:: JORDAN1A semantic presentation

begin

definition
let f be ( ( V21() Function-like FinSequence-like ) ( V21() Function-like FinSequence-like ) FinSequence) ;
func Center f -> ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) equals :: JORDAN1A:def 1
((len f : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) div 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
end;

theorem :: JORDAN1A:1
for f being ( ( V21() Function-like FinSequence-like ) ( V21() Function-like FinSequence-like ) FinSequence) st len f : ( ( V21() Function-like FinSequence-like ) ( V21() Function-like FinSequence-like ) FinSequence) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) is odd holds
len f : ( ( V21() Function-like FinSequence-like ) ( V21() Function-like FinSequence-like ) FinSequence) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (Center f : ( ( V21() Function-like FinSequence-like ) ( V21() Function-like FinSequence-like ) FinSequence) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:2
for f being ( ( V21() Function-like FinSequence-like ) ( V21() Function-like FinSequence-like ) FinSequence) st len f : ( ( V21() Function-like FinSequence-like ) ( V21() Function-like FinSequence-like ) FinSequence) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) is even holds
len f : ( ( V21() Function-like FinSequence-like ) ( V21() Function-like FinSequence-like ) FinSequence) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (Center f : ( ( V21() Function-like FinSequence-like ) ( V21() Function-like FinSequence-like ) FinSequence) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

begin

registration
cluster non empty being_simple_closed_curve compact non horizontal non vertical for ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: JORDAN1A:3
for D being ( ( non empty ) ( non empty ) Subset of )
for p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in N-most D : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = N-bound D : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:4
for D being ( ( non empty ) ( non empty ) Subset of )
for p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in E-most D : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = E-bound D : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:5
for D being ( ( non empty ) ( non empty ) Subset of )
for p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in S-most D : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = S-bound D : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:6
for D being ( ( non empty ) ( non empty ) Subset of )
for p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in W-most D : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = W-bound D : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:7
for D being ( ( ) ( ) Subset of ) holds BDD D : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) misses D : ( ( ) ( ) Subset of ) ;

theorem :: JORDAN1A:8
for p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) holds p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in Vertical_Line (p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:9
for r, s being ( ( real ) ( V11() real ext-real ) number ) holds |[r : ( ( real ) ( V11() real ext-real ) number ) ,s : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in Vertical_Line r : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:10
for s being ( ( real ) ( V11() real ext-real ) number )
for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) c= Vertical_Line s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
A : ( ( ) ( ) Subset of ) is vertical ;

theorem :: JORDAN1A:11
for p, q being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( real ) ( V11() real ext-real ) number ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = q : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) & r : ( ( real ) ( V11() real ext-real ) number ) in [.(proj2 : ( ( Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) ( V21() V24( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) V227( TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6(K7( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ,(proj2 : ( ( Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) ( V21() V24( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) V227( TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6(K7( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . q : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) .] : ( ( ) ( V165() V166() V167() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) holds
|[(p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ,r : ( ( real ) ( V11() real ext-real ) number ) ]| : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in LSeg (p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:12
for p, q being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( real ) ( V11() real ext-real ) number ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = q : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) & r : ( ( real ) ( V11() real ext-real ) number ) in [.(proj1 : ( ( Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) ( V21() V24( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) V227( TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6(K7( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ,(proj1 : ( ( Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) ( V21() V24( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) V227( TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6(K7( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . q : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) .] : ( ( ) ( V165() V166() V167() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) holds
|[r : ( ( real ) ( V11() real ext-real ) number ) ,(p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ]| : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in LSeg (p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:13
for p, q being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) )
for s being ( ( real ) ( V11() real ext-real ) number ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in Vertical_Line s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & q : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in Vertical_Line s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
LSeg (p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= Vertical_Line s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:14
for A, B being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) meets B : ( ( ) ( ) Subset of ) holds
proj2 : ( ( Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) ( V21() V24( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) V227( TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6(K7( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .: A : ( ( ) ( ) Subset of ) : ( ( ) ( V165() V166() V167() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) meets proj2 : ( ( Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) ( V21() V24( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) V227( TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6(K7( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .: B : ( ( ) ( ) Subset of ) : ( ( ) ( V165() V166() V167() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:15
for s being ( ( real ) ( V11() real ext-real ) number )
for A, B being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) misses B : ( ( ) ( ) Subset of ) & A : ( ( ) ( ) Subset of ) c= Vertical_Line s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & B : ( ( ) ( ) Subset of ) c= Vertical_Line s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
proj2 : ( ( Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) ( V21() V24( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) V227( TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6(K7( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .: A : ( ( ) ( ) Subset of ) : ( ( ) ( V165() V166() V167() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) misses proj2 : ( ( Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) ( V21() V24( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) Function-like V46( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) V227( TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6(K7( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .: B : ( ( ) ( ) Subset of ) : ( ( ) ( V165() V166() V167() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ;

begin

theorem :: JORDAN1A:16
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in LSeg ((G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:17
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in LSeg ((G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,(G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * ((len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:18
for j1, j2, i1, i2 being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= (G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:19
for i1, i2, j1, j2 being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= (G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:20
for t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board)
for f being ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) st f : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) >= N-bound (L~ f : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:21
for t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board)
for f being ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) st f : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= W-bound (L~ f : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:22
for t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board)
for f being ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) st f : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= S-bound (L~ f : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:23
for t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board)
for f being ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) st f : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * ((len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) >= E-bound (L~ f : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:24
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
not cell (G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is empty ;

