:: JORDAN23 semantic presentation

begin

theorem :: JORDAN23:1
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
len (L_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) >= 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN23:2
for f being ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) holds len (R_Cut (f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) >= 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN23:3
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p, q being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) holds B_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) <> {} : ( ( ) ( ) set ) ;

registration
let x be ( ( ) ( ) set ) ;
cluster <*x : ( ( ) ( ) set ) *> : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like constant V26() V33(1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) -> one-to-one ;
end;

definition
let f be ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) ;
attr f is almost-one-to-one means :: JORDAN23:def 1
 for i, j being ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) st i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) in dom f : ( ( V2() ) ( non empty V2() ) set ) : ( ( ) ( ) Element of K6(NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V26() ) set ) ) & j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) in dom f : ( ( V2() ) ( non empty V2() ) set ) : ( ( ) ( ) Element of K6(NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V26() ) set ) ) & ( i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) or j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> len f : ( ( V2() ) ( non empty V2() ) set ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & ( i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> len f : ( ( V2() ) ( non empty V2() ) set ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) or j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & f : ( ( V2() ) ( non empty V2() ) set ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) = f : ( ( V2() ) ( non empty V2() ) set ) . j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) holds
i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) = j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let f be ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) ;
attr f is weakly-one-to-one means :: JORDAN23:def 2
 for i being ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) < len f : ( ( V2() ) ( non empty V2() ) set ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( V2() ) ( non empty V2() ) set ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) <> f : ( ( V2() ) ( non empty V2() ) set ) . (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) ;
end;

definition
let f be ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) ;
attr f is poorly-one-to-one means :: JORDAN23:def 3
 for i being ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) < len f : ( ( V2() ) ( non empty V2() ) set ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( V2() ) ( non empty V2() ) set ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) <> f : ( ( V2() ) ( non empty V2() ) set ) . (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) if len f : ( ( V2() ) ( non empty V2() ) set ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) )
otherwise verum;
end;

theorem :: JORDAN23:4
for D being ( ( ) ( ) set )
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of D : ( ( ) ( ) set ) ) holds
( f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) is almost-one-to-one iff for i, j being ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) st i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) in dom f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V26() ) set ) ) & j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) in dom f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V26() ) set ) ) & ( i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) or j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> len f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & ( i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> len f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) or j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) /. i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) = f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) /. j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) holds
i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) = j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: JORDAN23:5
for D being ( ( ) ( ) set )
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of D : ( ( ) ( ) set ) ) holds
( f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) is weakly-one-to-one iff for i being ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) < len f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) /. i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) <> f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) /. (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN23:6
for D being ( ( ) ( ) set )
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of D : ( ( ) ( ) set ) ) holds
( f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) is poorly-one-to-one iff ( len f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) implies for i being ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) < len f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) /. i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) <> f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( ) ( ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( ) ( ) set ) ) /. (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) ) ) ;

registration
cluster V13() Function-like one-to-one FinSequence-like -> V13() Function-like FinSequence-like almost-one-to-one for ( ( ) ( ) set ) ;
end;

registration
cluster V13() Function-like FinSequence-like almost-one-to-one -> V13() Function-like FinSequence-like poorly-one-to-one for ( ( ) ( ) set ) ;
end;

theorem :: JORDAN23:7
for f being ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) st len f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
( f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) is weakly-one-to-one iff f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) is poorly-one-to-one ) ;

registration
cluster empty V13() Function-like FinSequence-like -> V13() Function-like FinSequence-like weakly-one-to-one for ( ( ) ( ) set ) ;
end;

registration
let x be ( ( ) ( ) set ) ;
cluster <*x : ( ( ) ( ) set ) *> : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like one-to-one constant V26() V33(1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like almost-one-to-one poorly-one-to-one ) set ) -> weakly-one-to-one ;
end;

registration
let x, y be ( ( ) ( ) set ) ;
cluster <*x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) *> : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) -> poorly-one-to-one ;
end;

registration
cluster non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like weakly-one-to-one for ( ( ) ( ) set ) ;
end;

registration
let D be ( ( non empty ) ( non empty ) set ) ;
cluster non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like circular weakly-one-to-one for ( ( ) ( ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) ;
end;

theorem :: JORDAN23:8
for f being ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) st f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) is almost-one-to-one holds
Rev f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) set ) is almost-one-to-one ;

theorem :: JORDAN23:9
for f being ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) st f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) is weakly-one-to-one holds
Rev f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) set ) is weakly-one-to-one ;

theorem :: JORDAN23:10
for f being ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) st f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) is poorly-one-to-one holds
Rev f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) set ) is poorly-one-to-one ;

registration
cluster non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like for ( ( ) ( ) set ) ;
end;

