::"R_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ))))) ; theorem :: JORDAN23:39 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)))) "holds" (Bool (Set ($#k3_jordan3 :::"R_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) ($#r1_jordan3 :::"is_S-Seq_joining"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Var "p"))))) ; theorem :: JORDAN23:40 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)))) "holds" (Bool (Set ($#k2_jordan3 :::"L_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) ($#r1_jordan3 :::"is_S-Seq_joining"::: ) (Set (Var "p")) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))))) ; theorem :: JORDAN23:41 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)))) "holds" (Bool (Set ($#k3_jordan3 :::"R_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ))) ; theorem :: JORDAN23:42 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)))) "holds" (Bool (Set ($#k2_jordan3 :::"L_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ))) ; theorem :: JORDAN23:43 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 2)) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "q"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)))) "holds" (Bool (Set ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r1_jordan3 :::"is_S-Seq_joining"::: ) (Set (Var "p")) "," (Set (Var "q"))))) ; theorem :: JORDAN23:44 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jordan23 :::"almost-one-to-one"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 2)) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "q"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)))) "holds" (Bool (Set ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) "," (Set (Var "q")) ")" ) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ))) ; theorem :: JORDAN23:45 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool "ex" (Set (Var "B")) "being" ($#m2_finseq_1 :::"S-Sequence_in_R2":::) "st" (Bool "(" (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k4_jordan3 :::"B_Cut"::: ) "(" (Set "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k1_jordan3 :::"Index"::: ) "(" (Set "(" ($#k2_jordan18 :::"South-Bound"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) "," (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ")" ) ")" ) ")" ")" ) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k1_jordan3 :::"Index"::: ) "(" (Set "(" ($#k2_jordan18 :::"South-Bound"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) "," (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ")" ) ")" ) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) "," (Set "(" ($#k2_jordan18 :::"South-Bound"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) "," (Set "(" ($#k1_jordan18 :::"North-Bound"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ")" )) & (Bool "ex" (Set (Var "P")) "being" ($#m2_finseq_1 :::"S-Sequence_in_R2":::) "st" (Bool "(" (Bool (Set (Var "P")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set ($#k2_goboard2 :::"GoB"::: ) (Set "(" (Set (Var "B")) ($#k4_graph_2 :::"^'"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set "(" ($#k1_jordan18 :::"North-Bound"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) "," (Set "(" ($#k2_jordan18 :::"South-Bound"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ($#k2_finseq_4 :::"*>"::: ) ) ")" ))) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set "(" ($#k1_jordan18 :::"North-Bound"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) "," (Set "(" ($#k2_jordan18 :::"South-Bound"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ($#k2_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "P")))) & (Bool (Set (Set (Var "P")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k1_jordan18 :::"North-Bound"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) & (Bool (Set (Set (Var "P")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "P")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_jordan18 :::"South-Bound"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" 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:::"L~"::: ) (Set (Var "B1")))) & (Bool (Set (Set (Var "B")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "B1")) ($#k7_partfun1 :::"/."::: ) (Num 1))) & (Bool (Set (Set (Var "B")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "B1")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "B1")) ")" ))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B1")))) & (Bool "ex" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "B1")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "P"))))) ")" )) ")" )) ")" ))))) ;