theorem :: JORDAN1A:25
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
cell (G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is connected ;

theorem :: JORDAN1A:26
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
not cell (G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is bounded ;

theorem :: JORDAN1A:27
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
not cell (G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is bounded ;

begin

theorem :: JORDAN1A:28
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) holds width (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1A:29
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1A:30
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1A:31
for m, n, i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) & ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) < len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JORDAN1A:32
for m, n, i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < width (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) & ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) < width (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JORDAN1A:33
for m, n, i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < width (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for i1, j1 being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) st i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) & j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) holds
(Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN1A:34
for m, n, i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) & ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JORDAN1A:35
for m, n, i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < width (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) & ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= width (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JORDAN1A:36
for i, n, j, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & ( ( n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) or ( n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
((Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = ((Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:37
for i, n, j, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & ( ( n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) or ( n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
((Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Center (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = ((Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Center (Gauge (D : ( ( non empty ) ( non empty ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:38
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = ((W-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + (E-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V34() V165() V171() ) set ) ) ;

theorem :: JORDAN1A:39
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(Center (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = ((S-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + (N-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V34() V165() V171() ) set ) ) ;

theorem :: JORDAN1A:40
for i, n, j, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for E being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= ((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:41
for i, n, j, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for E being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= ((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:42
for i, n, j, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for E being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= ((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:43
for i, n, j, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for E being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= ((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:44
for m, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for E being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
LSeg (((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= LSeg (((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:45
for m, n, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for E being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
LSeg (((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= LSeg (((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:46
for m, n, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for E being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
LSeg (((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= LSeg (((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((Center (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:47
for m, n, i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for E being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < width (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for i1, j1 being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) st ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) & ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) <= j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) holds
cell ((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ,i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= cell ((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:48
for m, n, i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for E being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < width (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for i1, j1 being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) st i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) & j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) = ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) * (j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) holds
cell ((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ,(i1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= cell ((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ,(i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:49
for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
cell ((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= UBD C : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:50
for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for E, C being ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
cell ((Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width (Gauge (E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= UBD E : ( ( compact non horizontal non vertical ) ( non empty bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

begin

theorem :: JORDAN1A:51
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds
north_halfline p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:52
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds
east_halfline p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:53
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds
south_halfline p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:54
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds
west_halfline p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) meets L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:55
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ex k, t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN1A:56
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ex k, t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN1A:57
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ex k, t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN1A:58
for k, n, t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
(Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in N-most (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:59
for k, n, t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
(Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in W-most (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:60
for k, n, t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
(Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in S-most (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:61
for k, n, t being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,t : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) holds
(Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) /. k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) in E-most (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1A:62
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds W-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = (W-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) - (((E-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) - (W-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) / (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V34() V165() V171() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:63
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds S-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = (S-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) - (((N-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) - (S-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) / (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V34() V165() V171() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:64
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds E-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = (E-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + (((E-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) - (W-bound C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) / (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V34() V165() V171() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:65
for n, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds (N-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + (S-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = (N-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + (S-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:66
for n, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds (E-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + (W-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = (E-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) + (W-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:67
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
E-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) < E-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:68
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
W-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) < W-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:69
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
S-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) < S-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:70
for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
N-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = ((Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:71
for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
E-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = ((Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * ((len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:72
for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
S-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = ((Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:73
for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
W-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = ((Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V21() non empty-yielding V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25(K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K358( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M14( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:74
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in (north_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) > x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:75
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in (east_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) > x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:76
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in (south_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) < x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:77
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in (west_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) < x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:78
for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in N-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in north_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in LSeg ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
LSeg ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is horizontal ;

theorem :: JORDAN1A:79
for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in E-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in east_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in LSeg ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
LSeg ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is vertical ;

theorem :: JORDAN1A:80
for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in S-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in south_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in LSeg ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
LSeg ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is horizontal ;

theorem :: JORDAN1A:81
for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in W-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in west_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) < len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in LSeg ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
LSeg ((Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) is vertical ;

theorem :: JORDAN1A:82
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in N-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in (north_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = N-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:83
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in E-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in (east_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = E-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:84
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in S-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in (south_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = S-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:85
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x, p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in W-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in (west_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) = W-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) ;

theorem :: JORDAN1A:86
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in N-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st (north_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( functional ) set ) ;

theorem :: JORDAN1A:87
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in E-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st (east_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( functional ) set ) ;

theorem :: JORDAN1A:88
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in S-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st (south_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( functional ) set ) ;

theorem :: JORDAN1A:89
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of )
for x being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) in W-most C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty bounded compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
ex p being ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) st (west_halfline x : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( connected convex ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) ( non empty V2() V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) V25( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) Function-like non constant FinSequence-like V231( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) special unfolded s.c.c. standard V250() ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact non horizontal non vertical ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V112() V177() V178() V210() V211() V212() V213() V214() V215() V216() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {p : ( ( ) ( V21() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V16() V99() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V34() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V157() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( functional ) set ) ;