registration
let f be ( ( V13() Function-like FinSequence-like almost-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like almost-one-to-one poorly-one-to-one ) FinSequence) ;
cluster Rev f : ( ( V13() Function-like FinSequence-like almost-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like almost-one-to-one poorly-one-to-one ) set ) : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) set ) -> V13() Function-like FinSequence-like almost-one-to-one ;
end;

registration
let f be ( ( V13() Function-like FinSequence-like weakly-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like weakly-one-to-one ) FinSequence) ;
cluster Rev f : ( ( V13() Function-like FinSequence-like weakly-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like weakly-one-to-one ) set ) : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) set ) -> V13() Function-like FinSequence-like weakly-one-to-one ;
end;

registration
let f be ( ( V13() Function-like FinSequence-like poorly-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like poorly-one-to-one ) FinSequence) ;
cluster Rev f : ( ( V13() Function-like FinSequence-like poorly-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like poorly-one-to-one ) set ) : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) set ) -> V13() Function-like FinSequence-like poorly-one-to-one ;
end;

theorem :: JORDAN23:11
for D being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is almost-one-to-one holds
for p being ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) holds Rotate (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is almost-one-to-one ;

theorem :: JORDAN23:12
for D being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is weakly-one-to-one & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is circular holds
for p being ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) holds Rotate (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is weakly-one-to-one ;

theorem :: JORDAN23:13
for D being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is poorly-one-to-one & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is circular holds
for p being ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) holds Rotate (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is poorly-one-to-one ;

registration
let D be ( ( non empty ) ( non empty ) set ) ;
cluster non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like circular for ( ( ) ( ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) ;
end;

registration
let D be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( almost-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like almost-one-to-one poorly-one-to-one ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) ;
let p be ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ;
cluster Rotate (f : ( ( almost-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like almost-one-to-one poorly-one-to-one ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) -> almost-one-to-one ;
end;

registration
let D be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( circular weakly-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like circular weakly-one-to-one ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) ;
let p be ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ;
cluster Rotate (f : ( ( circular weakly-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like circular weakly-one-to-one ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) -> weakly-one-to-one ;
end;

registration
let D be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( circular poorly-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like circular poorly-one-to-one ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) ;
let p be ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ;
cluster Rotate (f : ( ( circular poorly-one-to-one ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like circular poorly-one-to-one ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(D : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) -> poorly-one-to-one ;
end;

theorem :: JORDAN23:14
for D being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of D : ( ( non empty ) ( non empty ) set ) ) holds
( f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is almost-one-to-one iff ( f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) /^ 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is one-to-one & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) | ((len f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) -' 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is one-to-one ) ) ;

registration
let C be ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ;
let n be ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;
cluster Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) -> non empty non constant circular special unfolded s.c.c. standard V192() almost-one-to-one ;
end;

registration
let C be ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ;
let n be ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;
cluster Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) -> non empty non constant circular special unfolded s.c.c. standard V192() weakly-one-to-one ;
end;

theorem :: JORDAN23:15
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is weakly-one-to-one holds
B_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = <*p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) *> : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one constant V26() V33(1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like special almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ;

theorem :: JORDAN23:16
for f being ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) st f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) is one-to-one holds
f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) is weakly-one-to-one ;

registration
cluster V13() Function-like one-to-one FinSequence-like -> V13() Function-like FinSequence-like weakly-one-to-one for ( ( ) ( ) set ) ;
end;

theorem :: JORDAN23:17
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is weakly-one-to-one holds
for p, q being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
B_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = Rev (B_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ;

theorem :: JORDAN23:18
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) )
for i1 being ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is poorly-one-to-one & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is s.n.c. & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) < i1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) & i1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= len f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) = f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . i1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) holds
(Index (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) = i1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN23:19
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is weakly-one-to-one holds
for p, q being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
(B_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) /. 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ;

theorem :: JORDAN23:20
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is weakly-one-to-one holds
for p, q being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
(B_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) /. (len (B_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ;

theorem :: JORDAN23:21
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
L~ (L_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) c= L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN23:22
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p, q being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is weakly-one-to-one holds
L~ (B_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) c= L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN23:23
for f, g being ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) holds dom f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( ) ( ) Element of K6(NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V26() ) set ) ) c= dom (f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) ^' g : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) ) : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) set ) : ( ( ) ( ) Element of K6(NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V26() ) set ) ) ;

theorem :: JORDAN23:24
for f being ( ( non empty V13() Function-like FinSequence-like ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence)
for g being ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) holds dom g : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( ) ( ) Element of K6(NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V26() ) set ) ) c= dom (f : ( ( non empty V13() Function-like FinSequence-like ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) ^' g : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) ) : ( ( V13() Function-like FinSequence-like ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) set ) : ( ( ) ( ) Element of K6(NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V26() ) set ) ) ;

theorem :: JORDAN23:25
for f, g being ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) st f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) ^' g : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) set ) is constant holds
f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) is constant ;

theorem :: JORDAN23:26
for f, g being ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) st f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) ^' g : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) set ) is constant & f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) . (len f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) = g : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) <> {} : ( ( ) ( ) set ) holds
g : ( ( V13() Function-like FinSequence-like ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence) is constant ;

