JORDAN14

:: JORDAN14 semantic presentation

begin

for f being ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) holds
( BDD (L~ f : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) = RightComp f : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) or BDD (L~ f : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) = LeftComp f : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN14:2

for f being ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) holds
( UBD (L~ f : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) = RightComp f : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) or UBD (L~ f : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) = LeftComp f : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN14:3

theorem :: JORDAN14:4

for G being ( ( V15() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board)
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Point of ( ( ) ( non empty V21() ) set ) )
for i, j being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( V15() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= width G : ( ( V15() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Point of ( ( ) ( non empty V21() ) set ) ) in Int (cell (G : ( ( V15() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) iff ( (G : ( ( V15() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) `1 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) < p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Point of ( ( ) ( non empty V21() ) set ) ) `1 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Point of ( ( ) ( non empty V21() ) set ) ) `1 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) < (G : ( ( V15() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * ((i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) `1 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) & (G : ( ( V15() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) `2 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) < p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Point of ( ( ) ( non empty V21() ) set ) ) `2 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Point of ( ( ) ( non empty V21() ) set ) ) `2 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) < (G : ( ( V15() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) `2 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) ) ) ;

theorem :: JORDAN14:5

registration

let f be ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) ;

cluster BDD (L~ f : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) -> connected ;

end;

definition

let C be ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ;
let n be ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ;

func SpanStart (C,n) -> ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Point of ( ( ) ( non empty V21() ) set ) ) equals :: JORDAN14:def 1
(Gauge (C : ( ( non empty TopSpace-like V110() V156() V157() V158() V159() V160() V161() V162() add-continuous Mult-continuous ) ( non empty TopSpace-like V110() V156() V157() V158() V159() V160() V161() V162() add-continuous Mult-continuous ) RLTopStruct ) ,n : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) )) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * ((X-SpanStart (C : ( ( non empty TopSpace-like V110() V156() V157() V158() V159() V160() V161() V162() add-continuous Mult-continuous ) ( non empty TopSpace-like V110() V156() V157() V158() V159() V160() V161() V162() add-continuous Mult-continuous ) RLTopStruct ) ,n : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Y-SpanStart (C : ( ( non empty TopSpace-like V110() V156() V157() V158() V159() V160() V161() V162() add-continuous Mult-continuous ) ( non empty TopSpace-like V110() V156() V157() V158() V159() V160() V161() V162() add-continuous Mult-continuous ) RLTopStruct ) ,n : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ;

end;

theorem :: JORDAN14:6

theorem :: JORDAN14:7

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( right_cell ((Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ,k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) misses C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) & left_cell ((Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ,k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) meets C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ) ;

theorem :: JORDAN14:8

registration

cluster Cl (RightComp (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( closed ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) -> compact ;

end;

theorem :: JORDAN14:9

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) meets LeftComp (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:10

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) misses RightComp (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:11

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) c= LeftComp (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:12

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) c= UBD (L~ (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:13

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
BDD (L~ (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) c= BDD C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:14

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
UBD C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( non empty open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) c= UBD (L~ (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:15

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
RightComp (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) c= BDD C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:16

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
UBD C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( non empty open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) c= LeftComp (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:17

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
UBD C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( non empty open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) misses BDD (L~ (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:18

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
UBD C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( non empty open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) misses RightComp (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:19

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for P being ( ( ) ( ) Subset of )
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) & P : ( ( ) ( ) Subset of ) is_outside_component_of C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
P : ( ( ) ( ) Subset of ) misses L~ (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:20

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
UBD C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( non empty open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) misses L~ (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:21

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
L~ (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) c= BDD C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) : ( ( ) ( open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:22

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for i, j, k, n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) holds
i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) > 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN14:23

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for i, j, k, n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) holds
i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN14:24

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for i, j, k, n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) holds
j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) > 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN14:25

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for i, j, k, n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V136() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) holds
j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) < width (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN14:26

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) holds
Y-SpanStart (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) < width (Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN14:27

for C being ( ( compact non horizontal non vertical ) ( non empty closed bounded compact non horizontal non vertical ) Subset of )
for n, m being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st m : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) >= n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) holds
X-SpanStart (C : ( ( compact non horizontal non vertical ) ( non empty closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) = ((2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (m : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) -' n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) * ((X-SpanStart (C : ( ( compact non horizontal non vertical ) ( non empty closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() V37() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) ) : ( ( ) ( V11() V12() V37() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) + 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() V37() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) ;

theorem :: JORDAN14:28

theorem :: JORDAN14:29

theorem :: JORDAN14:30

for C being ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve)
for n, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) is_sufficiently_large_for C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) & Y-SpanStart (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= ((2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) -' (ApproxIndex C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) * ((Y-InitStart C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) -' 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(cell ((Gauge (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) ,((X-SpanStart (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ,k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) \ (L~ (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) c= BDD (L~ (Span (C : ( ( being_simple_closed_curve ) ( non empty closed connected bounded compact non horizontal non vertical being_simple_closed_curve ) Simple_closed_curve) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() non constant V26() FinSequence-like FinSubsequence-like V180( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V194() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty closed boundary nowhere_dense connected bounded V175() compact non horizontal non vertical being_simple_closed_curve ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( open connected V174( TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN14:31

for C being ( ( ) ( ) Subset of )
for n, m, i being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) st m : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (C : ( ( ) ( ) Subset of ) ,m : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(((2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() V37() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) ) : ( ( ) ( V11() V12() V37() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) + 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() V37() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() V37() ext-real ) Element of REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) < len (Gauge (C : ( ( ) ( ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V26() FinSequence-like FinSubsequence-like tabular ) FinSequence of K338( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V110() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V97() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V144() V145() V146() V147() V148() V149() V150() ) Element of K6(REAL : ( ( ) ( non empty V26() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN14:32

theorem :: JORDAN14:33

theorem :: JORDAN14:34

theorem :: JORDAN14:35

theorem :: JORDAN14:36