:: JORDAN1D semantic presentation

begin

theorem :: JORDAN1D:1
for A, B being ( ( ) ( ) set ) st ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in A : ( ( ) ( ) set ) holds
ex K being ( ( ) ( ) set ) st
( K : ( ( ) ( ) set ) c= B : ( ( ) ( ) set ) & x : ( ( ) ( ) set ) c= union K : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
union A : ( ( ) ( ) set ) : ( ( ) ( ) set ) c= union B : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;

registration
let m be ( ( non empty ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ;
cluster 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) |^ m : ( ( non empty ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative ) set ) -> even ;
end;

registration
let n be ( ( even ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let m be ( ( non empty ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ;
cluster n : ( ( even ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) |^ m : ( ( non empty ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative ) set ) -> even ;
end;

theorem :: JORDAN1D:2
for i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( real ) ( V14() real ext-real ) number ) st r : ( ( real ) ( V14() real ext-real ) number ) <> 0 : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( real ) ( V14() real ext-real ) number ) ) : ( ( ) ( V14() real ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) * (r : ( ( real ) ( V14() real ext-real ) number ) |^ (i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V14() real ext-real ) set ) : ( ( ) ( V14() real ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) = r : ( ( real ) ( V14() real ext-real ) number ) |^ i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V14() real ext-real ) set ) ;

begin

theorem :: JORDAN1D:3
for m, i, a, b being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= a : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= b : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & b : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,(i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,a : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) `1 : ( ( ) ( V14() real ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) = ((Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,(i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) -' 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,b : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) `1 : ( ( ) ( V14() real ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) ;

theorem :: JORDAN1D:4
for n, i, a, b being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= a : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= b : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & b : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,(i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (a : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) `2 : ( ( ) ( V14() real ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) = ((Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,(i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (b : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) -' 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) `2 : ( ( ) ( V14() real ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) ;

theorem :: JORDAN1D:5
for m, i, n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) st 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) holds
cell ((Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ,m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (((cell ((Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,(i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ,((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) -' 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) -' 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) \/ (cell ((Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,(i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ,((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) -' 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) \/ (cell ((Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,(i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ,((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) -' 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) \/ (cell ((Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,(i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ,((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1D:6
for m, i, n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of )
for k being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) holds
cell ((Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ,m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = union { (cell ((Gauge (D : ( ( compact non horizontal non vertical ) ( non empty compact non horizontal non vertical ) Subset of ) ,(i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + k : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ,a : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,b : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) where a, b is ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) |^ k : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) - (2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (k : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V14() real V16() ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) + 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V14() real V16() ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) <= a : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= (((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) |^ k : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * m : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) - (2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) |^ k : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V14() real V16() ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V14() real V16() ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) & (((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) |^ k : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) - (2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (k : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() even ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V14() real V16() ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) + 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V14() real V16() ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) <= b : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & b : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= (((2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) |^ k : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) - (2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) |^ k : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V14() real V16() ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V14() real V16() ext-real ) Element of REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) ) } : ( ( ) ( ) set ) ;

theorem :: JORDAN1D:7
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & N-max C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:8
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & N-max C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:9
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & E-min C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:10
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & E-min C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:11
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & E-max C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:12
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & E-max C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:13
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & S-min C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:14
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & S-min C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:15
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & S-max C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:16
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & S-max C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:17
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & W-min C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:18
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & W-min C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:19
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & W-max C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:20
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & W-max C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in right_cell ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:21
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & N-min (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:22
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & N-max (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:23
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in rng (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( V2() ) set ) ) ;

theorem :: JORDAN1D:24
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex j being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & E-min (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * ((len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:25
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex j being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & E-max (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * ((len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:26
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex j being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * ((len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in rng (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( V2() ) set ) ) ;

theorem :: JORDAN1D:27
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & S-min (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:28
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & S-max (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:29
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex i being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (i : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in rng (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( V2() ) set ) ) ;

theorem :: JORDAN1D:30
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex j being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & W-min (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:31
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex j being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & W-max (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1D:32
for n being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ex j being ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) & (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V18() non empty-yielding V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22(K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K354( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ) M17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) )) ) * (1 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V38(2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V95() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) in rng (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V14() real V16() ext-real non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) ( non empty V2() V18() V21( NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) V22( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) Function-like non constant FinSequence-like V198( the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) special unfolded s.c.c. standard V206() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V14() real V16() ext-real positive non negative V103() V104() V105() V106() V107() V108() V109() ) Element of NAT : ( ( ) ( V104() V105() V106() V107() V108() V109() V110() ) Element of K10(REAL : ( ( ) ( V104() V105() V106() V110() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V142() V167() V168() V169() V170() V171() V172() V173() strict ) RLTopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( V2() ) set ) ) ;