theorem :: JORDAN23:27
for f being ( ( special ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like special ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for i, j being ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds mid (f : ( ( special ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like special ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is special ;

theorem :: JORDAN23:28
for f being ( ( unfolded ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like unfolded ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for i, j being ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds mid (f : ( ( unfolded ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like unfolded ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is unfolded ;

theorem :: JORDAN23:29
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is special holds
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
L_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is special ;

theorem :: JORDAN23:30
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is special holds
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
R_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is special ;

theorem :: JORDAN23:31
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is special & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is weakly-one-to-one holds
for p, q being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
B_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is special ;

theorem :: JORDAN23:32
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded holds
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
L_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is unfolded ;

theorem :: JORDAN23:33
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded holds
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
R_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is unfolded ;

theorem :: JORDAN23:34
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is weakly-one-to-one holds
for p, q being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
B_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is unfolded ;

theorem :: JORDAN23:35
for f, g being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is almost-one-to-one & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is special & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is s.n.c. & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) & g : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) = (mid (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,(Index (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ^ <*p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) *> : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one constant V26() V33(1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like special almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
g : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is_S-Seq_joining f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) /. 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ;

theorem :: JORDAN23:36
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is poorly-one-to-one & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is s.n.c. & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) = f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . ((Index (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . (len f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) holds
((Index (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(Rev f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) + (Index (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) = len f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN23:37
for f being ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is weakly-one-to-one & len f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) >= 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
L_Cut (f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,(f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) /. 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ;

theorem :: JORDAN23:38
for f being ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is poorly-one-to-one & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is s.n.c. & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . (len f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) holds
L_Cut ((Rev f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = Rev (R_Cut (f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ;

theorem :: JORDAN23:39
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is almost-one-to-one & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is special & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is s.n.c. & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) holds
R_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is_S-Seq_joining f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) /. 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ;

theorem :: JORDAN23:40
for f being ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is almost-one-to-one & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is special & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is s.n.c. & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . (len f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) holds
L_Cut (f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is_S-Seq_joining p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) /. (len f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ;

theorem :: JORDAN23:41
for f being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is almost-one-to-one & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is special & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is s.n.c. & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) holds
R_Cut (f : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is being_S-Seq ;

theorem :: JORDAN23:42
for f being ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is almost-one-to-one & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is special & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is s.n.c. & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . (len f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) holds
L_Cut (f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is being_S-Seq ;

theorem :: JORDAN23:43
for f being ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p, q being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is almost-one-to-one & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is special & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is s.n.c. & len f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) & q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) holds
B_Cut (f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is_S-Seq_joining p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ;

theorem :: JORDAN23:44
for f being ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) )
for p, q being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is almost-one-to-one & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is special & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is unfolded & f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) is s.n.c. & len f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <> 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in L~ f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) & q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) <> f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) holds
B_Cut (f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( non empty V2() ) set ) ) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,q : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) is being_S-Seq ;

theorem :: JORDAN23:45
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of )
for p, q being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) in BDD (L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) holds
ex B being ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) st
( B : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) = B_Cut (((Rotate ((Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,((Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) /. (Index ((South-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,(Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) | ((len (Rotate ((Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,((Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) /. (Index ((South-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,(Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) -' 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like special unfolded ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,(South-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,(North-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & ex P being ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) st
( P : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) is_sequence_on GoB (B : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) ^' <*(North-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,(South-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) *> : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K304( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty functional FinSequence-membered ) M13( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K304( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty functional FinSequence-membered ) M13( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) & L~ <*(North-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,(South-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) *> : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) = L~ P : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & P : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = North-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & P : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) /. (len P : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = South-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & len P : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) >= 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ex B1 being ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) st
( B1 : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) is_sequence_on GoB (B : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) ^' <*(North-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,(South-Bound (p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Point of ( ( ) ( non empty V2() ) set ) ) ,(L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V192() ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard V192() almost-one-to-one weakly-one-to-one poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) *> : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like poorly-one-to-one ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K304( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty functional FinSequence-membered ) M13( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like V26() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K304( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty functional FinSequence-membered ) M13( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) & L~ B : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) = L~ B1 : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) : ( ( ) ( non empty compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( non empty ) set ) ) & B : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = B1 : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & B : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) /. (len B : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = B1 : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) /. (len B1 : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like V48() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & len B : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= len B1 : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) & ex g being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) st g : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) = B1 : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) ^' P : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like one-to-one V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq almost-one-to-one weakly-one-to-one poorly-one-to-one ) S-Sequence_in_R2) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like V26() FinSequence-like FinSubsequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V26() V31() ext-real positive non negative ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V26() V31() V32() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty V74() TopSpace-like V122() V147() V148() V149() V150() V151() V152() V153() strict ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) ) ) ;